A New Upper Bound for Max-2-Sat: A Graph-Theoretic Approach

A New Upp er Bound for Max-2-SA T: A Graph-Theoretic Approac h Daniel Raible & Henning F ernau Universit y of T rier, FB 4—Abteilung In forma tik , 54286 T rier , German y { raible,fernau } @info rmatik.un i-trier.de Abstract. In MaxSa t , we as k for an assignment which satisfies the maximum n u m b er of clauses for a b oolean formula in CNF. W e present an algorithm yielding a run time up p er b ound of O ∗ (2 K 6 . 2158 ) for Max- 2-Sa t ( each clause contains at most 2 literals), where K is the n u m b er of clauses. The run t ime has b een ac h ieved b y using heuristic priorities on the c hoice of the v ariable on whic h w e branch. The implemen tation of these heuristic priorities is rather simple, th ough they hav e a significant effect on the run time. Also th e analysis uses a non- standard measure. 1 In tro duction Our Problem . MaxSa t is an optimization version of t he w ell-known deci- sion pro blem SA T : given a bo olean for m ula in CNF, we ask for an ass ig nmen t which satisfies the maximum n umber of clauses. The applicatio ns for MaxSa t range over suc h fields as combinatorial optimization, ar tificial in telligenc e and database-s ystems as men tioned in [5 ]. W e put our fo cus on Max-2-Sa t , wher e every form ula is constrained to hav e a t most t wo literals pe r cla use, to which problems a s Maximum Cut and Maximum Independent Set are reducible. Therefore Max-2-Sa t is N P -complete. Results So F ar . The be st published upper bound of O ∗ (2 K 5 . 88 ) has been a chieved by Kulikov and Kutzov in [6 ] co nsuming only po lynomial s pace. They build up their algorithm o n the one of Ko jevniko v and Kuliko v [5] who w ere the fir st who used a non-standar d measure yielding a run time of O ∗ (2 K 5 . 5 ). If we mea- sure the co mplex it y in the num b er n of v a riables the curr en t fas tes t alg orithm is the o ne of R. Williams [10 ] having run time O ∗ (2 ω 3 n ), where ω < 2 . 37 6 is the matrix-multiplication exp onent . A drawback of this algo r ithm is its requir emen t of expo nen tial space. Scott and Sorkin [9] pres en ted a O ∗ (2 1 − 1 d +1 n )-algor ithm consuming po lynomial space, where d is the average degree of the v a riable gr a ph. Max-2-Sa t has a lso b een studied with respec t to approximation [3,7] and pa- rameterized algor ithms [1,2]. Our R esults . The ma jor result we prese n t is a n a lgorithm solving Max-2-Sa t in time O ∗ (2 K 6 . 2158 ). Basically it is a refinement of the algor ithm in [5], which also in tur n builds up on the r esults of [1]. The run time improv ement is tw ofold. In [5] an upp er b ound of O ∗ (1 . 1225 n ) is o btained if the v ariable graph is cubic. 2 Here n denotes the num be r of v ar iables. W e could improve this to O ∗ (1 . 1119 9 n ) by a more accurate a nalysis. Secondly , in the cas e where the maxim um degree of the v a riable gr aph is four, we cho ose a v ariable for branching acco rding to some heuristic priorities. These t wo improv ements already give a run time of O ∗ (2 K 6 . 1489 ). Moreover we like to point o ut that these heuristic prior ities can be implemen ted such that they only consume O ( n ) time. The author s of [6] impr o ve the a lgorithm of [5] b y having a new branching strategy when the v ariable g r aph has max im um degree five. Now combining o ur improv ements with the ones from [6] gives the claimed run time. Basic Definitions and T erminol ogy . L e t V ( F ) be the set o f v ariables of a given b o olean formula F . F or v ∈ V ( F ) by ¯ v w e deno te the negation of v . If v is set, then it will b e a s signed the v alues true or false . By the word liter al , we refer to a v aria ble o r its negation. A clause is a dis junction of literals . W e cons ider formulas in c onjunctive normal form (CNF) , that is a conjunction of claus es. W e allow o nly 1- and 2 -clauses, i.e., clauses with at most t wo literals. The weight of v , written # 2 ( v ), refers to the num be r of 2-cla uses in whic h v or ¯ v o ccurs. F or a set U ⊆ V ( F ) we define # 2 ( U ) := P u ∈ U # 2 ( u ). If v or ¯ v o ccurs in some clause C we write v ∈ C . A set A of litera ls is called assignment if for ev er y v ∈ A it holds that ¯ v 6∈ A . Lo osely sp eaking if l ∈ A for a literal l , than l r e ceiv es the v alue true . W e allow the for mula to contain truth-clauses of the form {T } tha t are alw ays satisfied. F urthermor e, we consider a Max-2-Sa t instance as m ultiset of clause s . A x ∈ V ( F ) is a neighb or of v , written x ∈ N ( v ), if they o ccur in a common 2-clause. Let N [ v ] := N ( v ) ∪ { v } . The varia ble gr aph G var ( V , E ) is defined a s follows: V = V ( F ) a nd E = {{ u , v } | u, v ∈ V ( F ) , u ∈ N ( v ) } . Observe that G var is a undire cted multigraph a nd that it neglects clauses o f size one. W e will no t distinguish b etw een the w or ds “v ar ia ble” a nd “vertex”. Every v aria ble in a formula corresp onds to a vertex in G var and vice versa. By wr iting F [ v ], we mean the formula which emerges from F by se tting v to true the following wa y: First, substitute all clauses containing v by {T } , then delete all o ccurrences o f ¯ v from any clause and finally delete all empty clauses fro m F . F [ ¯ v ] is defined analogo usly: w e set x to false . 2 Reduction Rules & Basic Observ ations W e state well-kno wn reduction rules from previous w or k [1,5]: RR-1 Replace any 2-clause C with l , ¯ l ∈ C , for a literal l , with {T } . RR-2 If for tw o claus es C, D and a literal l we hav e C \ { l } = D \ { ¯ l } , then substitute C and D b y C \ { l } and {T } . RR-3 A literal l o ccurr ing o nly positively (negatively , resp.) is set to true ( false ). RR-4 If ¯ l does no t o c c ur in more 2 -clauses than l in 1-clauses , such that l is a literal, then set l to true . RR-5 Let x 1 and x 2 be t wo v ariables, such that x 1 app ears at most once in another clause without x 2 . In this case, we call x 2 the c omp anion of x 1 . RR-3 or RR-4 will set x 1 in F [ x 2 ] to α and in F [ ¯ x 2 ] to β , where α, β ∈ { true , false } . Depending o n α a nd β , the following actions will b e carried out: 3 If α = false , β = false , set x 1 to false . If α = true , β = true , set x 1 to true . If α = true , β = fals e , substitute every occur r ence of x 1 by x 2 . If α = false , β = true , substitute every occurr ence of x 1 by ¯ x 2 . F ro m now o n w e will only consider reduced fo rm ulas F . This means that to a g iv en for mula F we apply the following pr ocedur e: RR-i is alwa ys applied befo re RR-i+1 , ea c h r eduction rule is carried out exha ustiv ely and after RR - 5 we start again with RR-1 if the formula changed. A formula for whic h this pro cedure do es not apply will b e called r e duc e d . Co nc e r ning the reduction rules we hav e the following lemma [5]: Lemma 1. 1. If # 2 ( v ) = 1 , then v wil l b e set. 2. F or any u ∈ V ( F ) in a r e duc e d formula with # 2 ( u ) = 3 we have | N ( v ) | = 3 . 3. If the variables a and x ar e neighb ors and # 2 ( a ) = 3 , t hen in at le ast one of the formulas F [ x ] and F [ ¯ x ] , the r e duction ru les set a . W e need some auxiliary notions: A sequence of distinct v ertices a 1 , v 1 , . . . , v j , a 2 ( j ≥ 0) is ca lled lasso if # 2 ( v i ) = 2 for 1 ≤ i ≤ j , a 1 = a 2 , # 2 ( a 1 ) ≥ 3 and G var [ a 1 , v 1 , . . . , v j , a 2 ] is a cycle. A quasi-lasso is a las so with the difference that # 2 ( v j ) = 3. A lasso is called 3-la sso (resp. 4 -lasso) if # 2 ( a 1 ) = 3 (# 2 ( a 1 ) = 4, resp.). 3-quasi-lasso a nd 4-quasi-lasso a re defined ana logously . Lemma 2. 1. L et v , u , z ∈ V ( F ) b e p airwise distinct with # 2 ( v ) = 3 such that ther e ar e clauses C 1 , C 2 , C 3 with u, v ∈ C 1 , C 2 and v, z ∈ C 3 . Th en either v is set or the two c ommon e dges of u and v wil l b e c ontra cte d in G var . 2. Th e r e duction rules d elete the variables v 1 , . . . , v j of a lasso (quasi-lasso, r esp.) and the weight of a 1 dr ops by at le ast two (one, r esp.). Pr o of. 1. If v is not set it will be substituted by u or ¯ u due to RR -5 . The emerging clauses C 1 , C 2 will be reduced either by RR-1 or b ecome 1-clauses. Also w e hav e an edge b etw een u and z in G var as now u, z ∈ C 3 . 