On the Structure of Linear Programs with Overlapping Cardinality Constraints

On the Stuctue of Linea Pogams with Ovelapping Cadinality Constaints Tobias Fische and Mac E. Pfetsch Depatment of Mathematics, TU Damstadt, Gemany tfische,pfetsch}@mathematik.tu-damstadt.de Januay 25, 2017 Abstact Cadinality constaints enfoce an uppe bound on the numbe of vaiables that can be nonzeo. This aticle investigates linea pogams with cadinality constaints that mutually ovelap, i.e., shae vaiables. We pesent the components of a banch-and-cut solution appoach, including new banching ules that exploit the stuctue of the coesponding conflict hypegaph. We also investigate valid o facet defining cutting planes fo the convex hull of the feasible solution set. Ou appoach can be seen as a continuous analogue of independence system polytopes. We study thee diffeent classes of cutting planes: hypeclique bound cuts, implied bound cuts, and flow cove cuts. In a computational study, we examine the effectiveness of an implementation based on the pesented concepts. 1 Intoduction This aticle deals with Cadinality Constained Linea Pogamming Poblems (CCLPs) of the fom (CCLP) max x c x s.t. Ax b, x C 0 C 1 fo all C C, x 0, whee A R m n, b R m, c R n, fo m, n Z +, and C 2 V is a family of subsets of V := 1,..., n}. By x C we denote the subvecto of x esticted to an index set C V. Moeove, x C 0 := i C : x i 0} denotes the cadinality of x C, i.e., the numbe of its nonzeo enties. The sets C C ae the hypeedges o cicuits of the so-called conflict hypegaph H = (V, C), whose nodes coespond to vaiable indices i V. We assume that H is simple, i.e., C C fo distinct C, C C. CCLPs fom a vey geneal class of optimization poblems and consequently have a lage vaiety of applications, e.g., in cash management (see, e.g., Galati [26]), lot sizing (see, e.g., Gade and Küçükyavuz [25]), compessed sensing (see, e.g., Sun et al. [43]), and potfolio selection (see e.g., Gao and Li [29]). This motivates the main goal of this aticle, which is to povide an efficient solution appoach fo CCLPs based on banch-and-cut. To this end, the pesent aticle investigates the two main components of banching ules and cutting planes. We will show in computational expeiments that this indeed geneates an effective solution method. Not supisingly, (CCLP) can be efomulated as a mixed intege pogam (MIP) if thee ae finite uppe bounds u R n + on the vaiables. Most ideas fo CCLPs ae inspied by thei 1

MIP countepats o have to be compaed to this fomulation altenative. This Mixed Intege Pogam with Packing Constaints (MIPPC) fomulation is: (MIPPC) max x,y c x (1.1) s.t. Ax b, (1.2) y i C 1 C C, (1.3) i C 0 x i u i y i i V, (1.4) y 0, 1} n. (1.5) Hee, the cadinality equiements on the vaiables ae enfoced via packing constaints (1.3), big-m constaints (1.4), and auxiliay binay vaiables (1.5). On the one hand, the MIPPC fomulation has the advantage that it can diectly be attacked using MIP-solve technology. Howeve, it contains twice as many vaiables as (CCLP) and it might be the case that the x-pat of a elaxation solution would be feasible fo (CCLP), but the y-pat still contains factional values, which makes it infeasible fo (MIPPC). The stength of its elaxation will depend on the size of the bounds u. On the othe hand, (CCLP) contains fewe vaiables, but yields a weake elaxation, since the cadinality constaints ae not epesented. Of couse, this can patly be alleviated by using cutting planes, as we will show. Ou computational expeiments will compae the two fomulations and we will discuss thei espective computational benefit. It tuns out that most of the techniques that we conside ae moe effective fo ovelapping cadinality constaints. 