Fixed-Parameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs

Fixed-P arameter Algorithms in Analysis of Heuristics for Extracting Networks in Linear Programs ∗ Gregory Gutin † Daniel Karapetyan ‡ Igor Razgon § November 15, 2018 Abstract W e consider the prob lem of ex tracting a maximu m-size reflected netwo rk in a linear program. This problem has been studied before and a state-of-the-art SGA heuristic with two v ariations hav e been proposed . In this pap er we apply a new approach to e v aluate the quality of SGA. In particular , we s olve majority of the instances in the testbed to optimality using a ne w fix ed-parameter algorithm, i.e., an algo rithm whose runtime is polynomial in the input size b ut e xponen tial in terms of an additional parameter associated with the gi ven problem. This analysis allo ws us to conclud e that the the e xisting SGA heuristic, in fact, produces solutions of a v ery high quality and often reach es the optimal objectiv e v alues. H o we ver , SGA contain two comp onents which leave some space for im- prov ement: building of a s panning tree and searching for an indepen dent set in a graph. In the hope of obtaining ev en better heuristic, we tried to replace both of these components with some equi v alent algorithms. W e tried to use a fixed-p arameter algorithm instead of a greedy one for search- ing of an indepen dent set. But e ven the exact solution o f this subproblem improved the whole heuristic insignificantly . Hence, the cruc ial part of SGA is building of a spanning tree. W e tried three differen t algorithms, and it appears that the Depth- First sea rch is clearly su perior to t he other ones in b uilding of the spanning tree fo r SGA. Thereby , by application of fixed-parameter algorithms, we managed to check that the existing SGA heuristic is of a high quality and selected the componen t which required an improvemen t. This allowed us t o i ntensify the research in a proper direction which yielded a superior variation of SG A. T his v ariation signif- icantly improve s the results of t he basic S GA solving most of the instances in our experimen ts to optimality in a short t ime. ∗ A preli minary version of this pap er will appear in the Proc eeding s of the 4th Inte rnation al W orkshop on Parame terize d and Exact Computation (IWPEC’09). † Departmen t of Computer Science, Royal Hollow ay , Univ ersity of London, Egham, Surrey TW20 0EX, England, UK, gutin@cs.rhul.ac.uk ‡ Departmen t of Computer Science, Royal Hollow ay , Univ ersity of London, Egham, Surrey TW20 0EX, England, UK, daniel.karapetyan@g mail.com § Departmen t of Computer Science , Unive rsity Colle ge Cork, Ireland, i.razgon@cs.u cc.ie 1 1 Introd uction, terminology and notation Large-scale LP models which arise in ap plications usually have sparse coefficient ma- trices with special structu re. If a special stru cture ca n b e r ecognized , it can often be used to co nsiderably speed u p the p rocess of so lving the LP p roblem and/or to he lp in understan ding the nature of the LP model. A well-known f amily of such special struc- tures is networks; a number of heuristics to e xtract ( reflected) network s in LP prob lems have been de veloped an d analyzed, see, e.g ., [3, 6 , 5, 7, 12, 13, 18] (a formal d efinition of a reflected network is giv en b elow). From the co mputation al point of view , it is worthwhile extracting a reflected network on ly if th e LP p roblem und er conside ration contains a relativ ely large r eflected network. W e consider an LP problem in the standard form stated as Minimize { p T x ; subject to Ax = b, x ≥ 0 } . LP prob lems ha ve a number of equivalent, in a sense, for ms that can be ob tained from each oth er b y various op erations. Often scalin g o peration s, tha t is multip lications of rows and column s of the matrix A of constraints by non -zero constants, are ap plied, see, e.g ., [ 3, 6, 7, 12]. In the sequel unless stated o therwise, we assume th at certain scaling operations on A have been carried out and will not be applied again apart from row reflectio ns defin ed below . A matrix B is a ne twork (matrix) if B is a ( 0 , ± 1 )- matrix (that is, entries o f B belo ng to the set { 1 , 0 , − 1 } ) and ev ery colum n of B has at most one entry equal to 1 an d a t most one entry equal to − 1 . The operatio n of reflection o f a row of a m atrix B ch anges the signs of all n on-zer o entries of this row . A matrix B is a reflected network (m atrix) if the re is a sequ ence of row reflection s that tr ansforms B