2. W e give the pro of by inductio n o n j . In the lasso case for j = 0, ther e must be a 2 -clause C = { a 1 , ¯ a 1 } , whic h will b e deleted b y RR-1 , so that the initial step is shown. So no w j > 0 . Then on a n y v i , 1 ≤ i ≤ j , we ca n a pply RR-5 with any neigh b or as companion, so, w.l.o .g., it is applied to v 1 with a 1 as companion. RR-5 e ither sets v 1 , then w e are done with Le mma 1.1, or v 1 will be substituted by a 1 . By applying RR-1 , this lea ds to the lasso a 1 , v 2 , . . . , v j , a 2 in G var and the claim follows by induction. In the quasi- lasso case for j = 0 , the a rgument s from ab ov e hold. F or j = 1, item 1 . is sufficient. F or j > 1, the induction step from above also a pplies here. ⊓ ⊔ 3 The Algorithm W e set d i ( F ) := |{ x ∈ V ( F ) | # 2 ( x ) = i }| . T o measure the run time, we cho ose a non standard mea sure a pproach with the measure γ defined as follows: γ ( F ) = P n i =3 ω i · d i ( F ) with ω 3 = 0 . 941 65 , ω 4 = 1 . 803 15 , ω i = i 2 for i ≥ 5 . 4 Clearly , γ ( F ) never exce e ds the n umber of clauses K in the cor resp onding for- m ula . So, by showing an upper bound of c γ ( F ) we can infer an upp er bo und c K . W e set ∆ 3 := ω 3 , ∆ i := ω i − ω i − 1 for i ≥ 4. Co ncerning the ω i ’s w e hav e ∆ i ≥ ∆ i +1 for i ≥ 3 and ω 4 ≥ 2 · ∆ 4 . The algor ithm presen ted in this pape r pro ceeds as follows: After applying the a bov e- mentioned reduction r ules exhaus- tively , it will branch on a v aria ble v . That is, we will reduce the problem to the t wo formulas F [ v ] and F [ ¯ v ]. In each of the t wo branches, we m ust determine by how m uch the orig inal formula F will b e reduced in terms of γ ( F ). Reduction in γ ( F ) can b e due to branching on a v ariable or to the subsequent applica tion of reduction rules. By an ( a 1 , . . . , a ℓ ) -br anch , we mean that in the i -th br a nc h γ ( F ) is reduced by a t le a st a i . The i-th c omp onent of a branch refer s to the search tree evolving from the i -th branch (i.e., a i ). By writing ( { a 1 } i 1 , . . . , { a ℓ } i ℓ )-branch we mean a ( a 1 1 , . . . , a i 1 1 , . . . , a 1 ℓ , . . . , a i ℓ ℓ )-branch wher e a s j = a j with 1 ≤ s ≤ i j . A ( a 1 , . . . , a ℓ )-branch dominates a ( b 1 , . . . , b ℓ )-branch if a i ≥ b i for 1 ≤ i ≤ ℓ . Heuristic Priorities If the maxim um degree of G var is four, v ariables v with # 2 ( v ) = 4 will b e calle d limite d if ther e is another v ariable u a pp earing with v in tw o 2-clauses (i.e., we hav e t wo edges b et ween v and u in G var ). W e call such u, v a limite d p air . Note that also u is limited and that at this p oin t by RR-5 no t wo weight 4 v aria bles can app ear in mo re than tw o clauses together. W e call u 1 , . . . u ℓ a limite d se quenc e if ℓ ≥ 3 and u i , u i +1 with 1 ≤ i ≤ ℓ − 1 are limited pairs. A li mite d cycle is a limited sequence with u 1 = u ℓ . T o obtain an asymptotically fast algo rithmic b eha vio r we intro duce he ur istic priorities ( HP ), concerning the c hoice of the v a riable used for branching. 1. Choos e any v with # 2 ( v ) ≥ 7. 2. Choos e any v with # 2 ( v ) = 6, preferably with # 2 ( N ( v )) < 36 . 3. Choos e any v with # 2 ( v ) = 5, preferably with # 2 ( N ( v )) < 25 . 4. Choos e any unlimited v with # 2 ( v ) = 4 and a limited neighbor. 5. Choos e the vertex u 1 in a limited se quence or cycle. 6. Pic k a limit ed pair u 1 , u 2 . Let c ∈ N ( u 1 ) \ { u 2 } wit h s ( c ) := | ( N ( c ) ∩ ( N ( u 1 )) \ { c, u 1 } ) | maxima l. If s ( c ) > 1, then cho ose the unique vertex in N ( u 1 ) \ { u 2 , c } , else choos e u 1 . 7. F rom Y := { v ∈ V ( F ) | # 2 ( v ) = 4 , ∃ z ∈ N ( v ) : # 2 ( z ) = 3 ∧ N ( z ) 6⊆ N ( v ) } choose v , pr eferably suc h that # 2 ( N ( v )) is maximal. 8. Choos e any v , with # 2 ( v ) = 4, preferably with # 2 ( N ( v )) < 16 . 9. Choos e any v , with # 2 ( v ) = 3, suc h tha t there is a ∈ N ( v ), which forms a triangle a, b, c and b, c 6∈ N [ v ] (we say v ha s p ending t riangle a, b, c ). 10. Choose any v , such that we hav e a (6 ω 3 , 8 ω 3 )- or a (4 ω 3 , 10 ω 3 )-branch. F ro m now o n v denotes the v ar iable pick ed according to HP . Key Ideas The main idea is to ha ve s ome priorities on the choice of a weight 4 v aria ble such that the branching behavior is b eneficial. F or example limited v ar iables tend to b e unstable in the following sense: If their weight is decreased due to bra nching they will b e reduce d due to Lemma 2.1. This means we can get an amount of ω 4 instead of ∆ 4 . In a graph lacking limited vertices we wan t a v ariable v with a weight 3 neighbor u suc h tha t N ( u ) 6⊆ N ( v ). In the branch 5 Algorithm 1 An algorithm for solving Max-2-Sa t . Procedure: SolMax2Sat( F ) 1: App ly SolMax2Sat on ever y component of G va r separately . 2: App ly the redu ction rules exhaustively to F . 3: Search exhaustively on any sub-formula b eing a comp onen t of at most 9 v ariables. 4: if F = {T } . . . {T } then 5: return | F | 6: el se 7: Choose a v ariable v according to HP . 8: return max { SolMax2SA T( F [ v ]) , SolMax2Sat( F [ ¯ v ]) } . on v wh er e u is set (Lemma 1.3) we ca n gain so me extra r eduction (at least ∆ 4 ) from N ( u ) \ N ( v ). If we fail to find a v ariable a ccording to priorities 5 -7 we show that either v as four w eight 4 v ariables and that the graph is 4-reg ular, o r otherwise we have tw o distinct situatio ns which can be handled quite efficiently . F urther , the most critical branches ar e when we hav e to c ho ose v such tha t all v ar iables in N [ v ] hav e weigh t ω i . Then the r eduction in γ ( F ) is minimal (i.e., ω i + i · ∆ i ). W e analyz e this r egular case together with its immediate preceding branch. Thereby we prove a better branc hing b ehavior compared to a separate analysis. In [9] similar ideas were us ed for Max-2-CSP . W e are now ready to present our algor ithm, see Alg. 1. Reaching step 7 w e ca n r ely on the fact that G var has at least 1 0 vertices. W e c a ll this the smal l c omp onent pr op erty (scp) which is crucial for some cases o f the ana ly sis. 4 The Analysis In this section we in vestigate the cases when we branch on vertices pic ked ac- cording to items 1-10 o f HP . F or each item we will derive a branching v ector which upp er b ounds this ca s e in terms of K . In the res t o f this section we show: Theorem 1. A lgorithm 1 has a run time of O ∗ (2 K 6 . 1489 ) . 4.1 G v ar has Minimum Degree F our Priorit y 1 If # 2 ( v ) ≥ 7, we first obtain a reductio n of ω 7 bec ause v will b e deleted. Secondly , we get an amount of at least 7 · ∆ 7 as the weigh ts of v ’s neighbors ea c h dr ops by at least one and we hav e ∆ i ≥ ∆ i +1 . Thus, γ is reduced by at least 7 in e ither of the tw o branches (i.e., we have a ( { 7 } 2 )-branch). R e gular Br anches W e call a branc h h-r e gular if w e branch on a v ariable v s uc h that for all u ∈ N [ v ] w e hav e # 2 ( u ) = h . W e will handle those in a separate part. During our cons iderations a 4-regula r branch will hav e exa c tly four neighbors as otherwise this situation is handled by priority 4 of HP . The following subsections handle non-r e gular branches, which means that we can find a u ∈ N ( v ) with # 2 ( u ) < # 2 ( v ). Note that we already ha ndled h - regular bra nc hes for h ≥ 7. 6 Priorities 2 and 3 Cho osing v ∈ V ( F ) with # 2 ( v ) = 6 there is a u ∈ N ( v ) with # 2 ( u ) ≤ 5 due to no n-regularity . Then by deletion of v , there is a reduction by ω 6 and another o f at least 5 ∆ 6 + ∆ 5 , r esulting from the dropping weigh ts of the neighbor s. Esp ecially , the weigh t of u must drop b y at least ∆ 5 . This leads to a ( { 6 . 19685 } 2 )-branch. If # 2 ( v ) = 5, the same observ ations as in the last choice lea d to a reduction of at least ω 5 + 4 · ∆ 5 + ∆ 4 . Th us we have a ( { 6 . 148 9 } 2 )-branch. Priorit y 4 Let u 1 ∈ N ( v ) b e the limited v aria ble. u 1 forms a limited pair with some u 2 . After branching on v , the v ariable u 1 has weight at most 3 . A t this po in t, u 1 app ears only with one other v a r iable z in a 2-cla us e. Then, RR-5 is applicable to u 1 with u 2 as its companion. According to Lemma 2.1, either u 1 is set or the t wo edges of u 1 and u 2 will b e contracted. In the first ca se, we receive a total r eduction of at least 3 ω 4 + 2 ∆ 4 , in the sec o nd o f at least 2 ω 4 + 4 ∆ 4 . Thus, a prop e r estimate is a ( { 2 ω 4 + 4 ∆ 4 } 2 )-branch, i.e., a ( { 7 . 