1.1 Relation to Independence Systems The just discussed elation to the MIP fomulation is highlighted by the analogue between cadinality constaints and independence systems. Given the conflict hypegaph H = (V, C), a subset I V is independent if I does not contain any cicuit C C. The independence system is given by I, the set of all independent sets. An independent set is maximal o a basis if it is not contained in any othe independent set. We denote the set of all bases by B. Evey cicuit C C and all of its supesets ae dependent. The set C := D V : C D, C C} of all dependent sets foms the so-called dependence system induced by C. Note that C = 2 V \ I. The incidence vectos x 0, 1} V of independent sets ae chaacteized by the cicuit inequalities x C 0 C 1 fo all C C. Thus, the solutions that satisfy cadinality constaints as in (CCLP) can be seen as a continuous analogue of independence systems. Indeed, it is not had to show that the convex hulls of the solution sets ae equal, if all vaiables ae bounded by 1, see Section 3. Simila to the case of independence systems, we note seveal fundamental obsevations about the stuctue of cadinality constaints that we need thoughout the aticle. Hee, fo C C, we call x C 0 C 1 a cicuit constaint and C its ode, while x K 0 k fo K V, k Z + epesents a cadinality constaint. Cicuit constaints of ode two ae complementaity constaints equivalent to x i x j = 0 with C = i, j}. If all cicuit constaints ae complementaity constaints, we speak of H as a conflict gaph instead of a conflict hypegaph. We then denote H by G and its edge set by E. We define H[K] with K V to be the induced hypegaph of H that has node set K and hypeedge set C C : C K}. An induced hypegaph H[K] is called k-hypeclique if ) := C K : C = k} C. Due to the equivalence ( K k x K 0 k x C 0 k C ( K k+1), 2

evey cadinality constaint can be epesented by cicuit constaints; in this case, H[K] is a (k + 1)-hypeclique. Thus, (CCLP) in fact allows to epesent abitay cadinality constaints, which explains its name. Cadinality constaints have a close elationship to special odeed set type k (SOSk) constaints, see Beale and Tomlin [6], fo which the nonzeo components of x K with x K 0 k have to be adjacent w..t. a given odeing. Note that fo the special case of SOS1 constaints, the adjacency condition is void such that SOS1 constaints exactly coespond to cadinality constaints of the fom x K 0 1. 1.2 Liteatue Oveview Thee exist many aticles that deal with the facial stuctue of the independence system polytope P IS := conv(x (I) 0, 1} n : I I}), whee X : 2 V 0, 1} n is the chaacteistic function. This polytope o its (affinely equivalent) complementay set coveing polytope was studied by, e.g., Balas and Ng [4], Sassano [39], and Sánchez-Gacía et al. [38]. In paticula, the woks of Eule et al. [21] and Lauent [33] investigate cutting planes belonging to hypecliques, odd cycles, anticycles, and antiwebs fo independence systems. Moeove, Easton et al. [19] and Maheshway [34] discussed a elationship between hypecliques and facet defining inequalities of the knapsack polytope. In both of these latte woks, the authos algoithmically exploit cetain hypegaph stuctues in a banch-and-cut appoach to solve paticula difficult knapsack instances. On the othe hand, the polyhedal popeties of cadinality constaints have been widely unexploed so fa. Most studies concentate on the special case that thee is only a single cadinality constaint pesent in the optimization poblem. Fo these poblems, Bienstock [8] intoduced classes of mixed-intege ounding inequalities, disjunctive cuts, and citical set inequalities. The latte ones wee genealized by de Faias and Kozyeff [15]. In futhe woks, de Faias et al. investigate disjunctive cuts [14] and flow cove cuts [17]. One algoithmic appoach fo solving (CCLP) is