Counting or producing all fixed cardinality transversals

Coun ting or pro ducing all fixed cardinalit y transv ersals M ARCEL W ILD A BSTRACT : An a lgorithm to coun t, or a lt ernativ ely generate, a ll k -elemen t transv ersals of a set sy stem is presen ted. F or special cases it w orks in output- linear t ime. 1 In tro duc tion Generating al l minimal transv ersals of a h yp ergraph H based on a set W is a prominen t researc h endea v our [EMG]. But also g enerating (and ev aluating) al l transv ersals of H ma y b e required [W3]. Like wise, the fo cus ma y b e on al l k -element transve rsals for some in teger k . F or instance in [W2] they need t o b e coun ted ( no t generated) f or k = 1 up to k = | W | . As to fixed cardinality constrain ts in general, see also [BEHM]. While [W2] and [W3] display particular applications of the so called transv ersal e -alg o rithm, the presen t pap er harks bac k to [W1] a nd pro vides additio na l theoretic results. Let u s b egin with a broader persp ectiv e and then zo om in on to transv ersals. Supp ose that a 1 up to a h denote “constrain ts” applying to subsets X of a finite set W . Man y kinds of com binatorial ob j ects X can b e mo delled as the sets X tha t satisfy h suitably c hosen constrain ts. The principle of inclusion-exclusion states that N ( a 1 ∧ · · · ∧ a h ) = 2 w − h X i =1 N ( a i ) + X 1 ≤ i 0. Of cours e B = r for t = 0. 6 of all k -elemen t tra nsve rsals of a h yp ergaph H on W , define Card( r , k ) := |{ X ∈ r : | X | = k }| for an y { 0 , 1 , 2 , e } -v alued r . Ob viously τ k is the sum of all Card( r , k ) where r ranges ov er all final ro ws pro duced b y the transve rsal e -algorit hm. F or r fixed, let us first determine the ra ng e of k ’s for which Card( r, k ) 6 = 0. With notat io n as in (1) set (4) c min ( r ) := min {| X | : X ∈ r } = β + t. Put X max = W \ zeros( r ). Then X max ∈ r and X ⊆ X max for all X ∈ r , whence c max ( r ) := max {| X | : X ∈ r } = | X max | = w − α. By (3) it is easy to compute | r | , but now w e fix k ∈ { c min ( r ) , . . . , c max ( r ) } and striv e for Card( r , k ). The extreme cases k ∗ = c min ( r ) and k ∗ = c max ( r ) are trivial: (5) Card( r, k ∗ ) = ε 1 , ε 2 · · · ε t and C ard( r, k ∗ ) = 1. Computing Card( r, k ) whe n k ∗ < k < k ∗ is more subtle. It is an exercise (carried out in [W4]) to a pply inclusion-exclusion and o btain Card( r, k ) as an alternating sum of 2 t binomial coefficien ts. Unless r is lo ng and t is small this metho d is inferior t o the following manner, particularly when Card( r, k ) is needed fo r subsequen t v alues of k . W e illustrate it on r 0 := ( e 1 , e 1 , e 2 , e 2 , e 2 , e 3 , e 3 , e 3 , e 4 , e 4 , e 4 , e 4 ) and f or w = 12 and 1 ≤ k ≤ 5: k = 1 2 3 4 5 2 1 0 0 0 0 6 9 5 1 0 0 18 45 48 0 0 0 72 288 T able 2: Calculating τ k recursiv ely The line 2 , 1 , 0 , 0 , 0 giv es the n umber o f sets in ( e 1 , e 1 ) ha ving cardinality 1 , 2 , 3 , 4 , 5 re- sp ectiv ely . The next line gives the n umber of sets in ( e 1 , e 1 , e 2 , e 2 , e 2 ) ha ving these car- dinalities, and so forth. In general, if c 1 , c 2 , · · · , c k − 1 are the num b ers of sets in the seg- men t ( e 1 , · · · , e 1 , · · · , e s − 1 , · · · , e s − 1 ) hav ing cardinalit y 1 , 2 , · · · , k − 1 respectiv ely , then the n um b er of sets in the extended s egmen t ( e 1 , · · · , e 1 , · · · , e s − 1 , · · · , e s − 1 , e s , · · · , e s ) having cardinalit y k equals (6)  ε s 1  c k − 1 +  ε s 2  c k − 2 + · · · +  ε s ε s  c k − ε s This also holds for k ≤ ε s pro vided w e put c i := 0 for i ≤ 0. F or instance, if w e tak e s = 3 7 and k = 5 in r 0 , t hen (6) ev aluates to  3 1  c 4 +  3 2  c 3 +  3 3  c 2 = 3 · 5 + 3 · 9 + 1 · 6 = 48 . As to the calculation of binomial co effic ients of ty p e  ε 1  ,  ε 2  , · · · ,  ε ε  , they are con ve nien tly calculated as follows :  ε 1  = ε,  ε j + 1  =  ε j  ε − j j + 1 for 1 ≤ j ≤ ε − 1 . By first m ultiplying with ε − j and then dividing b y j + 1 one stays in the realm of in tegers. Doing this for ε = ε 1 up to ε = ε s requires ( ε 1 − 1) + · · · + ( ε s − 1) < w multiplications and just as man y in teger-v alued divis ions. Applying the O ( w log w log lo g w ) = O ( w log 2 w ) (for shortness ) Sch ¨ onhage-Strassen algorithm for m ultiplying tw o w -digit n um b ers (se e Wikip edia), the at most w many required binomial co effic ients can b e readied in time O ( w 2 log 2 w ), and they o ccup y space O ( w 2 ). Theorem 1 : Let r b e a { 0 , 1 , 2 , e } -v alued ro w of length w and let K ≤ w . Then it costs space O ( w 2 ) and time O ( K w 2 log 2 w ) to compute the K num b ers Card( r, 1) up to Card( r , K ). Pr o of. W e assume that r consists only of t many e - bubbles , so α = β = γ = 0 in (1 ) . Other c hoices of α, β , γ only cause trivial adaptions. As seen, preparing the binomial coefficien ts o ccuring in (6) costs O ( w 2 log 2 w ). F or fixed s ≤ t consider an initial segmen t of e -bubbles ( e 1 , · · · , e 1 , · · · , e s , · · · e s ) of lengths ε 1 , · · · , ε s resp ectiv ely . If Card ′ ( r , k ) is the num b er of k -elemen t sets represen ted b y this segmen t then, as seen in (6), calculating Card ′ ( r , k ) in v olv es ε s man y m ultiplications of pairs of previously determined at most w -digit n um b ers (and ε s − 1 free additions), whenc e costs O ( ε s w lo g 2 w ). Do ing this f o r 1 ≤ k ≤ K giv es O ( K ε s w lo g 2 w ). Summing up yields O ( K ε 1 w lo g 2 w ) + · · · + O ( K ε t w lo g 2 w ) = O ( K w 2 log 2 w ).  It is easy to see that the describ ed metho d to calculate Card( r , k ) amounts 3 to expanding a pro duct of some ob vious p olynomials asso ciated to the e -bubbles of r . F or r 0 this giv es (2 x + x 2 ) (3 x + 3 x 2 + x 3 ) 2 (4 x + 6 x 2 + 4 x 3 + x 4 ) = 72 x 4 + 288 x 5 + 53 4 x 6 + 594 x 7 + 431 x 8 + 208 x 9 + 65 x 10 + 12 x 11 + x 12 . Here Card( r 0 , 4) = Card( r 0 , k ∗ ) = ε 1 ε 2 ε 3 ε 4 = 72 a nd Card( r 0 , 12) = Card( r 0 , k ∗ ) = 1 matc h (5), and Card( r 0 , 5) = 288 matche s T able 2. 3 The author ado pted this p olynomial p oin t o f view a nd the ma tc hing Mathematica command Expand [ · · · ] to g e t the num b ers Card( r, k ). Wha tev er the underlying metho d of Expa nd [ · · · ], for our small v alues of w that har dwir e d command likely be ats a high level Mathematica implemen tation of the O ( w 2 log 2 w ) method fr om Theo rem 1. 