into a network m atrix. The p r ob lem o f detecting a maximum embedded reflected network (DMERN) is to fin d t he maximum nu mber of rows that f orm a submatr ix B o f A such that B is a reflected network . This numb er is den oted by ν ( A ) . The DMERN problem is known to be NP-hard [4]. G ¨ ulpınar et al. [13] showed that the max imum size of an emb edded reflec ted net- work equals the maximu m ord er of a balanced induc ed subgrap h of a special signed graph associated with matrix A (for details, see Section 2). T his result led G ¨ ulp ınar et al. [13] to a heuristic n amed SGA for detection of reflected networks. Comp utational experiments in [1 3] with SGA and th ree other h euristics d emonstrated that SGA and another heu ristic, RSD, wer e of very similar quality and clear ly outp erform ed the tw o other heur istics in this respec t. Howe ver, SGA was abo ut 20 times faster , on average, than RSD. Mor eover , SGA h as an importan t theoretica l prop erty that RSD d oes n ot have: SGA alw ays solves th e DMERN problem to o ptimality when the whole m atrix A is a reflected network [1 3]. Sin ce SGA appeared to b e th e best choice for a heuristic for detection of reflected n etworks, Gutin and Zverovitch [14] inv estigated ‘ repetition’ ver- sions of SGA and found ou t th at three times repetition of SGA (SGA3) gi ves about 1% improvement, while 80 times repetition of SGA (SGA80) leads to 2% improvement. In this pap er we p ropo se a more refined analy sis of the SGA heu ristic. By usin g a fixed-p arameter algorithm, we man aged to find the optimal solutions f or majo rity of the instances in the testbed . This h elped us to understand th at SGA, in fact, obtains solutions of very h igh qu ality a nd some times even solves the instances to optima lity . 2 This mean s that even a small improvement of SGA quality is a si gnificant achiev ement. Having this result, we tried to improve SGA. SGA contains two c ompon ents wh ich leav e som e sp ace f or improvement. One is an in depend ent set searching algorith m. In the original version, a greedy algorithm was used f or this pur pose. W e replaced it with a fixed- parameter alg orithm. Here we used the well-known fact that the com plement of a n in depend ent set in a gr aph is a vertex cover . Howe ver , our experiments have shown that even the greedy alg orithm usually finds the op timal o r almost op timal so lutions and, thus, this mo dification o f SGA ap pears to be of little p ractical intere st. This demo nstrated that the in depend ent set e xtracting h euristic need not be rep laced by a more powerful heuristic or exact algorithm . Hence, th e cru cial co mpone nt in SGA is th e second comp onent, i.e., building of a spanning tree. In the or iginal version of SGA ([13, 1 4]) a rando m procedure was used for this pu rpose. W e tried to replace it with Breadth -First Search ( BFS) and Depth - First sear ch ( DFS) algo rithms. The expe riments show that th e final solution quality significantly depe nds on spanning tree an d that using DFS is clearly sup erior to both BFS and Random Search (RS). Moreover , we o bserve th at the cho ice o f the alg orithm for spann ing tree compu- tation do es no t essentially influence the run time of the heur istic un der consider ation. Thus we propo se three ne w heu ristics SGA(DFS), SGA3(DFS), and SGA80(DFS) th at take about the same time as their respecti ve counterparts SGA, SGA3, and SGA80 b ut are more precise. In or der to ev aluate the quality of the consider ed heur istics we com pare the ir out- puts with optima l solu tions of the considere d in stances. T o so lve th e in stances to op- timality , we desig n a fixed-p arameter algorithm for the DMERN p roblem . Such alg o- rithm is polyn omial in the input size but expo nential in term s of an additio nal p arameter associated with the gi ven problem. Pro blems that can be solved by fixed- parameter al- gorithms a re called fixed-parameter tractable (FPT). A fixed-param eter algo rithm is usually applied wh en the parameter is sm all. In th is situation the expo nential comp o- nent of the ru ntime become s a relatively s mall multiplicative or additive c onstant, that is the p roblem is solved by a poly nomial ( usually even a low polyno mial) algorith m. Nev ertheless, ev en if the parame ter is small, the researc hers often pr efer to use impr e- cise heu ristic m ethods simply b ecause they are faster . W e argu e th at in this situation a fixed-p arameter algo rithm may be still of a considerab le u se beca