0523 } 2 )-branch. Priorit y 5 If u 1 , . . . , u ℓ is a limited cycle, then ℓ ≥ 10 due to scp . By RR -5 this y ields a (10 w 4 , 10 w 4 )-branch. I f u 1 , . . . , u ℓ is a limited seque nce , then due to priority 4 the neighbors of u 1 , u ℓ lying outside the s e q uence ha ve w eig h t 3. B y RR-5 the br anc h on u 1 is a ( { 3 ω 4 + 2 ω 3 } 2 )-branch, i.e, a ( { 7 . 29 2 75 } 2 )-branch. Priorit y 6 At this point every limited v ar ia ble u 1 has tw o neig h b oring v ariables y , z with weigh t 3 and a limited neighbor u 2 with the same properties (due to priorities 4 a nd 5 ). W e now examine the lo c al structur e s aris ing from this fact and b y the v alues of | N ( y ) \ N ( u 1 ) | and | N ( z ) \ N ( u 1 ) | . 1. W e rule out | N ( y ) \ N ( u 1 ) | = | N ( z ) \ N ( u 1 ) | = 0 due to scp . 2. | N ( y ) \ N ( u 1 ) | = 0 , | N ( z ) \ N ( u 1 ) | = 1: Then, N ( y ) = { u 2 , z , u 1 } , N ( u 2 ) = { u 1 , y , s 1 } and N ( z ) = { u 1 , y , s 2 } , see Fig ure 1 (a ). In this case we branch on z as s ( y ) > 0 and s ( y ) > s ( z ). Then due to RR-5 y a nd u 1 disapp ear; either by being set or r eplaced. Thereafter due to RR-1 and Lemma 1.1 u 2 will be set. Additiona lly we get a n amount of min { 2 ∆ 4 , ω 4 , ω 3 + ∆ 4 } from s 1 , s 2 . This de p ends on whether s 1 6 = s 2 or s 1 = s 2 and in the second case on the weigh t of s 1 . I f # 2 ( s 1 ) = 3 we g et a re duction of ω 3 + ∆ 4 due to setting s 1 . In total we have at least a ( { 2 ω 4 + 2 ω 3 + 2 ∆ 4 } 2 )-branch. Analogo us is the case | N ( y ) \ N ( u 1 ) | = 1 , | N ( z ) \ N ( u 1 ) | = 0. 3. | N ( y ) \ N ( u 1 ) | = 1 , | N ( z ) \ N ( u 1 ) | = 1: Her e tw o p o ssibilities o ccur: (a) N ( y ) = { u 1 , u 2 , s 1 } , N ( z ) = { u 1 , u 2 , s 2 } , N ( u 2 ) = { u 1 , y , z } , s e e Fig- ure 1(b): Then w.l.o.g., we branch on z . Simila rly to item 2 . we obta in a ( { 2 ω 4 + 2 ω 3 + 2 ∆ 4 } 2 )-branch. (b) N ( y ) = { u 1 , z , s 1 } , N ( z ) = { u 1 , y , s 2 } , see Figure 1(c): W.l.o.g ., we branch on z . Basically we get a total reduction of ω 4 + 2 ω 3 + 2 ∆ 4 . Tha t is 2 ω 3 from y a nd z , ω 4 from u 1 and 2 ∆ 4 from s 2 and u 2 . In the branch wher e y is set (Lemma 1.3) we additionally get ∆ 4 from s 1 and ω 4 from u 2 as it will disapp ear (Lemma 1.2). This is a (2 ω 4 + 2 ω 3 + 2 ∆ 4 , ω 4 + 2 ω 3 + 2 ∆ 4 )-branch. 4. | N ( y ) \ N ( u 1 ) | = 1 , | N ( z ) \ N ( u 1 ) | = 2, see Figur e 1(d): W e br anc h on z yielding a ( { 2 ω 4 + 2 ω 3 + 2 ∆ 4 } 2 )-branch. Analogous is the case | N ( y ) \ N ( u 1 ) | = 2 , | N ( z ) \ N ( u 1 ) | = 1. 7 5. | N ( y ) \ N ( u 1 ) | = 2 , | N ( z ) \ N ( u 1 ) | = 2: In this case we chose u 1 for branching. Essentially we get a reduction of 2 ω 4 + 2 ω 3 . In the branch setting z w e receive an extra amount of 2 ∆ 4 from z ’s t wo neig h b ors o utside N ( u 1 ). Hence we hav e a (2 ω 4 + 2 ω 3 + 2 ∆ 4 , 2 ω 4 + 2 ω 3 )-branch. W e have at least a (2 ω 4 +2 ω 3 +2 ∆ 4 , ω 4 +2 ω 3 +2 ∆ 4 )-branch, i.e., a (7 . 212 6 , 5 . 40945)- branch. Priorit y 7 W e need further a uxiliary notions: A 3-p ath ( 4-p ath , resp.) for an unlimited w eight 4 vertex v is a sequence of vertices u 0 u 1 . . . u l u l +1 ( u 0 u 1 . . . u l , resp.) forming a path, s uc h that 1 ≤ l ≤ 4 (2 ≤ l ≤ 4, resp.), u i ∈ N ( v ) for 1 ≤ i ≤ l , # 2 ( u i ) = 3 for 1 ≤ i ≤ l (# 2 ( u i ) = 3 for 1 ≤ i ≤ l − 1 , # 2 ( u l ) = 4, resp.) and u 0 , u l +1 6∈ N ( v ) ( u 0 6∈ N ( v ), resp.). Due to the abs ence of limited vertices, every vertex v , chosen due to prior it y 7, m ust hav e a 3 - or 4- pa th. 3-path If u 0 6 = u l +1 we basica lly get a reduction of ω 4 + l ω 3 + (4 − l ) ∆ 4 . In the branch where u 1 is set, u 2 . . . u l will be also set due to Lemma 1.1. There- fore, we gain an extra amo un t of at lea s t 2 ∆ 4 from u 0 and u l +1 , leading to a ( ω 4 + l · ω 3 + (6 − l ) ∆ 4 , ω 4 + l · ω 3 + (4 − l ) ∆ 4 )-branch. If u 0 = u l +1 then in F [ v ] and in F [ ¯ v ], u 0 u 1 . . . u l u l +1 is a lasso . So b y Lemma 2.2, u 1 , . . . , u l are deleted and the w eight of u 0 drops b y 2. If # 2 ( u 0 ) = 4 this yields a reductio n o f l · ω 3 + ω 4 . If # 2 ( u 0 ) = 3 the re duction is ( l + 1) · ω 3 but then u 0 is set. It is not har d to see that this yields a b on us reduction o f ∆ 4 (see Appendix A). Thus, we hav e a ( { ω 4 + ( l + 1) · ω 3 + (5 − l ) ∆ 4 } 2 )-branch. 4-path W e get an amoun t of ω 4 + ( l − 1) ω 3 + (5 − l ) ∆ 4 by deleting v . In the branch wher e u 1 is set we get a b onus o f ∆ 4 from u 0 . F ur ther u l will b e deleted completely . Hence we have a (2 ω 4 + ( l − 1 ) ω 3 + (5 − l ) ∆ 4 , ω 4 + ( l − 1) ω 3 + (5 − l ) ∆ 4 )-branch. The fir st bra nc h is worst for l = 1 , the second a nd third for l = 2 (as l = 1 is im- po ssible). Th us, we have ( { 7 . 21 26 } 2 )-branch for the second and a (7 . 0523 , 5 . 329 3)- branch for the first and third ca se which is s ha rp. Priorit y 8 If we hav e c hose n a v a riable v with # 2 ( v ) = 4 according to prior it y 8, such that # 2 ( N ( v )) < 16, then we ha ve tw o distinct situations. By branching on v , we get at leas t a ( { 2 ω 4 + 2 ω 3 + 2 ∆ 4 } 2 )-branch. (See App endix B). The 4- 5- a nd 6-regular case The part of the alg o rithm when we bra nc h on v ariables o f w eig ht h 6 = 4 will be called h-phase . Bra nc hing acco rding to priorities 4-8 is the 4-phase , acco rding to priorities 9 and 10 the 3-phase . In the following we ha ve 4 ≤ h ≤ 6. Any h -regular branch which was preceded by a branch from the ( h + 1)-pha se can be neglected. This situation can only o ccur once o n e ac h path from the ro ot to a leaf in the sear c h tree. Hence, the run time is only affected by a constant m ultiple. W e now clas sify h -regula r bra nc hes: An internal h-r e gular br anch is a h -r egular branch such that a nother h -reg ular branch immediately follows in the search tree in at least one comp onent. A final h-r e gu lar br anch is a h -regular branch suc h that no h -r egular branch immediately succeeds in either o f the comp onents. When we are forced to do a n h -r egular branch, then accor ding to HP the whole graph m ust b e h -reg ular at this p o in t. 8 P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d q e f y z (a) P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d q e f y z (b) P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d q e f y z (c) P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d q e f y z (d) P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d q e f y z (e) P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d q e f y z (f ) Fig. 1. Observ ation 2. If a br anch is fol lowe d by a h -r e gular bra nch in one c omp onent, say in F [ v ] , then in F [ v ] any u ∈ V ( F ) with # 2 ( u ) < h wil l b e r e duc e d. Due to Obser v ation 2 every vertex in N ( v ) must be completely deleted in F [ v ]. Prop osition 1. O ∗ (1 . 1088 K ) u pp er b ou n ds any internal h -r e gular br anch. Pr o of. By Observ ation 2 for h = 4 this yields at lea st a (5 ω 4 , ω 4 + 4 ∆ 4 )-branch as v m ust hav e 4 different weight 4 neigh b ors due to HP . If bo th co mponents a r e follow ed b y an h -regular branc h we ge t a total reduction of 5 ω 4 in both cases . The same wa y we can analyze in ter nal 5- and 6-r egular branches. This yields (3 ω 5 , ω 5 + 5 ∆ 5 )-, ( { 3 ω 5 } 2 )-, (3 ω 6 , ω 6 + 6 ∆ 6 )- and ( { 3 ω 6 } 2 )-branches as for a n y v ∈ V ( F ) we hav e | N ( v ) | ≥ 2. ⊓ ⊔ W e now analyze a final h - regular ( { b } 2 )-branch with its preceding ( a 1 , a 2 )- branch. The fina l h -reg ular branch migh t follow in the first, the seco nd or bo th compo nen ts o f the ( a 1 , a 2 )-branch. So, the c ombine d analysis would be a ( { a 1 + b } 2 , a 2 ), a ( a 1 , { a 2 + b } 2 )- and a ( { a 1 + b } 2 , { a 2 + b } 2 )-branch. Prop osition 2. A ny final h - r e gular br anch ( h ∈ { 5 , 6 } ) c onsider e d to gether with its pr e c e ding br anch c an b e upp er b ounde d by O ∗ (1 . 