banch-and-cut: The efomulation of (CCLP) as (MIPPC) allows the use of state-of-the-at MIP-solves. Howeve, as mentioned above, big-m based modeling leads to a lage poblem size and can esult in numeical toubles if the uppe bounds of the vaiables ae lage, see, e.g., [23] fo a detailed discussion. To wok aound with this, Bienstock [8] was the fist to intoduce a specialized banching scheme, which allows a diect enfocement of the cadinality constaints without the use of auxiliay binay vaiables and big-m constaints. This method was taken up and genealized late by de Faias et al. [14]. Oiginally, the idea of specialized banching was poposed by Beale and Tomlin [6] in the ealy 1970s in the context of SOS1 constaints. The appoach diectly enfoces SOS1 constaints by banching on sets of vaiables. In [23], we focused on the case in which the SOS1 constaints ovelap and studied how to algoithmically exploit the coesponding conflict gaph, fo instance, to get impoved banching stategies. Computational esults confim the benefit of specialized banching w..t. banching on an individual binay vaiable of (MIPPC). Moeove, de Faias et al. [16] deived classes of flow cove cuts fo SOS1 constained poblems and used them in a banch-and-cut scheme; hee the constaint matix has only nonnegative coefficients. The case that also negative coefficients ae pesent is consideed in Section 8.3. Next to banch-and-cut, thee ae nonlinea pogamming based methods known fo finding local optima to optimization poblems that ae constained by a single cadinality constaint. Budakov et al. [10] intoduced a efomulation as a complementaity constained mixed-intege optimization poblem fo which the elaxation of the integality conditions peseves optimal solution popeties. This esults in solving a mathematical pogam with equilibium constaints (MPEC) fo which thee aleady exist a vaiety of solution appoaches, see e.g., Hoheisel et al. [31]. 3

1.3 Contibution of this Aticle In Section 2, we discuss a disjunctive pogamming efomulation of (CCLP). This efomulation esults fom a epesentation of the cicuit constaints via independent sets in the conflict hypegaph. The polyhedal popeties of the feasible solution set ae investigated in Section 3. We will see in Section 4 that CCLPs ae N P-had to solve, even fo simple hypegaph classes such as single paths o matchings. Howeve, we show in Section 5 that fo paticula classes, they can be solved in polynomial time. In Section 6, we investigate banching ules fo (CCLP). These ules constitute a genealization of a banching scheme that we aleady tested computationally in [23] fo the special case of complementaity constaints. To extend ou banch-and-bound algoithm to banch-and-cut, we pesent in Section 7 a pocedue to deive valid o facet defining inequalities. In Section 8, this pocedue is illustated fo thee classes of cutting planes: hypeclique bound cuts, implied bound cuts, and flow cove cuts. The latte ones genealize an ealie esult of de Faias et al. [16] that elies on the assumption that the constaint matix has only nonnegative enties. Ou poof coves the case of an abitay matix. In a computational study, we examine the diffeent components of ou implementation using SCIP [41, 1]. We demonstate that fo instances with a specific stuctue, ou algoithm outpefoms CPLEX 12.6.1. 2 Disjunctive Pogamming Fomulation Cadinality constaints have a close elationship to the theoy of disjunctive pogamming, intoduced by Balas [3]. This elationship is discussed in the following. 