use it c an help to ev aluate th e quality of the c onsidered h euristics. In particu lar , in our experiments the use of a fixed-p arameter algo rithm allowed to observe that th e heu ristic SGA80(DFS) almost alw ays returns an optimal solu tion. That is, altho ugh the h euristic is the slowest among the considered ones, this is s till not the reason to d iscard it : the heuristic can be the best choice when the quality is particularly crucial. Thus we represent a novel way of application of fixed-param eter algorithms which, we belie ve, w ould be a significant contribution to the applied research related to fixed-parameter computation. T o de sign a fixed-param eter algorithm fo r the DME RN p roblem we in fact design a fixed-p arameter a lgorithm for th e max imum balance d subgr aph prob lem using the equiv alence result from [13]. The fix ed-param eter a lgorithm for the latter prob lem is design ed by showing its equiv alence to the bipar tization p roblem and then u sing a fixed-parameter algorithm for the bipartization problem first proposed in [2 5] and then 3 refined in [19]. The rest o f th e pap er is organ ized as follows. In Section 2 we intro duce necessary notation, Section 3 p resents the SGA h euristic and its variants, and Section 4 in troduc es the fixed-p arameter algor ithms. Section 5 describes a fixed-parameter alg orithm for t he maximum balanced sub graph prob lem. In Section 6 we repo rt em pirical results an d analyze them. Conclu ding remarks are made in Section 7. 2 Embedded network s and signed graphs In this sectio n, we assume, fo r simp licity , that A is a ( 0 , ± 1 )- matrix itself ( since all rows containing entries n ot from the set {− 1 , 0 , +1 } canno t be part of a reflected n etwork). Here we allow graphs to have parallel edges, but no loops. A graph G = ( V , E ) along with a f unction s : E →{− , + } is called a signed graph . Signed gr aphs have been studied by many researchers, see, e.g., [15, 16, 17, 26]. W e assume that signed grap hs have no parallel ed ges o f the same sign, b ut may have parallel edg es of oppo site signs. An ed ge is positive ( negative ) if it is assigned plus ( minus). For a ( 0 , ± 1 )-matrix A = [ a ik ] with n ro ws, we co nstruct a signed graph G ( A ) as follows: the vertex set of G ( A ) is { 1 , 2 , . . . , n } ; G ( A ) ha s a positive (negative) edge ij i f and only if a ik = − a j k 6 = 0 ( a ik = a j k 6 = 0 ) for some k . Let G = ( V , E , s ) be a signed g raph. For a non- empty subset W of V , th e W -switch of G is the signed grap h G W obtained from G by cha nging the signs of the edges between W and V ( G ) \ W . A signed g raph G = ( V , E , s ) is b alanced if there exists a subset W of V ( W may co incide with V ) such that G W has no n egativ e edges. L et η ( G ) be the largest order of a balanced induced subgraph of G . The following impo rtant result was proved in [ 13]. This result allows us to search for a largest balanced induced subgraph of G ( A ) instead of a largest reflected network in A . Theorem 1. [13] Let A be a (0 , ± 1) -ma trix. A set R o f r o ws in A fo rms a r eflected network if and o nly if the vertices of G ( A ) corr espo nding to R in duce a balan ced subgraph of G ( A ) . In particular , ν ( A ) = η ( G ( A )) . 3 SGA and its V ariations The heuristic SGA introduce d in [13] is based on the following: Lemma 1. [13] Every signed tree T is a bala nced gr aph. Pr oo f. W e pr ove th e lemma by ind uction on the num ber of edges in T . The lemma is tr ue when the n umber of edges is one. Let x be a vertex of T of degree on e. By the in duction hy pothesis, there is a set W ⊆ V ( T ) − x such tha t ( T − x ) W has n o negativ e edge s. In T W the edge e incident to x is positi ve or negative. In the first case, let W ′ = W and the second case, let W ′ = W ∪ { x } . Then , T W ′ has n o n egativ e edges. 