1172 K ) . Pr o of. W e will a pply a combined a nalysis for bo th branches. Due to Obse r v a- tion 2 N ( v ) will be deleted in the cor resp onding co mponent o f the preceding branch. Due to App endices C.1 and C.2 the least amount w e can g et by deleting N ( v ) is ω 5 + ω 4 in case h = 5 a nd ω 6 + ω 4 in case h = 6. Hence, we get four differ- ent branches: A ( { 3 ω 5 + ω 4 + 5 ∆ 5 } 2 , ω 5 + 5 ∆ )-, a ( { 3 ω 6 + ω 4 + 6 ∆ 6 } 2 , ω 6 + 6 ∆ )-, a ( { 3 ω 5 + ω 4 + 5 ∆ 5 } 4 )- and a ( { 3 ω 6 + ω 4 + 6 ∆ 6 } 4 )-branch, res p ectively . ⊓ ⊔ Prop osition 3. A ny final 4 -r e gular br anch c onsider e d with its pr e c e ding br anch c an b e upp er b ounde d by O ∗ (2 K 6 . 1489 ) ≈ O ∗ (1 . 1193 3 K ) . Pr o of. W e m ust analyze a final 4 - regular br anc h together with an y possible predecessor . These are all bra nc hes der ived from priorities 4-8. See Appendix D for omitted cases. In ternal 4-regular branc h The t wo corresp onding branches ar e a ( { 6 ω 4 + 4 ∆ 4 } 2 , ω 4 + 4 ∆ 4 )-branch and a ( { 6 ω 4 + 4 ∆ 4 } 4 )-branch. Priorities 4, 5 and 8 ar e all do minated by a ( { 2 ω 4 + 4 ∆ 4 } 2 )-branch. Analyzing 9 these ca s es tog ether with a succee ding final 4-r egular br anc h gives a ( { 3 ω 4 + 8 ∆ 4 } 2 , 2 ω 4 + 4 ∆ 4 )-branch and a ( { 3 ω 4 + 8 ∆ 4 } 4 )-branch. Priorit y 7 Let o b e the n umber of weigh t 4 vertices from N ( v ) and the 3- or 4-path, resp ectively . If in one comp onent a final 4- regular branch follows then the worst ca se is when o = 0 as any weigh t 4 vertex w o uld be deleted completely and ω 4 > ω 3 . On the other hand if there is a comp onent without an immediate 4-regula r br anc h succeeding then the w or st case app ears when o is maximal as ω 3 ≥ ∆ 4 . So in the analys is we will co nsider for each case the particular w orst case even though bo th tog ether never appea r. 3-p ath with u 0 6 = u l +1 : First if there is a weight 4 v ariable in N ( u ) w e hav e at least the following branches: a ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ω 4 + ω 3 + 3 ∆ 4 ), b ) ( ω 4 + ω 3 + 5 ∆ 4 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) a nd c ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ). An y of those is upp er bounded b y O ∗ (2 K 6 . 1489 ). Now s uppose for all y ∈ N ( v ) we hav e # 2 ( y ) = 3 . T a ble 1 captur es the de r iv ed branches fo r certain combinations. Here we will also consider the weights of u 0 and u l . An y entry is upper bounded # 2 ( u 0 ), # 2 ( u l +1 ) left comp onent righ t component b oth components # 2 ( u 0 ) = 3 ( { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 , ( ω 4 + 6 ω 3 , ( { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 , # 2 ( u l +1 ) = 3 ω 4 + 4 ω 3 ) { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 ) { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 ) # 2 ( u 0 ) = 3 ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ( ω 4 + 5 ω 3 + ∆ 4 , ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , # 2 ( u l +1 ) = 4 ω 4 + 4 ω 3 ) { 2 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 ) { 2 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 ) # 2 ( u 0 ) = 4 ( { 4 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , ( ω 4 + 4 ω 3 + 2 ∆ 4 , ( { 4 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , # 2 ( u l +1 ) = 4 , ω 4 + 4 ω 3 ) { 2 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 ) { 2 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 ) T able 1. by O ∗ (2 K 6 . 1489 ) ex c ept α ) ( { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 , ω 4 + 4 ω 3 ) the left upper entry and β ) ( ω 4 + 4 ω 3 + 2 ∆ 4 , { 2 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 ) the middle en try of the la st row. F or U ⊆ V ( F ) we define E 3 ( U ) := {{ u , v } | u ∈ U, # 2 ( u ) = 3 , v 6∈ U } . Claim. 1. Su pp ose for a ll y ∈ Q := N ( v ) ∪ { u 0 , u l +1 } w e hav e # 2 ( y ) = 3 . Then there must b e s ome y ′ ∈ V \ ( N ( v ) ∪ { u 0 , u l +1 } ) with # 2 ( y ′ ) = 3. 2. Suppose for all y ∈ N ( v ) w e hav e # 2 ( y ) = 3 and # 2 ( u 0 ) = # 2 ( u l +1 ) = 4 . Then there m ust b e some y ′ ∈ V \ ( N ( v ) ∪ { u 0 , u l +1 } ) with # 2 ( y ′ ) = 3. Pr o of. 1. Assume the contrary . F o r any 1 ≤ l ≤ 4 we hav e | E 3 ( Q ∪ { v } ) | ≤ 10. Due to scp there is a weigh t 4 vertex r adjacent to so me vertex in Q . O bserve that we must hav e r ∈ Y as either there is u ∈ N ( v ) with u ∈ N ( r ) and v 6∈ N ( r ) or w.l.o.g. u 0 ∈ N ( r ) but u 1 6∈ N ( r ). Hence, r has 4 weigh t 3 neighbors from Q due to the choice of v . Hence we must hav e | E 3 ( Q ∪ { v , r } ) | ≤ 6. Using the s ame arguments a gain we find some r ′ ∈ Y with | E 3 ( Q ∪ { v , r , r ′ } ) | ≤ 2 . Again, due to scp we find a r ′′ ∈ Y with 4 weight 3 neighbors where at most t wo are from Q , a contradiction. 2. Assume the co n trar y . Observe that u 0 , u l +1 ∈ Y and due to the c hoice o f v bo th hav e 4 weigh t 3 neig h b ors which m ust be from N ( v ). F ro m | E 3 ( N [ v ]) | ≤ 8 follows that | E 3 ( N [ v ] ∪ { u 0 , u l +1 } ) | = 0 whic h contradicts scp . ⊓ ⊔ 10 Due to the last claim and Obser v ation 2 we hav e a ( { 2 ω 4 + 7 ω 3 + 4 ∆ 4 } 2 , ω 4 + 4 ω 3 )- branch for ca se α ) and a ( ω 4 + 4 ω 3 + 2 ∆ 4 , { 2 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 )-branch for ca se β ). Both are upper b ounded by O ∗ (2 K 6 . 1489 ). ⊓ ⊔ 4.2 The Cubic Case Priorit y 9 Obser v e that when we have arrived at this p oint, the graph G var m ust b e 3-regular and each v a r iable has three different neigh b ors, due to G var being reduced and due to Lemma 1.2. Also, an y 3 - regular gr aph has an even nu mber of v ertices, b ecause w e have 3 n = 2 m . Thus a n y br anc hing m ust be of the form (2 i · ω 3 , 2 j · ω 3 ) for some 1 ≤ i, j . Also, branching on any v ariable will at least result in a (4 ω 3 , 4 ω 3 )-branch (see Lemma 1.2). Note that any u ∈ N ( v ) will be either set in F [ v ] or in F [ ¯ v ], due to Lemma 1.3. Lemma 3. L et v have a p ending triangle a, b, c and N ( v ) = { a, p , q } . Then by br anching on v , we have an (8 ω 3 , 6 ω 3 ) -br anch. Pr o of. In F [ v ] and F [ ¯ v ], the v a riables a, b, c form a 3-quasi-las so. Hence, due to Lemma 2.2 w.l.o.g., only b remains in the reduced formula with # 2 ( b ) = 2 (Lemma 2 2). Also , in b oth branches, q and p are of w eight tw o and therefore deleted. Note that N ( { q , p } ) ∪{ q, p } ⊆ { v, a, b, c, q , p } , contradicts scp . Therefore, w.l.o.g., there is a v a riable z ∈ N ( q ) such that z 6∈ { v , a, b, c, q, p } . So, in the branch where q is set, also z will b e deleted. Thus, s e ven v ariable s will b e deleted. ⊓ ⊔ Priorit y 10 F r om now on, due to HP , G var is triang le-free and cubic. W e show that if we are forced to cho o se a v ertex v to whic h none of the priorities 1- 9 fits, we can cho ose v such that we o btain either a (6 ω 3 , 8 ω 3 )- or a (4 ω 3 , 10 ω 3 )-branch. Lemma 4. L et v b e a vertex in G var and N ( v ) = { a, b, c } . Supp ose that, w.l.o.g., in F [ v ] a, b and in F [ ¯ v ] c wil l b e set . Then we have a (6 ω 3 , 8 ω 3 ) -br anch. Pr o of. If | ( N ( a ) ∩ N ( b )) \ { v }| ≤ 1, then by setting a and b in F [ v ], five v aria bles will b e reduced. T o gether with v and c , this is a total of seven. If | ( N ( a ) ∩ N ( b )) \ { v }| = 2 , then situation 1(e) must o ccur (note the absence o f tr ia ngles). If z = y then also z 6 = c due to scp . Then in F [ v ] due to Lemma 1 .1 v , a, b, c, d, f , z will be deleted. If z 6 = y then v , a, b , d, f , z , y will be deleted. T o gether with F [ ¯ v ] where c is set, w e hav e a (6 ω 3 , 8 ω 3 )-branch. ⊓ ⊔ Lemma 5. If for any v ∈ V ( F ) all its neighb ors ar e set in one br anch (say, in F [ v ] ), we c an p erform a (6 ω 3 , 8 ω 3 ) - or a (4 ω 3 , 10 ω 3 ) -br anch due to cubicity. Pr o of. If | N ( a, b, c ) \ { v }| ≥ 5, then in F [ v ], 9 v ar iables are deleted, so that we hav e a (4 ω 3 , 10 ω 3 )-branch. Otherwise, either one of the tw o follo wing situations m ust o ccur: a ) There is a v aria ble y 6 = v , s uch that N ( y ) = { a, b, c } , see Fig- ure 1(f ). Then branch on b . In F [ ¯ b ], v , y , a, c, z will disapp ear (due to RR-5 and Lemma 2.1). In F [ b ], due to setting z , additiona lly a neighbor f 6∈ { a, b, c, v, y } 11 of z will b e deleted due to scp . This is a total of seven v ariables. b ) There are v aria bles p, q , suc h that | N ( p ) ∩ { a, b, c }| = | N ( q ) ∩ { a, b, c }| = 2. The last part o f The o rem 4.2 of [5] handles b ). See also App endix E. ⊓ ⊔ Due to the la st three lemmas, branchings according to priorities 9 and 10 are upper bounded by O ∗ (2 K 6 . 