2.1 Linea Disjunctive Pogamming Linea disjunctive pogamming is concened with a finite union of linea inequality systems (A i x b i ), (2.1) i C whee A i R m i n and b i R m i fo i C. If the polyheda x R n : A i x b i } do not all have the same ecession cone, then (2.1) is not MIP epesentable, see Jeoslow [32]. Howeve, using slack vaiables s i fo i C, (2.1) can be efomulated with the help of a cicuit constaint as A i x s i 1 b i, s i 0 i C, s 0 C 1. Hee, 1 denotes the vecto of all ones of appopiate dimension. Thus, evey linea disjunctive pogam can be tansfomed into a CCLP. Note that this efomulation can easily be genealized to nonlinea disjunctive pogams. Convesely, a natual question is whethe the feasible solution set of (CCLP) can be descibed in the fom of (2.1). This is discussed in the next section. 2.2 Modeling with Disjunctive Pogamming In this section, we deive a disjunctive pogamming efomulation of (CCLP). This efomulation esults fom the epesentation of cicuit constaints via bases in the conflict hypegaph. Let I denote the set of independent sets of H = (V, C) and B its bases, i.e., maximal independent sets. To make the tansfe fom the epesentation of (CCLP) with cicuits to an equivalent epesentation with bases, we intoduce the following notation. 4

Notation 2.1. Define X := x R n : x C 0 C 1 fo all C C} as the set of those x R n that satisfy all the cicuit constaints of (CCLP) and Q := x X : Ax b, x 0} as the feasible solution set of (CCLP). Moeove, fo a subset I V, define the sets X(I) := x R n : x i = 0 fo all i I} and Q(I) := x X(I) : Ax b, x 0}. Based on this notation, we pesent fou staightfowad obsevations, which we will use thoughout this section and late in the aticle. Obsevation 2.2. Let H = (V, C) be a hypegaph, I its independent sets, C := D V : C D, C C} its dependent sets, and X, Q, X(I), Q(I) as above. (i) Fo given I V and x R n, x X(I) if and only if supp(x) I. (ii) I V is an independent set, i.e., I I, if and only if X(I) X. (iii) C V is a dependent set, i.e., C C, if and only if x C 0 < C fo all x X. (iv) The sets X(I), I I, ae affine subsets of X and the sets Q(I), I I, ae polyhedal subsets of Q. We have the following esult: Lemma 2.3. The set X can be witten as X = B B X(B). (2.2) This epesentation is minimal in the sense that thee do not exist affine spaces A 1,..., A k, with k < B and X = A 1... A k. Poof. Immediately fom the definition, we get that x X if and only if x satisfies C C j C (x j = 0). Using the basic ules of Boolean algeba, this expession can be efomulated equivalently as I F j V \I (x j = 0) x I F X(I), fo some F 2 V. In summay, this shows that X = I F X(I). Futhemoe, we get that F I due to Obsevation 2.2 (ii). Consequently, X = X(I) X(I) = X(B), I F I I whee the last equation esults fom the fact that X( ) is inclusion peseving. This shows the -inclusion of (2.2). B B 5

Fo the convese inclusion, let x B B X(B). Then, fo some B B, we get that x X(B ), and futhe that x X by Obsevation 2.2 (ii). This completes the poof of the fist assetion. Fo the poof of the second assetion, we may assume without loss of geneality that B > 2, since in the othe case X is an affine space (possibly empty). Assume that thee exist affine spaces A 1,..., A k, with k < B and X = A 1... A k. Let X : 2 V 0, 1} n be the chaacteistic function. Due to the fact that X (B) X = A 1... A k, fo evey B B, thee must exist some j 1,..., k} such that X (B 1 ), X (B 2 ) A j fo at least two distinct sets B 1, B 2 B. We can conclude that as an affine space the set A j also contains p λ = (1 λ)x (B 1 ) + λx (B 2 ) fo evey λ R. Theefoe, p λ X and supp(p λ ) must be an independent set. Due to p λ 0 = B 1 B 2 fo λ R \ 0, 1}, we get a contadiction to the maximality of B 1 and B 2. As an immediate consequence, Lemma 2.3 establishes a fomulation of Q as a union of polyheda: Coollay 2.4. The feasible solution set Q of (CCLP) can be coveed as Q = Q(B). (2.3) B B Coollay 2.5. Thee exists an optimal solution of (CCLP) that is a basic feasible solution of Q(B) fo some B