4 − + M + − − − + − − + + + + + − + − + 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 G T T { 2 } G { 2 } Figure 1: Illustratio n for SGA; M is the subg raph G { 2 } induced by the negative edges of G { 2 } . Heuristic SGA: Step 1: Con struct signed graph G = G ( A ) = ( V , E , s ) . Step 2: Find a spa nning forest T in G . Step 3: Using a rec ursive a lgorithm based on the pr oof of Lemm a 1, compute W ⊆ V such that T W has no negati ve ed ges. Step 4: Let N be th e subgr aph of G W induced b y th e negative edges. Apply the following g reedy-d egree algorith m [23] to find a maximal independen t set I in N : starting from empty I , append to I a verte x of N of minimum degree, delete this vertex tog ether with its neighb ors fro m N , and repeat the ab ove proced ure till N h as no vertex. Step 5: Outp ut I . For a gr aph H with conn ectivity components H 1 , . . . , H p , a span ning for est is the union o f spanning trees T 1 , . . . , T p of H 1 , . . . , H p , respecti vely . Th e secon d step of the algorithm ( i.e., fin ding of spanning fo rest) can be implemented in a numb er of different ways. W e tried the following ones: 1. Rand om Searc h (RS) starts fr om markin g some vertex. On every iteratio n, it finds some edge (positi ve or negative; d ouble edges are not co nsidered) between a marked vertex v and an unmarked vertex u . It marks u and include s or e xcludes it from th e set W accord ing to the sign of the edge u v . If no edges between marked and unm arked vertices are found, the algorithm ma rks some rand om vertex again. This was the meth od of c omputin g a spanning forest used fo r the experiments reported in [13]. 2. BFS is a well known alg orithm for c onstructing of span ning forests. It maintains a FIFO queu e. On e very iteration, it takes a verte x from this queue, marks all its unmarked neighbors into the queue and adds these n eighbor s to the queue. If the queue is emp ty , so me u nmarked vertex is ma rked and added in to th e q ueue. In our implemen tation, we take a vertex of maximu m degree in this case. 3. DFS is anoth er well known alg orithm f or constructing of span ning f orests. It starts from some v ertex and th en calls itself recursi vely f or every of its ne ighbor s which still are not included in the spannin g forest. Proposition 1. [13] If G is bala nced, then I = V . 5 Pr oo f. It is well-known ( see, e.g., Th eorem 2.8 in [13]) that a signed graph is balanced if and on ly if it d oes no t contain cycles with od d numb er o f n egati ve edges. Let T be a a span ning fo rest in G . Since T W has no negative edges, G W cannot have negative edges. Indeed , if xy was a negative ed ge in G W , it would be the un ique negative edge in a cycle formed by xy and the ( x, y ) -path of T W , a con tradiction. Gutin a nd Zverovitch [1 4] investigated a repetition versio n of SGA where Step s 2-4 were r epeated several time (each tim e the vertices o f G were p seudo-r andom ly permuted and a new sp anning fo rest of G was built). Th ey fou nd ou t that th ree times repetition of SGA g iv es about 1% improvemen t, while 80 times repetition of SGA leads to 2% improvement, on average. In our experim ents we used a larger test bed and better scaling procedure t han in [14] and, thus, we run SGA and its 3 a nd 80 times repetitions o n the new set o f instances of the DMERN pr oblem ( see Section 6). W e will denote these repetition versions of SGA by SGA3 and SGA80, respecti vely . In Section 6 we also re port r esults on another mod ification of SGA, SGA+VC, where we replace Step 4 w ith finding a vertex cover C of G W and setting I = V ( G W ) \ C. Since the vertex cover prob lem is well studied in the area o f pa rameterized complex- ity [1, 8, 22], to find C we c an use a fixed-parameter algorithm for the problem. 4 Fixed-Parameter Al gorithmics W e recall so me most basic no tions of fixed-param eter alg orithmics (FP A) here, for a more in-d epth treatment of th e top ic we r efer the reader to th e mon ograp hs [10, 11, 22]. FP A is a relatively new ap proach for dea ling with intra ctable c omputatio nal prob- lems. In the f ramework of FP A we introdu ce a parameter k , wh ich is often a positive integer (but may b e a vector , g raph, or any o ther object f or some prob lems) such tha t the problem at hand ca n b e solved in time O ( f ( k ) n c ) , where n is the size of the p roblem instance, c is a con stant not dep endent on n or k , a nd f ( k ) is an arbitrar y comp utable function not d epend ent on n . The ultimate go al is to obtain f ( k ) and c such th at for small or even mod erate values of k the problem under con sideration can b e completely solved in a reasonable amount of ti me. As an example, con sider the V ertex Cover pr oblem (VC) : given an undirected gr aph G (with n vertices a nd m edg es), find a m inimum nu mber o f vertices such that every edge is in cident to at least one of these vertices. In th e (naturally ) parameterize d version of VC, k -VC, given a gra ph G , we ar e to ch eck whether G has a vertex cover with at most k vertices. k -VC admits an algor ithm o f running time O (1 . 