1489 ). Esp ecially , the (4 ω 3 , 10 ω 3 )-branch is sharp. 5 Com bining Tw o Approac hes Kuliko v and Kutzov [6] achiev ed a run time of O ∗ (2 K 5 . 88 ). This was obtained by sp e e ding up the 5 -phase by a concept c a lled ’clause learning’. As in our approach the 3- and 4-phase was improv ed we will show that if we use both strategies we ca n e ven beat o ur previous time bound. This means that in HP we substitute pr iorit y 3 by their strategy with one exception: we prefer v ariables v with a non weigh t 5 neighbor. F orced to violate this preference we do a simple branching of the fo r m F [ v ] and F [ ¯ v ]. F or the analysis we redefine the measure γ ( F ): w e set ω 3 = 0 . 9521, ω 4 = 1 . 8320, ω 5 = 2 . 488 and keep the other weigh ts. W e call this meas ur e ˜ γ ( F ). W e will repro duce the ana ly sis of [6] briefly with resp ect to ˜ γ ( F ) to show that their derived branches for the 5-phase are upper bo unded b y O ∗ (2 K 6 . 2158 ). It also can be chec ked that this is als o true for the branches derived for the other phases b y mea suring them in terms o f ˜ γ ( F ), se e Appendix F.2 . Let k ij denote the n umber of w eig h t j v ariables o ccurring i times in a 2-clause with so me v ∈ V ( F ) chosen for branching. Then we must have: k 13 + k 14 + k 15 + 2 k 24 + 2 k 25 + 3 k 35 = 5. If F ′ is the the for m ula obtained b y assigning a v alue to v and b y applying the reduction rules a fterw ar ds we hav e: ˜ γ ( F ) − ˜ γ ( F ′ ) ≥ 5 ∆ 5 + ω 5 + ( ω 3 − ∆ 5 ) k 13 + ( ∆ 4 − ∆ 5 ) k 14 + ( ω 4 2 − ∆ 5 )2 k 24 + ( ∆ 4 − ∆ 5 ) k 25 + ( ω 5 2 − 3 2 ∆ 5 )2 k 35 = 5 . 7 68 + 0 . 2 961 k 13 + 0 . 2239( k 14 + k 25 )+ 0 . 26 · 2( k 24 + k 35 ) Basically w e reduce ˜ γ ( F ) by at least ω 5 + 5 ∆ 5 . No w the co efficients of the k ij in the ab ov e equation express how the reduction grows if k ij > 0. If k 13 + k 14 + 2 k 24 + k 25 + 2 k 35 ≥ 2 we are done as ˜ γ ( F ) − ˜ γ ( F ′ ) ≥ 6 . 2158. If k 13 = 1 and k 15 = 4 then [6] stated a (5 ∆ 5 + ω 5 + ( ω 3 − ∆ 5 ) , 5 ∆ 5 + ω 5 + ( ω 3 − ∆ 5 ) + 2 ∆ 5 )-branch and for k 25 = 1 and k 15 = 3 a (5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) , 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + ω 3 )-branch. If k 14 = 1 a nd k 15 = 4 a bra nc hing of the kind F [ v ] , F [ ¯ v , v 1 ] , F [ ¯ v , ¯ v 1 , v 2 , v 3 , v 4 , v 5 ] is applied, where { v 1 , . . . , v 5 } = N ( v ). F ro m this follow a (5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) , 4 ∆ 5 + ω 5 + ∆ 4 + ω 4 + 3 ∆ 4 + ∆ 5 , 5 ω 5 + ω 4 )- and a ( ω 5 + 4 ∆ 5 + ∆ 4 , ω 5 + 4 ∆ 5 + ∆ 4 + ω 4 + 4 ∆ 5 , 5 ω 5 + ω 4 + 3 ω 3 )-branch. This depe nds on whether v 1 has at least thr e e neigh b ors of weight less than 5 in F [ ¯ v ] or not. W e obser v ed that w e can g et a additiona l r eduction of ∆ 5 in the third comp onen t of the first branch a s N [ v ] ca nnot be a co mponent in V ( F ) after step 3 of Alg. 1 yielding a (4 ∆ 5 + ω 5 + ∆ 4 , 5 ∆ 5 + ω 5 + 4 ∆ 4 + ω 4 , 5 ω 5 + ω 4 + ∆ 5 )-branch. F or the analysis of the 5-re gular branch (i.e. k 15 = 5) w e refer to Appendix F.1. It pro ceeds the same wa y as in the simple version of the algorithm except that we hav e to tak e int o acco un t the newly in tro duced branches. Theorem 3. Max-2-SA T c an b e solve d in time O ∗ (2 K 6 . 2158 ) ≈ O ∗ (1 . 118 K ) . 12 6 Conclusion W e presented an algo rithm s olving Max-2-Sa t in O ∗ (2 K 6 . 2158 ), with K the num- ber of clauses of the input formula. This is curr en tly the end of a sequence o f po lynomial-space algo r ithms each improving on the run time: be g inning with O ∗ (2 K 2 . 88 ) which w a s achiev ed by [8], it was subsequently improv ed to O ∗ (2 K 3 . 742 ) by [2], to O ∗ (2 K 5 ) b y [1], to O ∗ (2 K 5 . 217 ) b y [4], to O ∗ (2 K 5 . 5 ) b y [5] and finally to the hitherto fastest upper b ound of O ∗ (2 K 5 . 88 ) b y [6]. Our improv ement has b een achiev ed due to heuristic prior ities concerning the c hoice of v ariable fo r branch- ing in case of a maxim um degree four v ar iable graph. As [6] improv ed the case where the v ariable gr aph has maximum deg ree five, it seems that the o nly w ay to sp eed up the gener ic br anc hing algorithm is to impro ve the maximum degree six case. O ur analysis also implies that the situation when the v ariable graph is re gular is not that harmful. The reason for this that the preceding br anc h m ust hav e r educed the problem size mor e than exp ected. Th us considered to- gether these tw o bra nc hes balance ea ch other. Though the ana lysis is to some extent s ophisticated and quite detailed the algor ithm has a cle ar structure. The implemen tatio n o f the heuristic priorities for the weight 4 v ar iables should b e a straightforward ta sk. Actually , we ha ve alrea dy an implementation o f Alg. 1. It is still in a n ea rly pha se but nevertheless the p erforma nce is promising. W e ar e lo oking forward to r epor t on these results on another occa sion. References 1. J. Gramm, E. A . Hirsc h , R. Niedermeier, and P . Rossmanith. W orst-case up- p er b ounds for MAX-2-S A T with an application to MAX-CUT. Discr ete Applie d Mathematics , 130:139–1 55, 200 3. 2. J. Gramm and R. Niedermeier. F aster exact solutions for MAX2SA T. In CI A C , vol u me 1767 of LNCS , pages 174–186. Springer, 2000. 3. T. Hofmeister. An approximation algorithm for MAX-2-SA T with cardinalit y con- strain t. In ESA , v olume 283 2 of LNCS , pages 301–3 12. Sp ringer, 2003. 4. J. Kneis, D. M¨ olle, S. Rich ter, an d P . R ossmanith. Algorithms based on the treewidth of sparse graphs. In WG , v olume 3787 of LNCS , pages 385–39 6. Springer, 2005. 5. A. Ko jevniko v and A. S. Kuliko v. A new ap p roac h to pro ving up per b ounds for MAX-2-SA T. In SODA , p ages 11–17. ACM Press, 2006. 6. A. S. K ulik ov and K. Kutzko v. New boun ds for max-sat b y clause learning. In CSR , volume 4649 of LNCS , pages 194–204. Springer, 2007. 7. M. Lewin, D. Livnat, and U. Zwic k. I mpro ved rounding techniques for the MAX 2-SA T and MAX D I-CUT problems. In IPCO , volume 2337 of LNCS , pages 67–8 2. Springer, 2002. 8. R. N iedermeier and P . Rossmanith. New upp er b ounds for maximum satisfiability . Journal of Algorithms , 36:63–88, 2000 . 9. A. Scott and G. Sorkin. Linear-programming d esign and analysis of fast algorithms for Max 2-CSP . Discr ete Optimization , 4(3-4): 260-28 7, 2007. 10. R. Williams. A new algorithm for optimal 2-constraint satisfaction and its impli- cations. T he or etic al Computer Scienc e , 348(2 - 3):35 7–365, 2005. 13 A Additional Argumen ts Concerning 3-paths in the Non-regular Case Proposition 4. L et v ∈ V ( F ) b e chosen due to HP such that # 2 ( v ) = 4 and v has a 3-p ath of length l such that u 0 = u l +1 . Then we have at le ast a ( { ω 4 + 3 ω 3 + 3 ∆ 4 } 2 ) - br anch. Pr o of . In F [ v ] and in F [ ¯ v ], u 0 u 1 . . . u l u l +1 is a lass o. So by Lemma 2.2, u 1 , . . . , u l are deleted and the we ight of u 0 drops by 2. If # 2 ( u 0 ) = 4 this yields a reduction of l · ω 3 + ω 4 . If # 2 ( u 0 ) = 3 the redu ct ion is ( l + 1) · ω 3 but then u 0 is set. I f N ( u 0 ) \ N ( v ) is non-empty then we obtain a reduction of ∆ 4 in addition due to setting u 0 . Ot herwise there is a unique r ∈ N ( u 0 ) \ { u 1 , . . . , u l } with r ∈ N ( v ) \ { u 1 , . . . , u l } . If # 2 ( r ) = 4 w e get a ( { 2 ω 4 + ( l + 1) ω 3 + (3 − l ) ∆ 4 } 2 )-branch. If # 2 ( r 1 ) = 3 t hen r is set. As (4 − l ) ≤ 2 and by applying the same argumen ts to r which prev iously where applied to u 0 w e get at least a ( { ω 4 + ( l + 1) · ω 3 + (5 − l ) ∆ 4 } 2 )-branch. Observe that w e used the fact that ω 4 ≥ 2 ∆ 4 . ⊓ ⊔ B Pro of of the S t atemen t in Priority 8 (Non-regular Case) Pr o of . Note that when we are forced to pick a va riable v a ccording t o priority 8, then either v has four n eigh b ors of w eight 4 or for every w eight 3 neighbor z we hav e N ( z ) ⊆ N ( v ). F rom # 2 ( N ( v )) < 16 follo ws that, for every w eight 3 neighbor z , we ha ve N ( z ) ⊆ N ( v ) due to the choic e of v according to HP . Let N 4 (resp. N 3 ) be th e set of w eight four (three) neigh b ors of v . W e analyze different cases induced by k := | N 3 | . Let k = 1. If N 3 = { b } , then there are vertices a, c ∈ N 4 , suc h that b ∈ N ( a ) and b ∈ N ( c ). W e must hav e a ∈ N ( c ), or else a w ould violate our assumption. Thus, we get the situation of Figure 2(a). Let k = 2. Then N 3 = { b, c } and assume that b and c are n eigh b ors. If b, c ∈ N ( a ) for a ∈ N 4 , we ha ve situation depicted in Figure 2(b). Otherwise b ∈ N ( a ) and c ∈ N ( d ) for a, d ∈ N 4 . But th en, priorit y 7 app lies to b oth a and d , which is a con tradiction. In the case where b and c are not neighbors, it can b e easily observ ed that we must hav e the situation in Figure 2(c), where p riori ty 7 applies to a and d . If k = 3 it is easy to v erify that w e must hav e situation 2(d) in Figure 1. But then priorit y 7 applies to a . If k = 4 then clea rly N [ v ] forms a comp on en t of five vertice s whic h cannot app ear after step 3 of Alg. 1. In Figure 2(a) in either branch F [ v ] or F [ ¯ v ], the v ariables a, b, c form a 3-quasi-lasso, so by Lemma 2.2 we get a reduction of ω 3 + 3 ω 4 + ∆ 4 = 4 ω 4 . In Figure 2(b ) in b oth branches the v ariables a, b, c form a 3-las so, so by Lemma 2.2 b, c are deleted and a is set due to Lemma 1. 