B. We obtain the following disjunctive pogamming efomulation of (CCLP): (DP) max c x : x R n B B } (x Q(B)). By Obsevation 2.2 (i), the convex hull of the feasible solution set of (CCLP) can be witten as the following continuous analogue to the independence system polytope: conv(q) = conv(x R n : (x Q(B))}) B B = conv(x R n : Ax b, x 0, supp(x) B, B B}) = conv(x R n : Ax b, x 0, supp(x) = I, I I}). This suggests an intuitive exact appoach to detemine an optimal solution of (CCLP). Afte having identified the basis set B, it is enough to solve a numbe of B linea pogamming subpoblems, each belonging to an B B. Unfotunately, B may be exponential in n: Fo instance, a pefect matching as conflict gaph, in which evey node is contained in exactly one edge, has exactly 2 n/2 bases. To deal with this exponential behavio, banch-and-bound may be the basis fo an adequate solution stategy. We discuss seveal banching ules in Section 6. Howeve, if B is small o we ae only inteested in an estimation of the optimal solution, explicit enumeation may be an option, see Section 5. 3 Polyhedal Popeties of Sets Associated with Independence Systems In this section, we study the polyhedal popeties of conv(q) and its elation to the independence system polytope P IS. We will late use this to deive valid inequalities fo a banch-and-cut appoach. 6

3.1 Continuous Independence System Set We fist investigate, in which cases conv(q) is a polyhedon. To give an example whee this is not the case, we conside the system x 1 0, 0 x 2 1, x 1 x 2 = 0. The set of feasible solutions is Q = Q 1 Q 2, whee Q 1 := x R 2 Q 2 := x R 2 : x 1 = 0, 0 x 2 1}. Then : x 1 0, x 2 = 0} and conv(q) = x R 2 : x 1 0, 0 x 2 < 1} (0, 1) } is not a polyhedon, because it is not closed. Figue 3.1 shows a visualization. x 2 1 conv(q) Q 2 Q 1 x 1 Figue 3.1: Convex hull of the union of the two polyheda Q 1 := x R 2 : x 1 0, x 2 = 0} and Q 2 := x R 2 : x 1 = 0, 0 x 2 1}. Nevetheless, the closue conv(q) of conv(q) is always a polyhedon as shown by Confoti et al. [12]. To state this esult, we denote Q 0 (B) as the vetices and Q (B) as the exteme ays of Q(B) fo B B. Lemma 3.1 (Confoti et al. [12]). The following complete desciption of conv(q) as polyhedon holds: ( ) ( conv(q) = conv Q 0 ) (B) + cone Q (B). B B Moeove, if the sets Q (B) fo B B ae all identical, then conv(q) = conv(q). In ode to continue the polyhedal study of conv(q), we show the equivalence between (CCLP) and the MIP fomulation (MIPPC). The latte is only well-defined if thee exist finite uppe bounds u R n + of x. It is woth to mention that conv(q) = conv(q) in this case, since each Q(B), B B, must then be a polytope with an empty exteme ay set Q (B). We define S as the feasible solution set of (MIPPC), i.e., the set all those (x, y) R n 0, 1} n with B B Ax b, (3.1) 0 x i u i y i i V, (3.2) y i C 1 C C. (3.3) i C Moeove, let S(B) with B B define the subset of S aising fom vaiable fixings y i = 0 and x i = 0 fo evey i B. By Π x (S), we denote the pojection of S on the x-vaiables. 7

Lemma 3.2. Π x (S) = Q. Poof. Analogously to the poof of Lemma 2.3, one can obseve that S = B B S(B). Thus, it is enough to show that Π x (S(B)) = Q(B) fo evey B B: Fo a given B B let (x, y) S(B ). Then, by (3.2) and (3.3), x must satisfy all the cicuit constaints x C 0 C 1, C C, and we get that Π x (S(B )) Q(B ). Fo the convese inclusion, let x Q(B ). We define y := X (B ) as the chaacteistic vecto of B. Then (x, y) S(B ) and the assetion follows. We have seen that fo the special case that conv(q) is bounded, it can be completely chaacteized as a polyhedon. Lemma 3.2 suggests to deive valid o facet defining inequalities fo conv(q) fom the pojection of valid inequalities fo conv(s). We will demonstate this on the example of independence system inequalities in the next section. 