8 3.3 Generating all k -elemen t transv ersals within a ro w As to gener ating all k -elemen t mem b ers of a { 0 , 1 , 2 , e } -v alued ro w r , let us lo ok at r = (2 , e 2 , e 1 , 2 , 1 , e 2 , e 1 , 0 , e 2 ) and k = 6 . Similar to b efore w e apply recursion according to the partition { 5 } = ones( r ) , { 1 , 4 } = tw os ( r ) , { 3 , 7 } (for e 1 ) , { 2 , 6 , 9 } (for e 2 ) . Additionally we employ a last in first out (LIF O) stac k managemen t. Namely , the stack starts out with a single “ro ot ob ject” x = ( { 5 } , { 1 , 4 } , [0 , 2]). This is a cryptic command that in the next step x needs to split into four sons whose first comp onen ts a r e, resp ec- tiv ely , the subsets of { 1 , 4 } with cardina lity b et w een 0 and 2 joined to { 5 } . Eac h son’s second comp onen t is the next blo c k o f the partition (here { 3 , 7 } ). This give s rise to the heigh t four stac k in Fig. 1 . Notice that [1 , 2] rather than [0 , 2] occurs three times be- cause e 1 e 1 (as opp osed to 22) forbids the empt y set. More subtle, in the b ottom ob ject ( { 5 } , { 3 , 7 } , [2 , 2]) the entry [2 , 2] demands that only { 3 , 7 } itself may ev en tually b e added to { 5 } (b ecause ot herwise the final cardinalit y k = 6 cannot b e reac hed). The philosophy of LIF O b eing tha t alw a ys only the top record of the stac k is tr eated, the second stack gives rise to the third stac k in Fig. 1. Its top ob ject giv es rise to the final k -sets { 5 , 1 , 4 , 3 , 7 , 2 } , { 5 , 1 , 4 , 3 , 7 , 6 } , { 5 , 1 , 4 , 3 , 7 , 9 } . After the next t w o new top ob jects ha v e eac h giv en rise to three final k -sets, t he stac k has ( { 5 , 4 } , { 3 , 7 } , [1 , 2]) as its top ob ject. Splitting it yields the fourth stack in Fig. 1. And so on and so fort h. { 5 } , { 1 , 4 } , [0 , 2] → { 5 , 1 , 4 } , { 3 , 7 } , [1 , 2] { 5 , 4 } , { 3 , 7 } , [1 , 2] { 5 , 1 } , { 3 , 7 } , [1 , 2] { 5 } , { 3 , 7 } , [2 , 2] → { 5 , 1 , 4 , 3 , 7 } , { 2 , 6 , 9 } , [1 , 1] { 5 , 1 , 4 , 7 } , { 2 , 6 , 9 } , [2 , 2] { 5 , 1 , 4 , 3 } , { 2 , 6 , 9 } , [2 , 2] { 5 , 4 } , { 3 , 7 } , [1 , 2] { 5 , 1 } , { 3 , 7 } , [1 , 2] { 5 } , { 3 , 7 } , [2 , 2] → · · · → { 5 , 4 , 3 , 7 } , { 2 , 6 , 9 } , [2 , 2] { 5 , 4 , 7 } , { 2 , 6 , 9 } , [3 , 3] { 5 , 4 , 3 } , { 2 , 6 , 9 } , [3 , 3] { 5 , 1 } , { 3 , 7 } , [1 , 2] { 5 } , { 3 , 7 } , [2 , 2] → · · · Fig. 1: Generating all k -elemen t transv ersals with LIF O Theorem 2: Let r b e a { 0 , 1 , 2 , e } -v alued ro w of length w and let k ≤ w b e fixed. Then the sets X ∈ r with | X | = k can b e generated in time O ( w 2 C ar d ( r, k )). Pr o of. W e firs t mak e precise ho w the t o p ob ject ( A, B , [ i, j ]) in the ske tche d LIFO al- gorithm is to b e split. Here A ⊆ W is the accumulated target set, and B ⊆ W is the 9 e -bubble to some e m e m · · · e m (see (1)), and [ i, j ] b y induction is the appropriate subin ter- v al of the in teger in terv al [1 , | B | ]. The “sons” of ( A, B , [ i, j ]) m ust b e of t yp e ( C , D , [ ∗ , ∗ ]) where D is the e -bubble 4 to e m +1 · · · e m +1 , and C can b e any o f the sets A ∪ B ′ where B ′ ranges o v er all subsets of B with cardinality b etw een i and j . What is the in terv al [ ∗ , ∗ ] for a particular fixed C ? Recalling that k is the final cardinality to b e ac hiev ed, and putting δ := k − | C | , a momen t’s thought sho ws that [ ∗ , ∗ ] = [max(1 , δ − ε m +2 − · · · − ε