27 38 k + k n ) obtained in [8] that allows us to solve VC with k up to se veral hund reds. W ithout using FP A, we would be likely to end up with the obvious alg orithm of complexity O ( mn k ) . The last algorithm is far too slo w e ven for small v alues of k such as k = 10 . Parameterized problems that admit algo rithms of complexity O ( f ( k ) n c ) (we refer to such algorithms as fixed-parameter ) are called fixed-pa rameter tractable (FPT) . No- tice that not e very parameterized problem is F PT , b ut there are many pr oblems that are FPT [10, 11, 22]. 6 5 Minimum Balanced Deletion pr oblem By our discussions above, we are interested in the following parameterized problem. The minimum balanced deletion problem (MBD) Input: A sign ed graph G = ( V , E , s ) , an integer k . P arameter: k . Output: A set of at most k vertices whose rem oval makes G balanced o r ’NO’ if no such set exists. W e show that the MBD prob lem is FPT by tran sforming it in to the Bipartization problem defined as follows. The Bipartization problem Input: A g raph G , an integer k P arameter: k Output: A set of at m ost k vertices whose remov al makes G bipartite or ’NO’ if no such set exists. The transform ation is described in the following theorem. Theorem 2. The MBD pr oblem is FPT a nd can be solved in time O ∗ (3 k ) . Pr oo f. It is well-known ( see, e.g., Th eorem 2.8 in [13]) that a signed graph is balanced if and only if it does not contain cycles inv olving od d number of negative e dges. Hen ce, the MBD prob lem in fact asks for at mo st k vertices whose rem oval breaks all cycles containing an odd number of negativ e ed ges. Let G ′ be the (u nsigned ) graph obtained fro m G by subdividing each positiv e edge . In other words, for each positi ve edge { u, v } , we introduce a new vertex w and replace { u, v } by { u, w } and { w , v } . W e claim that G has a set of at most k vertices bre aking all c ycles with an odd number of negative edges if and only if G ′ can be mad e bipartite by removal of at mo st k vertices. Assume the fo rmer an d let K be a set of at m ost k vertices wh ose removal breaks all cycles with an od d n umber of negati ve edge s. It follows th at G ′ − K is bipartite. Ind eed, each cycle C ′ of G ′ − K can b e o btained from a cycle C of G − K by subdivision of its p ositiv e edg es. Hen ce, C ′ can b e of an od d leng th o nly if C has a n o dd nu mber o f negativ e ed ges which is impossible according to our assumption about K . Con versely , let K b e a set o f at mo st k vertices such that G ′ − K is bip artite. W e may safely assume th at K does not con tain the new vertices sub dividing positiv e edges: otherwise each such vertex can be replaced by one of its neig hbor s. Thu s, K ⊆ V ( G ) . Observe that G − K d oes not have cycles with odd number of negati ve edg es. I ndeed, by sub dividing positive edges, any su ch cycle translates into an o dd cycle o f G ′ − K in contradiction to our assumption about K . It fo llows fro m th e above argumentation that the MBD prob lem can be solved as follows. T ransform G into G ′ and run on G ′ the O ∗ (3 k ) algo rithm solving the b i- partization prob lem [19]. I f the algorith m retu rns ’ NO’ then return ’NO’. Other wise, replace each subd ividing vertex by on e o f its neig hbors an d retu rn the resulting set of vertices. Clearly , the complexity o f the resulting algorithm is O ∗ (3 k ) . 7 Remarks. The usual trick to av oid the subd ivided vertices to b e selected to the resulting solution would be to make k + 1 co pies of each vertex. Howe ver , such ap- proach would increase the runtime of the resulting imp lementation and henc e we ha ve used a slightly m ore sophisticated m ethod. Unfortun ately , it is not k nown yet whether the Bipartization prob lem has a polynom ial-size pro blem kernel [1 9]. Thu s, it is no t known yet whether the MBD prob lem h as a poly nomial-size pr oblem kernel. (If one was known, we co uld try to use it to speed up our fixed-parameter algorithm.) Note that a version of the MBD prob lem, where edge- deletions rather than vertex- deletions are used was considered in [20, 9]. 6 Experimental Evaluation In this section we provide and discuss our exper iment results for th e heuristics SGA, SGA3, SGA8 0, SGA+VC descried in Section 3 and the exact alg orithm given in Sec- tion 5 . Note th at in our experime nts we u se a larger test bed an d b etter scalin g p roce- dure than in [14]. 