3.2 Relation to Independence System Polytopes In this section, we discuss the polyhedal elationship between (CCLP) and the independence system poblem (ISP). The convex hull of the feasible solution set of (ISP) is given by the independence system polytope P IS := conv(x (I) 0, 1} n : I I}), whee the chaacteistic vectos X (I) of the independent sets I I epesent the vetices. The polytope P IS is down monotone, i.e., z P IS implies z P IS fo all 0 z z. Thee exists an equivalent fomulation of P IS using cicuit constaints and continuous vaiables. In the following, let C denote the minimal dependent sets (cicuits) of (ISP) and X be defined accoding to Notation 2.1. Lemma 3.3. P IS = conv(x [0, 1]). Poof. Let x P IS. Then x can be witten as a convex combination of the vetices X (I), I I, of P IS. Since by Obsevation 2.2 (i), X (I) X [0, 1] fo evey I I, it follows that x conv(x [0, 1]). On the othe hand, if x conv(x [0, 1]), then I = supp(x) is an independent set by Obsevation 2.2 (ii). Thus, X (I) P IS. Since P IS is down monotone, x P IS follows. As an immediate consequence, we obtain the following coollaies: Coollay 3.4. If Ax b equals x 1, then conv(q) = P IS. Coollay 3.5. Let 0 < u i < fo i V. The inequality i V g i y i β is valid (facet defining) fo P IS if and only if i V g i u i x i β is valid (facet defining) fo conv(x [0, u]). Poof. The poof follows fom Lemma 3.3 using the fact that (x 1,..., x n ) (u 1 x 1,..., u n x n ) defines an affine tansfomation mapping conv(x [0, 1]) to conv(x [0, u]). We call inequalities that ae deived fom this one-to-one coespondence independence system inequalities. One impotant class of independence system inequalities ae (hypeclique) bound inequalities: Coollay 3.6. Let H[K] with K V be an induced (k + 1)-hypeclique of H = (V, C), whee 2 k < K. If 0 < u i < fo i K, then the (hypeclique) bound inequality i K x i u i k is valid fo conv(x [0, u]). Moeove, if H is (k + 1)-unifom, i.e., C = k + 1 fo all C C, and H[K] is maximal, i.e., H[K v}] is not a (k + 1)-hypeclique fo all v V \ K, then k is facet defining fo conv(x [0, u]). i K x i u i Poof. The validity of the bound inequality esults fom Coollay 3.5. Fo the poof of the facet defining popety, we efe to Eule et al. [21]. Futhe independence system inequalities include epesentations of odd cycles, anticycles, and antiwebs (see Eule et al. [21] and Lauent [33]). 8

4 Complexity In this section, we investigate complexity esults fo the decision poblem of detemining whethe (CCLP) has a feasible solution. We focus on the special case that the conflict hypegaph is a standad gaph, i.e., all the cicuit constaints ae complementaity constaints. Most of the pesently-known complexity esults concening (CCLP) deal with the Linea Complementaity Poblem (LCP). This fundamental poblem in mathematical optimization (see, e.g., Muty [35]) is the feasibility poblem of finding a vecto (x, w) R 2n, such that (LCP) w = q + Mx, x, w 0, x i w i = 0 i 1,..., n}, whee q R n and M R n n. Chung [11] showed that the stongly N P-complete poblem of detemining whethe a linea equation system has a binay solution educes to (LCP). His poof uses the agument that evey binay constaint y 0, 1} can be tansfomed into a complementaity constaint of the fom y (1 y) = 0. In paticula, this shows that the feasibility poblem of (CCLP) is stongly N P-complete fo matchings as conflict gaphs, i.e., gaphs fo which evey node is contained in at most one edge. In the following, we deive complexity esults fo futhe gaph classes. Poposition 4.1. Detecting whethe (CCLP) is feasible is stongly N P-complete, even fo instances fo which all cicuit constaints ae complementaity constaints and the conflict gaph defines a cycle, o