t ) , min( ε m +1 , δ − σ )] where σ is the cardinalit y of { m + 2 , m + 3 , · · · , t } . Running the LIFO alg orithm amoun ts to building a ro oted tree T whose lea ve s corresp ond to the C ard ( r , k ) sets X ∈ r with | X | = k . The unique pat h from a leaf X t o the ro ot hence tra ces t + 2 no des. F or instance: X = { 5 , 1 , 4 , 3 , 7 , 2 } → {{ 5 , 1 , 4 , 3 , 7 } , { 2 , 6 , 9 } , [1 , 1]) → ( { 5 , 1 , 4 } , { 3 , 7 } , [1 , 2]) → ( { 5 } , { 1 , 4 } , [0 , 2]) . These nodes correspo nd to the o b jects that were split to c reate X . The claim fo llo ws from | T | ≤ ( t + 2) C ard ( r, k ) ≤ w Card( r, k ) and the fact that eac h ob ject in T requires w ork O ( w ), as is clear from the ab ov e.  It is easy to see that O ( w 2 ) is the maxim um size of the LIF O stack in Fig.1; this heigh t can b e m uc h smaller than C ar d ( r, k ). 3.4 The sp ecial case k = k min The imp ortan t tr ansversal numb er of a set system H is defined a s k min ( H ) := min {| X | : X ∈ T r ( H ) } F o r instance, finding the minimum n um b er of pieces necessary in a set cov ering problem amoun ts to determine k min = k min ( H ) for some asso ciated h yp ergraph H . Note that k min as wel l as τ min := τ k min can be gleaned at once fr om a represen ta tion of T r ( H ) b y { 0 , 1 , 2 , e } -v a lued rows. F or instance, with resp ect to T able 1 w e get from (4 ) that : k min = min { c min ( r 1 ) , · · · , c min ( r 7 ) } = min { 0 + 4 , 2 + 2 , 2 + 2 , 4 + 1 , 3 + 1 , 3 + 2 , 4 + 1 } = 4 . Using (5) that giv es τ min = τ 4 = Card( r 1 , 4) + Card ( r 2 , 4) + Card ( r 3 , 4) + Card( r 5 , 4) = (2 · 2 · 4 · 3) + (4 · 3) + (2 · 2) + 2 = 66 . 4 F or conv enience we as s ume that m + 1 , m + 2 are s till ≤ t . Otherwise sp e cial cases arise that a re similarly handled. 10 It is eviden t that als o g e ner ating all transv ersals X with | X | = k min can b e done more smo othly than in Section 3 .3. The minim um-cardinalit y transv ersals constitute a sub- family of the p opular [EMG] inc lusion-minimal transv ersals. The e -algorithm seems to b e predestined to handle that subfamily , although it isn’t easy to formally assess its p erformance (w ork in progress). 4 The transv ers al e -algorithm in t heory If a sev en th constraint corresp onding to sa y H 7 = { 3 , 4 , 5 } w ere to b e imp osed in T able 1, this would cause the cancellation o f r 3 to r 7 , and so the work to pro duce these (m ul- tiv alued) row s w ould ha v e been in v ain. F ortunately suc h costly deletion s of r ows c an b e pr evente d b y lo oking a head. Sp ecific ally , an y POE-pro duced row is called fe asible if it con tains at least one mo del X 0 . Because r is the disjoint union of its “candidate sons” r [ e ] , r [0 , e ] , r [0 , 0 , e ] and so forth (Section 2), a t least o ne of them will remain f easible . As opp osed to ot her a pplicatio ns of t he POE, here feasibilit y is easily tested. Namely , r is feasible if and only if (7) ( ∀ 1 ≤ i ≤ h ) H i 6⊆ zeros( r ). Ob viously (7 ) is necess ary , a nd it is sufficien t b ecause then X max = W \ zeros( r ) is a mo del. T he non-feasible sons can hence b e deleted right a w a y . More generally , fix k ∈ [ w ] and call r extr a fe as i b l e if it con tains a mo del o f cardinalit