1 . W e get a redu ction of ω 4 + 2 ω 3 from this. If d 6∈ N ( a ) we additionally get 2 ∆ 4 , otherwise ω 4 . Altogether, w e reduce γ ( F ) by at least 2 ω 4 + 2 ω 3 + 2 ∆ 4 . ⊓ ⊔ C Analysis of In t ernal 5- and 6-regular Branc hes W e will consider branches whic h are immediately follo wed by a h -regular branch in at least one comp onent. In th is comp onen t of the branc h w e can delete any v ariable 14 P S f r a g r e p l a c e m e n t s v a b c d (a) P S f r a g r e p l a c e m e n t s v a b c d (b) P S f r a g r e p l a c e m e n t s v a b c d (c) P S f r a g r e p l a c e m e n t s v a b c d (d) Fig. 2. in N ( v ) additionally due to Observ ation 2. The Subsections C.1 and C.2 will explore by h o w muc h we additionally can decrement γ ( F ) in th e corresponding component in case h ∈ { 5 , 6 } . Let k ij denote the num b er of w eight j var iables o ccurring i times in a 2-clause with some v ∈ V ( F ) chosen for branching. C.1 In ternal 5-regular Branc hes Proposition 5. L et v ∈ V ( F ) b e the variable chosen due to HP such that # 2 ( v ) = 5 . If this br anch is fol lowe d by a 5 -r e gular br anch i n one c omp onent, then we c an de cr ement γ ( F ) by at le ast ω 5 + ω 4 in addition to the weight of v in that c omp onent. Pr o of . According t o [6] we m ust have the foll owing relation: k 13 + k 14 + k 15 + 2 k 24 + 2 k 25 + 3 k 35 = 5 (1) W e now hav e to d etermine an integer solution to (1) such t hat ω 3 k 13 + ω 4 k 14 + ω 5 k 15 + ω 4 k 24 + ω 5 k 25 + ω 5 k 35 is minimal. W e can assume k 14 = k 15 = 0 as w e ha ve ω 3 < ω 4 < ω 5 . F or any solution viol ating this property w e ca n fi nd a smal ler solution by setting k ′ 13 = k 13 + k 14 + k 15 , k ′ 14 = 0 and k ′ 15 = 0 and keeping th e other co efficien ts. The same w ay we fi nd t h at k 25 = 0 must be th e case as ω 4 < ω 5 . If k 13 ≥ 2 we set k ′ 13 = k 13 − 2 ⌊ k 13 2 ⌋ , k ′ 24 = k 24 + ⌊ k 13 2 ⌋ and keep t h e other co efficients. By 2 ω 3 > ω 4 this is a smalle r solution. No w suppose k 13 = 1, then we hav e k 24 = 0 in a minimal solution as ω 3 + ω 4 > ω 5 (i.e., if k 24 ≥ 1 w e set k ′ 13 = 0, k ′ 24 = k 24 − 1 and k ′ 35 = k 35 + 1) . But then no k 35 could satisfy (1 ). Thus, we hav e k 13 = 0. Then the only solution is k 24 = 1 and k 35 = 1. Hence, the minimal reduction w e get from N ( v ) is ω 5 + ω 4 . ⊓ ⊔ C.2 In ternal 6-regular Branc hes Proposition 6. L et v ∈ V ( F ) b e the variable chosen due to HP such that # 2 ( v ) = 6 . If this br anch is fol lowe d by a 6 -r e gular br anch i n one c omp onent, then we c an de cr ement γ ( F ) by at le ast ω 6 + ω 4 in addition to the weight of v in that c omp onent. Pr o of . In this case the follo wing relation holds: k 13 + k 14 + k 15 + k 16 + 2 k 24 + 2 k 25 + 2 k 26 + 3 k 35 + 3 k 36 + 4 k 46 = 6 (2) W e now hav e to d etermine an integer solution to (2) such t hat ω 3 k 13 + ω 4 k 14 + ω 5 k 15 + ω 6 k 16 + ω 4 k 24 + ω 5 k 25 + ω 6 k 26 + ω 5 k 35 + ω 6 k 36 + ω 6 k 46 is minimal. As ω 3 < ω 4 < ω 5 < ω 6 w e conclude that k 1 ℓ = 0 for 4 ≤ ℓ ≤ 6, k 2 ℓ ′ = 0 for 5 ≤ ℓ ′ ≤ 6 an d k 36 = 0. W e also must hav e k 13 ≤ 1 as in t he section ab ov e. By 2 ω 4 > ω 6 w e must ha ve k 24 ≤ 1. By (2) 15 w e also hav e k 35 ≤ 2 and k 46 ≤ 1. If k 13 = 0 th e only solutions under the given restrictions are k 35 = 2 and k 24 = 1 , k 46 = 1. If k 13 = 1 th e on ly solution is k 35 = 1 , k 24 = 1. Thus, the minimal amount w e get by redu ction from N ( v ) is ω 6 + ω 4 . ⊓ ⊔ D Omitted Cases of the Analysis of the Final 4-regular Case Proposition 7. A ny final 4 -r e gular br anch c onsider e d wi th its pr e c e ding br anch c an b e upp er b ounde d by O ∗ (2 K 6 . 1489 ) . Pr o of . Here we must an alyze a final 4-regular branc h together with any possible pre- decessor. These are all branc hes derived from priorities 4-8: Let o be the num b er of w eight 4 vertices from N ( v ) or the 3- or 4-path, respectively . If in one component a final 4-regular branc h follo ws then the w orst case is when o = 0 as any suc h vertex w ould b e deleted completely and ω 4 > ω 3 . On the other hand if t h ere is a component without an immediate 4-regular branch succeeding then the w orst case app ears when o is maximal as ω 3 ≥ ∆ 4 . So in the analysis w e will consider for each case the particular w orst case even th ough b oth together never app ear. Priorit y 6 S ubcases 2, 3( a ) and 4 of our non-regular priorit y-6 analysis can be ana- lyzed similar t o priorities 4, 5 and 8. W e now analyze the remaining subcases. Sub c ase 1 Here we d eal with small comp onen ts which are directly solved without any branching. Therefore we get a ( { 3 ω 4 + 2 ω 3 + 4 ∆ 4 } 2 ) -branch in the com bin ed analysis. Consider no w cases 5 and 3( b ). Let u 1 , u 2 b e the pick ed limited pair. Due to HP the va riable u 2 has tw o w eight 3 neigh b ors. Thus, if a fin al 4-regular b ranc h is follo wing in these cases we get a reduction of 2 ω 3 in addition (with respect to th e compon ent of the branch). F or both cases we derived a non symmetric branch, e.g., an ( a, b )-branch with a 6 = b . Dep ending wheth er th e final 4-regular branch follo ws in t he first, the second or b oth comp onen ts w e derive three combined branches: a ) ( { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , 2 ω 3 + ω 4 + 2 ∆ 4 ), b ) (2 ω 3 + 2 ω 4 + 2 ∆ 4 , { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 )- and c ) ( { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 ). As O ∗ (2 K 6 . 1489 ) not prop er up- p erbound s a ) w e need a further d iscussio n for the t wo sub cases. R ememb er that in the fi rst component of a ) some w eight 3 neighbor t of v is set. Sub c ase 3b Firs t supp ose that N ( z ) \ ( N ( u 1 ) ∪ N ( u 2 )) = ∅ and N ( y ) \ ( N ( u 1 ) ∪ N ( u 2 )) = ∅ , see Figure 3(a) . Then by e ither branching on y or z we get a ( { 2 ω 4 + 4 ω 3 } 2 )-branch. In this case the combined analysis is similar to priorities 4, 5 and 8. S econdly , w.l.o.g. we hav e N ( z ) \ ( N ( u 1 ) ∪ N ( u 2 )) 6 = ∅ , see Figure 3(b ) and 3(c). In Figure 3(b) we might hav e pic ked y = v or z = v . But observ e that in b oth cases in the branch where the particular w eight 3 n eigh b or t is set ( t = s if v = z and t = z if v = y ) such that in th is comp onen t a 4-regular branch follo ws w e have a ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ω 4 + 2 ω 3 + 2 ∆ 4 )-branch in th e combined analysis instead of a ). I f th e case in Figure 3(c) matc hes then w e hav e t = s . Then in the branch w ere s is set y and u 1 will b e redu ced due to RR-5 and N [ u 2 ] \ { u 1 } due to the fact that a 4-regular branch follo ws. Th us, the derived branch is the same as for th e case of Figure 3(b). Sub c ase 5 As th e vertices in N ( u 1 ) ∪ N ( u 2 ) can not form a comp onen t w.l.o.g. we hav e that N ( z ) \ ( N ( u 1 ) ∪ N ( u 2 )) 6 = ∅ . In this case we branch on u 1 . No w in t he 16 branch where we set z (i.e., z = t ) suc h that a 4-regular branch follow s in that compon ent we h ave a ( { 3 ω 4 + 5 ω 3 + 4 ∆ 5 } 2 , 2 ω 4 + 2 ω 3 )-branch in the combined analysis insted of a ) Both branches replacing a ) have an u pper b ound of O ∗ (2 K 6 . 