6.1 Scaling Proc edur e Recall that we consider an LP problem in the standard form stated as Minimize { p T x ; subject to Ax = b, x ≥ 0 } . In Section 2, to simplify ou r notation we assumed that A is a ( 0 , ± 1 )-ma trix. Howe ver , in general, in real LP problems A is n ot a ( 0 , ± 1 )- matrix. Ther efore, in rea lity , the fi rst phase in solving the DMERN problem is applying a scaling procedur e whose aim is to increase the numb er of (0 , ± 1) -r ows by scaling rows and colu mns. Here we describ e a scaling proced ure that we h av e used . Our co mputation al expe riments indicate that this scaling is often better th an the scaling proced ures we fou nd in the literature. Let us describe our scaling procedu re. Let A = [ a ij ] n × m . First we app ly simple row scaling, i.e., scale all th e rows which contain only zero s and ± x , where x > 0 is s ome constant: f or e very i ∈ { 1 , 2 , . . . , n } set a ij = a ij /x for j = 1 , 2 , . . . , m if a ij ∈ { 0 , − x, + x } for every j ∈ { 1 , 2 , . . . , m } . Then we apply a mor e sophisticated pr ocedur e. Let [ r i ] n be a n array of boolean values, where r i indicates whether the i th r ow is a (0 , ± 1) -r ow . Let [ b j ] m be an a rray of boolean v alues, where b j indicates whether t he j th column is bo unded , i.e., whe ther it has at least one non zero value in a (0 , ± 1) -row: for s ome j ∈ { 1 , 2 , . . . , m } the value b j = tr ue if and only if there exists some i such that r i = tru e and a ij 6 = 0 . Next we do the following for ev ery non (0 , ± 1 ) -row (note that at this stage any non (0 , ± 1 ) -row co ntains at least two nonze ro elements). Let J b e th e set o f indices of b ound ed colum ns with no nzero elements in the current row c : J = { j : a cj 6 = 0 and b j = tr ue } . I f J = ∅ , i.e ., a ll the column s c orrespon ding to n onzero elem ents in the curren t ro w are unbound ed, then we simply scale every of these columns: a ij = a ij /a cj for every i = 1 , 2 , . . . , n an d for e very j such that a cj 6 = 0 . If J 6 = ∅ and a cj ∈ { + x, − x } for e very j ∈ J , where x is some constant, then we s cale according ly the curr ent row ( a cj = a cj /x fo r every j ∈ { 1 , 2 , . . . , m } ) and scale the u nbou nded 8 columns: a ij = a ij /a cj for every j / ∈ J if a cj 6 = 0 . Otherwise we do n othing for the current row . Every time when we scale ro ws or columns we update the arrays r and b . Since th e matrices processed by this heuristic ar e u sually sp arse, we u se a special data structu re to store them. In particular, w e store only n onzero elemen ts providing the row and co lumn ind ices for each of them. W e also store a list of refere nces to the correspo nding n onzero elements for ev ery row an d for e very column of the matrix. 6.2 Computational Experience The com putational results for all h euristics apart from SGA+VC a s well as for th e ex- act a lgorithm ar e provided in T able 1. As a test bed we use all the instance s p rovided in Netlib ( http://netl ib.org/lp/d ata/ ). All algo rithms were implemen ted in C++ and th e ev aluation p latform is b ased on an AMD Athlon 6 4 X2 3 .0 GHz p rocessor . For the exact algo rithm we used a co de o f H ¨ u ffner h ttp://theinf 1.informati k.uni- jena.de/ ˜ hueffner / . In SGA+VC we used a vertex cover code ba sed o n [ 2]. Since the exact algorith m can potentially take a very long time, we introd uced a timeout: if after on e hour of work the algorithm is unable to compute the output, it terminates. Denote by n th e num ber of (0 , ± 1 ) -rows in the instance, i.e., the num ber of vertices in the c orrespon ding signed grap h G . The column k repo rts the difference between n and the numb er o f vertices in a maximum induce d b alanced subgra ph of G fo und by the exact algorithm . The rows where k is n ot given correspon d to the instances where the algorithm terminated after on e hou r withou t com puting the outpu t. Th e n ine colu mns following column k repo rt the same differences fou nd by heur istics SGA, SGA3, an d SGA80. Th e first th ree column s are r elated to SGA. The colu mns RS, BFS, and DFS are r elated to the way o f co mputin g th e spa nning tree (see Section 3 for th e d etailed explanation). Th e next thr ee colum ns are related to SGA3 and the last three colu mns are r elated to SGA8 0 with the ana logous m eaning o f p articular c olumns. The column t rep orts the run time taken by th e exact algo rithm. The time provided in th e colum n ‘ t 1 ’ is the a verage time required for SGA(RS), SGA(BFS) and SGA( DFS) 1 to proceed once. Our experimen ts show that for SGA3 this tim e is appro ximately 3 tim es larger and fo r SGA80 80 times larger . It also ap pears tha t there is no significant difference between running times of SGA based on RS, BFS or DFS. The ’A verage ’ row shows th e average value of the respective column s. The ‘ A vg. diff. ’ row shows how far , on av erage, is a par ticular modification of SGA fr om the optimal solution s. ‘# exact sol. ’ shows the n umber of in stances for which a particu lar modification of SGA o btained an optimal solution . Both ‘ A v g. d iff. ’ an d ‘# exact sol. ’ consider only the instances with the knows o ptimal solutions. 