y ≥ k . The abov e rem arks constitute the essence of the pro of of Theorem 3. Theorem 3 : Let H b e a ( w , h )-hypergraph, and let k ∈ [ w ]. Then the transv ersal e -algorithm can b e adapted to calculate: a) The nu mber N of all tra nsv ersals of H in time O ( N h 2 w 2 ); b) The n umber of N of all at least k - elemen t transv ersals of H in time O ( N k h 2 w 2 log 2 w ). Pr o of. As b efore w e think of r 0 = (2 , 2 , · · · , 2), with comp onen ts lab elled by the ele men ts of W = [ w ], as the p o w erset of W . Initially the “working stac k” solely comprises the ro w r 0 with the p oin ter P C ( r 0 ) = 1 (where P C stands for p ending constrain t). Note that r 0 is extra feasible since W ∈ r 0 . Generally , the top ro w r of the w orking stac k is treated as follows . If P C ( r ) = j (for some j ∈ [ h ]) then the h yp eredge H j ∈ H is “imp osed” up on r , whic h m eans that the set U of all X ∈ r with X ∩ H j 6 = ∅ is represen ted as a disjoin t union of s ≤ w many row s r 1 , · · · , r s . According to [W1, Section 5], this is alw ays p ossible. (Section 2 of the presen t article illustrates the most subtle case.) W riting U as r 1 ∪ r 2 ∪ · · · ∪ r s costs O ( sw ) = O ( w 2 ). Because r w as extra feasible b y induction, at least one of it s candidate sons r j will b e as w ell. Since the extra feasibilit y of r j amoun ts to 11 the truth of b oth (7) a nd | X max | ≥ k , it costs O ( shw ) = O ( hw 2 ) to siev e the sons of r , i.e. the extra feasible ro ws amoung r 1 , · · · , r s . Altogether the cost of one imp osition of a constrain t up on a ro w is O ( w 2 ) + O ( hw 2 ) = O ( hw 2 ). The R final rows can b e view ed as the lea v es of a t r ee with ro ot (2 , 2 , · · · 2) that has height h ; eac h imp osition triggers all sons of some node. Therefore the num b er o f imp ositions is at most R h (distinct final rows p ossibly ha ving some of their h forfathers coinciding). It follo ws that producing the R final row s costs O ( Rh · hw 2 ) = O ( N h 2 w 2 ) in view of R ≤ N , b y the disjointne ss of final rows. Coun ting a ll transv ersals within a row costs O ( w ) b y (3), whence doing it for all ro ws costs O ( N w ) = O ( N h 2 w 2 ). This yields claim (a). As to (b), b y Theorem 1 it costs O ( k w 2 log 2 w ) to coun t the | r | − Card ( r , 1) − Card( r, 2) − · · · − Card( r, k − 1) man y transv ersals X ∈ r with | X | ≥ k . Do ing it for all final ro ws costs O ( N k w 2 log 2 w ). Claim (b) thus follo ws from O ( N h 2 w 2 ) + O ( N k w 2 log 2 w ) = O ( N k h 2 w 2 log 2 w ).  As is clear from the pro of, the O ( N k h 2 w 2 log 2 w ) b ound can b e improv ed to O ( Rk h 2 w 2 log 2 w ) where R ≤ N is the men tio ned n um b er of final { 0 , 1 , 2 , e } -v alued row s. Alb eit in practise R is often m uc h smaller than N , t he only ob vious theoretic upp er b ound of R is N . If rather than counting w e must 5 gener ate all relev an t transv ersals one b y one, then w e hav e no c hoice b et we en R and N but are stuc k with the latter. Let s max b e the maxim um num b er of sons of a m ultiv alued ro w that o ccurs in an y fixed run o f the POE (whether e -algorithm or something else). According to [W1, Thm.6] using a LIF O stac k managemen t (akin to Section 3.3) reduces the sp ac