1489 ). P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d s (a) P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d s (b) P S f r a g r e p l a c e m e n t s z y u 1 u 2 s 1 s 2 v a b c d s (c) Fig. 3. Priorit y 7 4-p ath In this case we ha ve the follo wing branc hes: a ) ( { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , ω 4 + ω 3 + 3 ∆ 4 ), b ) (2 ω 4 + ω 3 + 3 ∆ 4 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ), c ) ( { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ). The cases a ) and c ) are not up per boun ded by O ∗ (2 K 6 . 1489 ) and h ence n eed furt h er discussion. Supp ose there is a vertex y ∈ D := N ( v ) ∪ { u 0 , . . . , u l − 1 } with we ight 4. Then by O b serv ation 2 w e ha ve branches a ′ ) ( { 4 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 , ω 4 + ω 3 + 3 ∆ 4 ) and c ′ ) ( { 4 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) which are b oth upp erb ounded b y O ∗ (2 K 6 . 1489 ). F or th e remaining case we need the nex t prop osition. F or U ⊆ V ( F ) w e define E 3 ( U ) = {{ u, v } | u ∈ U, # 2 ( u ) = 3 , v 6∈ U } . Claim. Su ppose for all y ∈ D we hav e # 2 ( y ) = 3. Then th ere m u st b e some y ′ ∈ V \ ( D ∪ { v, u l } ) with # 2 ( y ′ ) = 3. Pr o of . Assume the con trary . Observ e that if l ≥ 3 then u l ∈ Y du e to u l − 2 6∈ N ( u l ). If l = 2 and u 0 6∈ N ( u 2 ) t hen also u l ∈ Y holds. Let us assume this case as the other one wil l b e treated separately . Now due to t he choi ce of v w e ha ve that u l must be adjacen t to v , u l − 1 and to t wo further weigh t 3 v ertices in D . Therefore and as D ∪ { v , u l } can not b e a comp onen t w e hav e l < 4. Also for any 2 ≤ l ≤ 3 we alw a ys have | E 3 ( D ∪ { v , u l } ) | ≤ 8 − 2 l . There must some weigh t 4 vertex r 6∈ D ∪ { v , u l } adjacent to some w eight 3 vertex in D as w e ha ve no small comp onents. Note that r ∈ Y , as v 6∈ N ( r ) or w. l.o.g. u 0 ∈ N ( r ) but u 1 6∈ N ( r ). Due to the choice of v , r must have at least three w eight 3 neigh b ors. Hence l = 2. If r has 4 w eight 3 neighbors then ( D ∪ { v , u l , r } ) forms a comp onent whic h is a contradiction. Hence, w e hav e | E 3 ( D ∪ { v , u l , r } ) | = 1 and therefore w e find again some r ′ ∈ Y \ ( D ∪ { v , u l , r } ) whic h is adjacen t to at least 3 w eight 3 vertices where at most one is from D . Th us th ere must b e some weigh t 3 vertex in V \ ( D ∪ { v, u l } ), a con tradiction. Now supp ose l = 2 and u 0 ∈ N ( u 2 ). Let N ( u 0 ) = { z , u 1 , u 2 } . If z 6∈ N ( u 2 ) then u 2 ∈ Y and the fi rst part of the proof applies. Now supp ose z ∈ N ( u 2 ) and # 2 ( z ) = 3. Then it follo ws that z ∈ N ( v ) and | E 3 ( { D ∪ { v , u 2 }} ) | ≤ 2. No w due to scp w e can find an r ∈ Y \ ( D ∪ { v , u l } ) whic h is adjacen t to at least th ree weigh t 3 vertices where only tw o can b e from D ∪ { v , u l } , a con tradiction. Supp ose z ∈ N ( u 2 ) and # 2 ( z ) = 4. As u 1 6∈ N ( z ) we ha ve z ∈ Y . Th u s, z is adjacen t to the tw o weigh t 3 v ertices in N ( v ) \ { u 2 , u 1 } . As D ∪ { v, u 2 , z } is not a 17 compon ent we hav e | E 3 ( { D ∪ { v, u 2 , z }} ) | = 2. Similarly , a contradiction follo ws. ⊓ ⊔ If for all y ∈ D we ha ve # 2 ( y ) = 3 from t he last claim and Lemma 2.2 w e can derive tw o branches a ′′ ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ω 4 + ω 3 + 3 ∆ 4 ) and c ′′ ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) whic h are upp er b ounded by O ∗ (2 K 6 . 1489 ). 3-p ath In the case of 3-path such that u 0 = u l +1 the branch with l = 2 is dominated by all other choi ces. S ince t h is is a ( { 7 . 21 } 2 )-branch w e refer t o p rioriti es 4, 5 and 8 from ab ov e. ⊓ ⊔ E F ull Pro of of Lemma 5 Pr o of . If | N ( a, b, c ) \ { v }| ≥ 5, then in F [ v ], 9 v ariables are deleted, so that we ha ve a (4 ω 3 , 10 ω 3 )-branch. Otherwise, either one of th e tw o fol lowing situations m ust o ccur: a ) There is a v ariable y 6 = v , such that N ( y ) = { a, b, c } , see Figure 1(f ). Then b ranc h on b . I n F [ ¯ b ], v , y , a, c, z will disapp ear (due to RR-5 and Lemma 2.1). In F [ b ], due to setting z , additionally a neighbor f 6∈ { a, b, c, v , y } of z will b e d eleted as the vertices a, b, c, v , y , z do not form a comp onen t. This is a total of seven vari ables. b ) There are v ariables p, q , suc h that | N ( p ) ∩ { a, b, c }| = | N ( q ) ∩ { a, b, c }| = 2, see Figure 4(a) and 4(b). In F [ v ], the v ariables a, b, c, p, q will be set. Then, at least 3 additional v ariables will be deleted (even if there are 1 ≤ i < j ≤ 4 with h i = h j ). Theorem 4.2 of [5] contains also an alternativ e pro of of b ) . ⊓ ⊔ P S f r a g r e p l a c e m e n t s v a b c d q e f y z v a b c d y p q z h 1 h 2 h 3 h 4 (a) P S f r a g r e p l a c e m e n t s v a b c d q e f y z v a b c d y p q z h 1 h 2 h 3 h 4 (b) Fig. 4. F Additional Analysis of t he Com bined Approach F.1 5-regular Branc hes in the Combined Approac h Internal 5-regular branches yield the same recurren ces as in th e simple approach. Final 5-regular branches must b e analyzed together with th eir immediate p receding b ran ch. Thus they ha ve to be analyzed toge th er with the in tro duced branc hes of [6]. T ab le 2 cap- tures some cases ( k 15 = 5; k 13 = 1 , k 14 = 4; k 25 = 1 , k 15 = 3). F or th e case k 14 = 1 and k 15 = 4 there are tw o recurrences for the branching F [ v ] , F [ ¯ v , v 1 ] , F [ ¯ v , ¯ v 1 , v 2 , v 3 , v 4 , v 5 ]. 18 case one comp onent b oth comp onen ts upp er b ound k 15 = 5 ( { 7 ω 5 + 5 ∆ 5 } 2 , ω 5 + 5 ∆ 5 ) ( { 7 ω 5 + 5 ∆ 5 } 4 ) O ∗ (1 . 0846 K ) k 13 = 1, k 15 = 4 ( { 6 ω 5 + ω 3 + 5 ∆ 5 } 2 , ω 5 + 4 ∆ 5 + ω 3 ) { 6 ω 5 + ω 3 + 5 ∆ 5 } 4 ) O ∗ (1 . 0878 K ) k 25 = 1, k 15 = 3 ( { 6 ω 5 + 5 ∆ 5 } 2 , ω 5 + 3 ∆ 5 + ( ω 5 − ω 3 ) ( { 6 ω 5 + 5 ∆ 5 } 4 ) O ∗ (1 . 0914 K ) T able 2. The first recurrence assumes that v 1 has at least three neigh b or of wei ght less than five in F [ ¯ v ]: (A) (5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) , 4 ∆ 5 + ω 5 + ∆ 4 + ω 4 + 3 ∆ 4 + ∆ 5 , 5 ω 5 + ω 4 + ∆ 5 ). The other (B) ( ω 5 + 4 ∆ 5 + ∆ 4 , ω 5 + 4 ∆ 5 + ∆ 4 + ω 4 + 4 ∆ 5 , 5 ω 5 + ω 4 + 3 ω 3 ) captures the remaining case. Both branches ha ve th ree components. T able 3 captures the combined branches of a immediately follo wing final 5- b ranc h and branches (A) and (B) . This dep ends on whether the final 5-regular branch follo ws after the first (1), the second (2) or the third (3) component or in an y combination of th em. Branc h- t yp e comp onents com bined branch upp er b ound (A) (1) ( { 6 ω 5 + ω 4 + 5 ∆ 5 } 2 , O ∗ (1 . 0912 K ) 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + ω 4 + 3 ∆ 4 + ∆ 5 , 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + 4 ω 4 + ω 3 + ∆ 5 ) (A) (2) (5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) , { 6 ω 5 + ω 4 + 5 ∆ 5 } 2 , O ∗ (1 . 1094 k ) 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + 4 ω 4 + ω 3 + ∆ 5 ) (A) (3) (5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) , O ∗ (1 . 1175 K ) 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + ω 4 + 3 ∆ 4 + ∆ 5 , { 6 ω 5 + ω 4 + 5 ∆ 5 + ω 3 } 2 ) (B) (1) ( { 6 ω 5 + ω 4 + 5 ∆ 5 } 2 , O ∗ (1 . 0894 K ) 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + ω 4 + 4 ∆ 5 , 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + 4 ω 4 + ω 3 + 3 ω 3 ) (B) (2) (5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) , { 6 ω 5 + ω 4 + 5 ∆ 5 } 2 , O ∗ (1 . 1052 K ) 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + 4 ω 4 + ω 3 + 3 ω 3 ) (B) (3) (5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) , O ∗ (1 . 1159 K ) 5 ∆ 5 + ω 5 + ( ∆ 4 − ∆ 5 ) + ω 4 + 4 ∆ 5 , { 6 ω 5 + ω 4 + 5 ∆ 5 + 3 ω 3 } 2 ) (A) or (1)(2)/(1)(3) ( { 6 ω 5 + ω 4 + 5 ∆ 5 } 4 , ω 5 + 4 ∆ 5 + ∆ 4 ) O ∗ (1 . 1126 K ) (B) /(2)(3) (A) or (1)(2)(3) ( { 6 ω 5 + ω 4 + 5 ∆ 5 } 6 ) O ∗ (1 . 0936 K ) (B) T able 3. The second column indicates after whic h co mponents a final 5- regular branch immediately follows. W e w ould like to commen t the recurrences in the third and sixth row. Here w e get a reduction of ω 3 and 3 ω 3 in addition to 5 ω 4 + ω 4 from v , v 1 , . . . , v 5 . This add itional amount comes from clauses C such that | C ∩ { v , v 1 . . . , v 5 }| = 1. Especially in the first case due to scp N [ v ] is not comp onen t and thus at least one further va riable must b e deleted. 