1 the acron ym in the parenthesis correspo nd to the algorithm of computing the spanning tree. 9 T able 1: E xperiment results. SGA SGA3 SGA80 Time , s Instance k RS BFS DFS RS B FS DFS RS BFS DFS t t 1 25FV47 15 25 38 21 25 38 18 22 38 16 4.40 0.004 80B A U3B — 42 40 40 40 40 40 40 40 40 > 1 h 0.106 ADLITTLE 1 1 1 1 1 1 1 1 1 1 0.02 0.000 AFIR O 0 0 0 0 0 0 0 0 0 0 0.00 0.000 A GG — 107 108 104 106 108 104 104 108 103 > 1 h 0.001 A GG2 — 85 91 83 85 91 83 83 91 83 > 1 h 0.001 A GG3 — 85 91 83 85 91 83 83 91 83 > 1 h 0.001 B ANDM 23 24 24 24 23 24 23 23 24 23 1493.12 0.001 BEA CONFD 3 3 3 3 3 3 3 3 3 3 0.00 0.000 BLEND 1 1 1 1 1 1 1 1 1 1 0.00 0.000 BNL1 1 4 17 19 15 17 18 14 14 18 14 1.83 0.003 BNL2 — 1 27 109 103 110 107 101 99 105 86 > 1 h 0.061 BOEING1 — 49 61 42 49 61 42 48 60 42 > 1 h 0.001 BOEING2 15 17 17 16 17 17 16 15 17 15 0.05 0.000 BORE3D 12 14 15 13 14 15 12 12 15 12 0.14 0.001 BRAND Y 6 7 8 6 6 8 6 6 8 6 0.00 0.000 CAPRI — 40 43 37 37 40 34 34 40 33 > 1 h 0.001 CYCLE — 34 36 34 34 36 34 34 36 34 > 1 h 0.020 CZPR OB 1 1 1 1 1 1 1 1 1 1 0.27 0.018 D2Q06C — 67 97 67 67 96 67 67 94 65 > 1 h 0.042 D6CUBE — 61 50 50 52 50 50 46 47 50 > 1 h 0.003 DEGEN2 — 234 237 219 234 237 219 226 234 218 > 1 h 0.011 DEGEN3 — 822 819 769 822 819 769 815 819 769 > 1 h 0.197 DFL001 — 2818 2997 26 03 2818 2956 2603 2802 2903 258 5 > 1 h 1.871 E226 15 18 19 15 17 19 15 16 19 15 1.09 0.001 ET AMA CR O 12 20 31 19 20 31 19 20 26 16 0.47 0.001 FFFFF800 — 50 69 46 50 69 46 41 65 39 > 1 h 0.002 FINNIS — 121 127 120 121 127 120 119 127 119 > 1 h 0.003 FIT1D 6 6 7 6 6 7 6 6 7 6 0.00 0.000 FIT1P 0 0 0 0 0 0 0 0 0 0 0.00 0.000 FIT2D 6 7 7 6 6 7 6 6 7 6 0.00 0.004 FIT2P 2 2 2 2 2 2 2 2 2 2 0.00 0.007 FORPLAN 1 1 1 1 1 1 1 1 1 1 0.00 0.000 GANGES — 83 84 82 83 84 82 77 84 77 > 1 h 0.021 GFRD-PNC — 68 68 95 68 68 86 68 68 80 > 1 h 0.008 GREENBEA — 48 60 46 48 60 46 45 57 41 > 1 h 0.043 GREENBEB — 48 60 46 48 60 46 45 57 41 > 1 h 0.044 GR O W15 0 0 0 0 0 0 0 0 0 0 0.00 0.000 GR O W22 0 0 0 0 0 0 0 0 0 0 0.00 0.000 GR O W7 0 0 0 0 0 0 0 0 0 0 0.00 0.000 ISRAEL 8 9 10 8 8 10 8 8 10 8 0.02 0.000 KB2 1 1 3 2 1 3 1 1 3 1 0.00 0.000 LO TFI 18 24 24 19 22 20 19 19 20 19 11.23 0.001 MAR OS-R7 0 0 0 0 0 0 0 0 0 0 0.00 0.011 MAR OS 11 17 14 21 15 14 18 12 14 11 0.23 0.005 MODSZK1 — 237 267 237 237 267 237 237 267 235 > 1 h 0.004 NESM 10 13 13 13 11 13 11 10 13 10 0.03 0.003 PER OLD — 28 29 25 26 29 24 24 27 23 > 1 h 0.003 PILO T .J A 16 18 19 18 17 19 16 16 19 16 11.72 0.004 PILO T — 45 45 44 42 44 41 41 44 41 > 1 h 0.008 10 SGA SGA3 SGA80 Time , s Instance k RS BFS DFS RS B FS DFS RS BFS DFS t t 1 PILO T .WE — 34 36 31 30 36 30 28 36 27 > 1 h 0.003 PILO T4 3 3 3 3 3 3 3 3 3 3 0.00 0.001 PILO T87 — 77 87 74 76 87 71 70 87 69 > 1 h 0.015 PILO TNO V 19 21 24 21 21 24 20 19 24 19 201.29 0.005 RECIPE 0 0 0 0 0 0 0 0 0 0 0.00 0.000 SC105 16 17 32 22 17 31 21 17 31 19 12.56 0.000 SC205 — 36 68 50 36 67 47 36 67 41 > 1 h 0.000 SC50A 8 8 12 8 8 11 8 8 11 8 0.02 0 .000 SC50B 6 6 9 8 6 9 6 6 9 6 0.02 0.000 SCA GR25 0 0 0 0 0 0 0 0 0 0 0.03 0.002 SCA GR7 0 0 0 0 0 0 0 0 0 0 0.02 0.000 SCFXM1 12 13 14 13 12 14 13 12 14 12 0.30 0.001 SCFXM2 — 26 28 26 26 28 26 24 28 26 > 1 h 0.003 SCFXM3 — 39 42 39 38 42 39 36 42 39 > 1 h 0.006 SCORPION 1 1 1 1 1 1 1 1 1 1 0.02 0.001 SCRS8 9 9 9 9 9 9 9 9 9 9 0.06 0.002 SCSD1 0 0 0 0 0 0 0 0 0 0 0.03 0.000 SCSD6 0 0 0 0 0 0 0 0 0 0 0.14 0.001 SCSD8 0 0 0 0 0 0 0 0 0 0 0.59 0.001 SCT AP1 0 0 0 0 0 0 0 0 0 0 0.00 0.000 SCT AP2 0 0 0 0 0 0 0 0 0 0 0.00 0.006 SCT AP3 0 0 0 0 0 0 0 0 0 0 0.00 0.012 SEBA — 274 285 267 274 285 267 269 285 265 > 1 h 0.021 SHARE1B 4 5 4 4 4 4 4 4 4 4 0.00 0.000 SHARE2B 6 6 6 6 6 6 6 6 6 6 0.00 0.000 SHELL 2 2 2 2 2 2 2 2 2 2 1.51 0.006 SHIP04L — 36 36 36 36 36 36 36 36 36 > 1 h 0.006 SHIP04S — 36 36 36 36 36 36 36 36 36 > 1 h 0.005 SHIP08L — 64 64 64 64 64 64 64 64 64 > 1 h 0.023 SHIP08S — 64 64 64 64 64 64 64 64 64 > 1 h 0.015 SHIP12L — 96 96 96 96 96 96 96 96 96 > 1 h 0.046 SHIP12S — 96 96 96 96 96 96 96 96 96 > 1 h 0.030 SIERRA — 400 455 420 400 455 397 387 444 388 > 1 h 0.030 ST AIR 8 11 12 9 9 12 9 8 12 8 0.02 0.000 ST AND A T A — 53 57 59 53 57 59 53 57 59 > 1 h 0.002 ST ANDGUB — 53 57 59 53 57 59 53 57 59 > 1 h 0.002 ST ANDMPS — 54 58 73 54 58 73 54 58 70 > 1 h 0.004 STOCFOR1 0 0 0 0 0 0 0 0 0 0 0.00 0.000 STOCFOR2 — 258 311 202 258 307 202 243 305 197 > 1 h 0.050 TUFF 16 26 16 18 17 16 16 16 16 16 0.58 0.001 VTP .BASE 4 6 6 6 4 6 4 4 6 4 0.00 0.000 WOOD1P 0 0 0 0 0 0 0 0 0 0 0.02 0.001 WOOD W 0 0 0 0 0 0 0 0 0 0 0.00 0.007 A verage 79.3 84.85 75.57 7 8.55 84.2 74.82 7 6.91 83.19 73.54 32.26 0.030 A vg. dif f. 1.28 2.15 0.93 0.78 2.02 0.52 0.35 1. 