e requiremen t of POE- coun ting to O ( hws max ). It is easy to see that for the e -a lgorithm one has s max ≤ min { d, w 2 } where d := max {| H i | : 1 ≤ i ≤ h } , and so O ( hw s max ) = O ( hw 2 ) is indep enden t of N . Notice that X is a transve rsal of H 1 , · · · , H h if and only if its complemen t X c = W \ X is a nonc ove r in the sense that X c 6⊇ H i for all 1 ≤ i ≤ h . Although the e -algor ithm can th us coun t (or generate) nonco ve rs, it pa ys to in tro duce the sym b olism nn · · · n := “at least one 0” and a corresp onding nonc over n -algorithm which pro duces the noncov ers “directly”, not as X c . The noncov er n -a lgorithm in turn generalizes to the Horn n - algorithm of [W1] which counts the mo dels of an y giv en Horn form ula. Because Theorem 3a and Theorem 3b ab o v e corresp ond to not so ob vious special (and dualized) cases of [W1, Thm.2] respective ly [W1, Thm.7], w e deemed it worth wile to offer a fresh pro of. Ev en more so b ecause (7 ) is m uc h smo other than the corresponding feasibilit y t est for general Horn form ulae. Th eorem 4 below transfers f urther results of [W1] ab out fixe d cardinalit y mo dels to our framew ork. Its pro of is omitt ed (b eing along the lines of the pro of ab o v e) but w e men tion that Theorem 1 and The orem 2 are used thro ughout. They 5 In pr actise, genera ting all of them is ma inly necessary for exact optimization, but then one ra ther generates them bunch-wise in multiv alued ro ws. 12 app eared already as statemen ts (16) and (15) in [W1], but their pro ofs w ere p ostp oned 6 to the presen t article. Theorem 4: Let H b e a ( w , h )-hypergraph and let k ∈ [ w ]. T o av oid trivial sp ecial cases we assume that t he nu mber N of v arious mo dels considered b elow , is > 0. Define R ≤ N as the n um b er of final rows deliv ered by the tra nsv ersal e -algorithm when a pplied to H . (a) [W1, Thm.10] The num b er N of tra nsv ersals of H with | X | = k can b e calculated in time O ( R 2 h hw 4 k ). ( a ′ ) [W1, remark to Thm.10] The N transv ersals of H with | X | = k can b e generated in time O ( N 2 h hw 5 ). (b) [W1, Thm.8] Supp ose that h ≤ k ≤ w . Then the n um b er N of tr ansv ersals X of H with | X | = k can b e calculated in time O ( R k h 2 w 3 ). ( b ′ ) [W1, Thm.4] Supp ose that h ≤ k ≤ w . Then the N transv ersals of H with | X | = k can b e generated in time O ( N h 2 w 2 ). (c) [W1, Thm.9] Supp ose the n um b er of k ′ -elemen t transv ersals increases as k ′ ranges from w down to k . Then t he num b er N of H -tra nsv ersals X w ith | X | = k can b e calculated in time O ( N h 2 w 5 ). 5 Conclus ion In [W4], whic h is a somewhat v erb o se preliminary v ersion of the presen t article, a Mathe- matica implemen tation of the e -algorithm is pitted against Mathematica impleme ntations of ( a ) inclusion-exclusion, (b) lexicographic generation, and (c) the “ hardwired” whence adv antaged Mathematica command Satisfiabil ityCount . The latter is based o n binar y decision dia g rams (BDD’s). Broadly sp eaking, the e -algorithm com bines the a dv an tages of inclusion-exclusion and Satisfiabil ityCount without adopting their disadv an tages. Let τ b e the n um b er of all transv ersals. The adv an tage of inclusion-exclusion is that