19 Proposition 8. L et v ∈ V ( F ) b e the variable chosen for br anching by Alg. 1 such that # 2 ( v ) = 5 . Assume v induc es a solution to e quation (1) such that it is differ ent fr om k 13 = 1 , k 15 = 4 ; k 15 = 5 ; k 25 = 1 , k 15 = 3 ; k 14 = 1 , k 15 = 4 ( ⋆ ). I f a 5 -r e gular br anch fol lows in one c omp onent we have at le ast a ( { 3 ω 5 + ω 4 + 5 ∆ 5 } 2 , ω 5 + 3 ∆ 5 + 2 ∆ 4 ) -br anch and if it fol lows in b oth a ( { 3 ω 5 + ω 4 + 5 ∆ 5 } 4 ) -br anch. Pr o of . If a component is follow ed by a final 5-regular branch the least amoun t w e get by redu ction from N ( v ) is ω 5 + ω 4 . This refers to the case k 35 = 1 and k 24 = 1 which follo ws analogo usly from S ection C.1. The least reduction from N ( v ) without a follo wing final 5-regular branch can b e found as follo ws: Co n sider any solution of eq uation (1) expect the ones in ( ⋆ ). Among them find one which minimizes ∆ 3 k 13 + ∆ 4 k 14 + ∆ 5 k 15 + ( ∆ 4 + ∆ 3 ) k 24 + ( ∆ 5 + ∆ 4 ) k 25 + ( ∆ 5 + ∆ 4 + ∆ 3 ) k 35 (3) W e can assume k 24 = 0 as w e ha ve ∆ 4 + ∆ 3 > ∆ 5 + ∆ 4 . As w e are excluding ( ⋆ ) w e must ha ve k 15 ≤ 4. If k 15 = 4 w e conclude th at either k 14 = 1 oder k 13 = 1. Both solutions are forbidden (see ( ⋆ )). Thus w e m ust ha ve k 15 ≤ 3. If k 35 = 1 then there is a b etter solution as ∆ 5 + 2 ∆ 4 < ∆ 5 + ∆ 4 + ∆ 3 : set k ′ 35 = 0 , k ′ 15 = k 15 + 1 , k ′ 14 = k 14 + 2 and keep the other coefficients. Note that in this case w e must have k 15 ≤ 2 which assures t hat the new solutions is differen t from the ones in ( ⋆ ). Therefore it follo ws that k 35 = 0. Now supp ose k 25 = 2, th en k 15 = 1 h olds. But then k ′ 25 = k 25 − 1 , k ′ 15 = k 15 + 1 , k ′ 14 = 1 is a no w orse solution. Thus k 25 ≤ 1. If k 25 = 1 then with k 15 = 2 and k 14 = 1 this is minimal (since k 15 = 3 is forbidden ( ⋆ )) . Supp ose k 25 = 0, then clearly the b est solution is k 15 = 3 and k 14 = 2. Both solutions provide a reduction of 3 ∆ 5 + 2 ∆ 4 whic h is minimal. Now we analyze a final 5-regular branch and a b ranc h different form ( ⋆ ) satisfying equation (1). If t he final 5-regular branch follo ws in only one comp onen t then we hav e at least a ( { 3 ω 5 + ω 4 + 5 ∆ 5 } 2 , ω 5 + 3 ∆ 5 + 2 ∆ 4 )-branch in th e combined analysis. If it follo ws in b oth then a ( { 3 ω 5 + ω 4 + 5 ∆ 5 } 4 )-branch upp er boun ds correctly . ⊓ ⊔ Due to Proposition 8 we can up per b ound the 5-regular branches whose predecessors are different from k 13 = 1 , k 15 = 4; k 15 = 5; k 25 = 1 , k 15 = 3; k 14 = 1 , k 15 = 4 by O ∗ (1 . 1171 K ) in their com bined analysis. F.2 Analysis of the 6- 4- and 3-phase in the Com bined Approac h Here w e pro vide the ru n times under ˜ γ ( F ) for the cases w e did not consider in S ec- tion 5. The run time has b een estimated with resp ect to ˜ γ ( F ). Names will refer to the correspondin g ones in the analysis of Alg. 1 . Non-regular B ranc hes In T able 4 w e find t h e derived recurrences for each priority of HP if w e hav e a n on-regular branch. Y ou can fi nd them together with t h eir run times. Priorit y 3 is not considered as the 5-ph ase has been analyzed in Sections 5 and F.1. Regular Branc hes T able 5 captures the run times of any in ternal 6, 5 or 4-regular branch. T able 6 considers final 4 or 6-regular branc hes together with their preceding branches. The case where w e h a ve chosen v due to priorit y 7 suc h that v has a 3-path with u 0 6 = u l is treated separately . 20 Priorities branch upp er b ound Priorit y 1 (7 , 7) O ∗ (1 . 1042 K ) Priorit y 2 ( { ω 6 + 5 ∆ 6 + ∆ 5 } 2 ) O ∗ (1 . 118 K ) Priorit y 4 ( { 2 ω 4 + 4 ∆ 4 } 2 ) O ∗ (1 . 102 K ) Priorit y 5 ( { 3 ω 4 + 2 ω 3 } 2 ) O ∗ (1 . 1099 K ) Priorit y 6 (2 ω 4 + 2 ω 3 + 2 ∆ 4 , ω 4 + 2 ω 3 + 2 ∆ 4 ) O ∗ (1 . 1143 K ) Priorit y 7 ( ω 4 + ω 3 + 5 ∆ 4 , ω 4 + ω 3 + 3 ∆ 4 ) O ∗ (1 . 1172 K ) ( { ω 4 + 3 ω 3 + 3 ∆ 4 } 2 ) O ∗ (1 . 1 K ) (2 ω 4 + ω 3 + 3 ∆ 4 , ω 4 + ω 3 + 3 ∆ 4 ) O ∗ (1 . 1165 K ) Priorit y 8 ( { 2 ω 4 + 2 ω 3 + 2 ∆ 4 } 2 ) O ∗ (1 . 1 K ) Priorit y 9 (8 ω 3 , 6 ω 3 ) O ∗ (1 . 1105 K ) Priorit y 10 (8 ω 3 , 6 ω 3 ) O ∗ (1 . 1105 K ) (4 ω 3 , 10 ω 3 ) O ∗ (1 . 118 K ) T able 4. The non-regular cases case branc h upp er b ound In ternal 6-r egul ar (3 ω 6 , ω 6 + 6 ∆ 6 ) O ∗ (1 . 0978 K ) ( { 3 ω 6 } 2 ) O ∗ (1 . 0802 K ) In ternal 5-r egul ar (3 ω 5 , ω 5 + 5 ∆ 5 ) O ∗ (1 . 1112 K ) ( { 3 ω 5 } 2 ) O ∗ (1 . 0974 K ) In ternal 4-r egul ar (5 ω 4 , ω 4 + 4 ∆ 4 ) O ∗ (1 . 103 K ) ( { 5 ω 4 } 2 ) O ∗ (1 . 079 K ) T able 5. Internal h -r e gular cases ( h ∈ { 4 , 5 , 6 } ) an their upper b ounds. 3-p ath Finally we consider the case when a v ariable chosen to priority 7 has a 3- path with u 0 6 = u l +1 . The cases a ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ω 4 + ω 3 + 3 ∆ 4 ), b ) ( ω 4 + ω 3 + 5 ∆ 4 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) and c ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) are upp er b ounded by O ∗ (1 . 1152 K ), O ∗ (1 . 1159 K ) and O ∗ (1 . 1147 K ). T able 7 captures the branches together with th eir run times in the com bined algorithm if for all y ∈ N ( v ) w e ha ve # 2 ( y ) = 3. W e have a ( { 2 ω 4 + 7 ω 3 + 4 ∆ 4 } 2 , ω 4 + 4 ω 3 )-branch for case α ) whic h O ∗ (1 . 1132 K ) prop erly upp er b ounds. W e also hav e a ( ω 4 + 4 ω 3 + 2 ∆ 4 , { 2 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 )-branch for case β ) such that it is upp er b ounded by O ∗ (1 . 1142 K ). 21 Preceding branch branch upp er b ound Final 6-regular Branc h Any 6-phase branch ( { 3 ω 6 + ω 4 + 6 ∆ 6 } 2 , ω 6 + 6 ∆ 6 ) O ∗ (1 . 11 K ) ( { 3 ω 6 + ω 4 + 6 ∆ 6 } 4 ) O ∗ (1 . 105 K ) Final 4-regular Branc h In ternal 4-r egul ar ( { 6 ω 4 + 4 ∆ 4 } 2 , ω 4 + 4 ∆ 4 ) O ∗ (1 . 1115 K ) ( { 6 ω 4 + 4 ∆ 4 } 4 ) O ∗ (1 . 1003 K ) Priorities 4,5 and 8 ( { 3 ω 4 + 8 ∆ 4 } 2 , 2 ω 4 + 4 ∆ 4 ) O ∗ (1 . 1115 K ) Cases 2,3( a ),4 of Priorit y 6 ( { 3 ω 4 + 8 ∆ 4 } 4 ) O ∗ (1 . 117 K ) Priorit y 6 Case 1 ( { 3 ω 4 + 2 ω 3 + 4 ∆ 4 } 2 ) O ∗ (1 . 07 K ) Case 5,3( b ) ( { 2 ω 4 + 2 ω 3 + 2 ∆ 4 , { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 ) b ) O ∗ (1 . 109 K ) b ) and c ) of the analysis ( { 3 ω 4 + 4 ω 3 + 4 ∆ 4 } 4 ) c ) O ∗ (1 . 1143 K ) Case 3 b ), case a ) of the analysis ( { 3 ω 4 + 4 ω 3 + 5 ∆ 4 } 2 , ω 4 + 2 ω 3 + 2 ∆ 4 ) O ∗ (1 . 1151 K ) Case 5, case a ) of the analysis ( { 3 ω 4 + 4 ω 3 + 5 ∆ 4 } 2 , 2 ω 4 + 2 ω 3 ) O ∗ (1 . 1145 K ) Priorit y 7 Case of a 4-path Case b ) (2 ω 4 + ω 3 + 3 ∆ 4 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) O ∗ (1 . 1155 K ) Case a ′ ) ( { 4 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 , ω 4 + ω 3 + 3 ∆ 4 ) O ∗ (1 . 1156 K ) Case c ′ ) { 4 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 , { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) O ∗ (1 . 115 K ) Case a ′′ ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ω 4 + ω 3 + 3 ∆ 4 ) O ∗ (1 . 1152 K ) Case c ′′ ) ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ( { 3 ω 4 + 3 ω 3 + 4 ∆ 4 } 2 ) O ∗ (1 . 1147 K ) Case of a 3-p ath with u 0 = u l +1 similar to priorities 4,5 and 8 T able 6. The final h -regular cases ( h ∈ { 4 , 6 } ) and their com bined analysis # 2 ( u 0 ), # 2 ( u l +1 ) left comp onent righ t component b oth components # 2 ( u 0 ) = 3 case α instead ( ω 4 + 6 ω 3 , ( { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 , # 2 ( u l +1 ) = 3 { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 ) { 2 ω 4 + 6 ω 3 + 4 ∆ 4 } 2 ) upp er b ounds O ∗ (1 . 1075 K ) O ∗ (1 . 1136 K ) # 2 ( u 0 ) = 3 ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , ( ω 4 + 5 ω 3 + ∆ 4 , ( { 3 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 , # 2 ( u l +1 ) = 4 ω 4 + 4 ω 3 ) { 2 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 ) { 2 ω 4 + 5 ω 3 + 4 ∆ 4 } 2 ) upp er b ounds O ∗ (1 . 1136 K ) O ∗ (1 . 1138 K ) O ∗ (1 . 1143 K ) # 2 ( u 0 ) = 4 ( { 4 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , case β instead ( { 4 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 , # 2 ( u l +1 ) = 4 , ω 4 + 4 ω 3 ) { 2 ω 4 + 4 ω 3 + 4 ∆ 4 } 2 ) upp er b ounds O ∗ (1 . 1088 K ) O ∗ (1 . 1088 K ) T able 7. c a e q f d b x a c x d y z f b