93 0.17 # exac t sol. 33 31 36 40 31 44 48 31 50 Observe th at the exact algor ithm completed its co mputation s fo r 54 instances out of the total o f 93, and for 52 instances the run ning time was at mo st 1 m inute. Note that SGA(DFS) achiev ed the optimal solution in 36 out of 5 4 cases, SGA3(DFS) in 44 cases and SGA80(DFS) in 50 cases. The DFS-based version of SGA clea rly ou tperfor ms the RS- and BFS-based ver- 11 sions. T his is also tr ue for SGA3 and SGA80 . W e are not able to justify this result but it is strongly confirmed by our extensi ve exp erimentation . The results with SGA(DFS)+VC are not p rovided in T able 1 since SGA( DFS)+VC managed to improve SGA(DFS) only for 4 instances: Instance k k SGA(DFS)+VC k SGA(DFS) k SGA80(DFS) BOEING2 15 15 16 15 DEGEN2 — 218 219 218 DEGEN3 — 764 769 769 DFL001 — 2599 2 603 2585 SGA(DFS)+VC was n ot able to pr oceed in 30 minutes for A GG, AGG2, AGG3, FINNIS, MODSZK1 and SIERRA. Note tha t k SGA(DFS)+VC < k SGA80(DFS) for only one instance while k SGA(DFS)+VC > k SGA80(DFS) for 39 instances and re call that for 6 in- stances SGA(DFS)+VC did not term inate in the given time. Since the ru nning time of SGA(DFS)+VC usu ally exceed s that o f SGA8 0(DFS) and the q uality of SGA(DFS)+VC is not much different even from that o f SGA(DFS), SGA+VC ap pears to be o f little practical inter est. Howe ver , SGA+VC demonstra tes that the re is no need to rep lace Step 4 of SGA by a more powerful heur istic o r exact algor ithm and that in o rder to improve results one sho uld pay attention to the Step 2 of SGA. In oth er words, this negativ e resu lt pointed out the pro per ar ea for furth er resear ch. I ndeed, rep lacement of RS algorithm for the spanning tree building with DFS dramatically impr oved the heuristic quality . Although SGA80 is slo wer than SGA and SGA3, it is much m ore precise and its r unnin g time is still rea sonable. It follows tha t SGA80(DFS) is t he best choice with respect to the tradeoff between running time and solution quality . O bserve that in almost all (50 out of 54) the c ases fea sible for the exact algorith m, SGA80(DFS), being much faster than the exact algor ithm, manag ed to com pute an o ptimal solution ! Note that th is co nclusion can be made o nly given the knowledge ab out the optimal solution and without the con sidered fixed- parameter algo rithm such kn owledge would be very hard to ob tain (fo r example, the instance PILO TNO V with n = 329 and k min = 19 would hardly be feasible to a brute-force exploration o f all  329 19  possibilities). 7 Conclusions One o f co ntributions of th is paper is demo nstrating a novel way o f use of fixed-param eter algorithm s where they do not substitute heuristic methods but are used to ev aluate them. As a case study , we con sidered heur istics for the problem o f extractin g a maximu m- size reflected network in an L P problem. The use of fixed-parameter algor ithms help ed us to in vestigate a state-of-th e-art h euristic, improve it and allowed u s to arrive at an interesting obser vation that the improved version of the considered heuristic a lmost always returns an optimal solution. 12 W e believe that this way of apply ing fixed-parame ter algorithms can be useful for other prob lems as well. One candid ate might be the p roblem of finding whethe r th e giv en CNF for mula h as at most k variables so that th eir removal makes the re sulting formu la Renameable Ho rn. This is called the Renameable Horn d eletion b ackdoo r problem and w as r ecently shown FPT [ 24]. Heur istics for this problem ar e widely used in modern SA T solvers f or identif ying a sm all subset of variables on wh ich an exponential-tim e bran ching is to be perfor med [2 1]. Currently it is unclear wh ether substituting a heu ristic ap proach b y the exact fixed-p arameter alg orithm would result in a b etter SA T solver . 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