calculating all τ k (1 ≤ k ≤ w ) do esn’t take muc h longer than calculating τ (f o r fixed h time scales ab out prop ortio na l to w ), its disadv antage the ominous factor 2 h . The adv a n tage of Satisfiabi lityCount is its b enign exp onen tial dep endenc e on h . Its disadv an tage is the inabilit y of BDD’s to 6 The O ( K w 2 log 2 w ) b ound in Theorem 1 actually improves upon the O ( K w 3 ) b ound in [W1, (16)]. This en tails that ( a ′ ) in Theorem 4 a bov e could b e slightly improved ac c ordingly; we omitted it in order to minimize confusio n. 13 handle fixed-cardinality constrain ts. Alb eit some of the exp erimen t ial results in [W4] remain in teresting, the author also accepts the fo llowing criticism of o ne Referee: Satisfiabil ityCount is a function to count t he solutions of a satisfiability problem, and transv ersals are only a sp ecial case, so the f unction is “abused” (in particular when lots of artificial constrain ts are added to find t r a nsv ersals of a certain size!) to p erform a task it w as not programmed for. But then again, the principle of exclusion (Section 1) con tinues t o tease Satisfiabi lityCount when the issue is coun ting (let alone generating) the mo dels of an arbitr ary Bo olean f unc- tion in CNF, provid ed it happ ens to ha v e few or no mo dels. This is work in progress, and so ar e other applications of POE. If Mathematica co de algorithms compare fav orably with corresp onding hardwired Mathematica commands, o bviously the former algor it hms are inheren tly sup erior. It has b een suggested (fairly or not) that Mathematica commands aren’t state of the art, and hence the author’s POE-alg orithms should be implemen ted in C + (sa y) a nd compared to existing C + -implemen tations. Being not familiar with C + (and to o lazy to learn), I leav e that worth wile task to others. See also Section 9 in [W1] for furt her analysis of the pros and cons o f POE. Ac kno wledg emen t. I am grateful to Stephan W agner a nd Andrew Odlyzk o for p oin ting out references [IS] resp ectiv ely [O]. References [BEHM] M. Bruglieri, M. Ehrgot t, H.W. Hamac her, F . Maffioli, An annotated bibliography of com binatorial optimization problems with fixed cardinality constrain ts, Disc. Appl. Math. 154 (2 006) 1344- 1357. [EMG] T. Eiter, K . Makino, G. Gottlob, Computationa l asp ects of monotone dualization: A brief surv ey , Discrete Appl. Math. 156 (2008) 2035-2 049. [IS] The In teger-Sequences-W ebpage, http://o eis.org/A005493. [O] A. Odlyzk o, Asymptotic en umeration metho ds, Ha ndb o ok of Com binator ics V o l.2 , 1063-12 29, Elsevier 199 5. 14 [W1] M. Wild, Compactly generating all satisfying truth assignmen ts of a Horn form ula, Journal on Satisfiability , Bo olean Mo deling and Computation 8 (2012) 63-82. [W2] M. Wild, S. Janson, S. W agner, D. Laurie, Coup ons collecting and transv ersals of h yp ergraphs, to app ear in DMTCS. [W3] M. Wild, Computing the o utput distribution and selection probabilities of a stac k filter from the DNF of its p ositiv e Bo olean function, Journal of Math. Imaging and Vision, online, 1 August 2012. [W4] M. Wild, Counting or pro ducing all fixed cardinalit y transv ersals, preliminary v er- sion of the presen t article, a rXiv : 110 6 .0141v1. 15