path (tee). Poof. We educe the LCP as follows: Fo each vaiable pai (x 2k 1, x 2k ) with k 1,..., n 2 } and (w 2k, w 2k+1 ) with k 1,..., n 2 1}) of (LCP), we intoduce additional vaiables x k and w k, espectively. We link the new vaiables with the oiginal ones by adding complementaity constaints x k x 2k 1 = 0, x k x 2k = 0, and w k w 2k = 0, w k w 2k+1 = 0, which yields, togethe with the complementaity constaints x i w i = 0, i 1,..., n}, of (LCP), a path as conflict gaph. Using the linea constaints x k = 0 and w k = 0, we fix the new vaiables to zeo, and the eduction of LCP to the feasibility poblem of (CCLP) with paths and tees as conflict gaphs is complete. In the same way, a cycle can be established by additionally linking the pai (w 1, x n ) if n is odd o the pai (w 1, w n ) if n is even with a new vaiable. In the next section, we will pove that (CCLP) can be solved in polynomial time fo cotiangulated conflict hypegaphs. 5 Polynomially Solvable Special Cases The most fequently used type of elaxation to estimate the optimal solution of a subpoblem in a banch-and-bound algoithm is the LP-elaxation. Fo (CCLP) this means to elax all the cicuit constaints. Howeve, we will see in this section that sometimes it is enough to elax only a couple of them in ode to get a polynomially solvable poblem. Coollay 5.1. Thee exists an algoithm that solves (CCLP) in polynomial time if thee exists an algoithm that enumeates all bases in polynomial time in the size of (CCLP). Poof. The assetion follows fom Coollay 2.4 and the fact that evey linea poblem can be solved in polynomial time in the numbe of vaiables and constaints (see Götschel et al. [30]). 9

To the best of ou knowledge, thee is no algoithm known that enumeates all bases B B of a geneal hypegaph in polynomial time in B. Howeve, thee exists one fo hypegaphs with bounded edge-intesections, see Boos et al. [9]. Thus, if the sizes of the edge-intesections ae bounded by a constant and in addition B is polynomial in the size of (CCLP), then (CCLP) can be solved in polynomial time. In Section 6.2, we will povide a second poof of this claim. The emainde of this section investigates a hypegaph class fo which the condition of Coollay 5.1 is always satisfied. We stat by intoducing some notation, which is mainly taken fom Bege [7] and Emtande [20]: Fo a given hypegaph H = (V, C) and node v V, we define Γ(v) := u V : u, v C fo u v and some C C} as the neighbohood of v. Moeove, we define Γ[v] := Γ(v) v} as its closed neighbohood. A hypegaph H = (V, C) is called d-unifom fo some d N, if C = d fo evey C C. A d-unifom hypegaph H is tiangulated if fo evey nonempty set U V eithe the induced hypegaph H[U] is the empty gaph without hypeedges o thee exists some node v U such that the induced hypegaph H[Γ[v] U] is a d-hypeclique. Tiangulated gaphs, as a special case, ae also called chodal. The d-complement of a d-unifom hypegaph H = (V, C) is defined as H c := (V, C c ), whee C c := C 2 V \ C : C = d}. We call a d-unifom hypegaph co-tiangulated if its d-complement is tiangulated. The next esult by Emtande [20] shows that thee exists a polynomial-time algoithm fo computing all maximal hypecliques in a d-unifom tiangulated hypegaph. Lemma 5.2 (Emtande [20]). A d-unifom hypegaph H = (V, C) is tiangulated if and only if it has a pefect elimination ode, i.e., an odeing v 1,..., v n of its nodes such that fo evey i 1,..., n} eithe the node set K i := Γ[v i ] v i,..., v n } induces a d-hypeclique H[K i ] o v i has no adjacent hypeedges in H[v i,..., v n }]. By definition evey induced subhypegaph of a tiangulated hypegaph is again tiangulated. Theefoe, a pefect elimination odeing in a tiangulated hypegaph can be computed iteatively in polynomial time: At each iteation i 1,..., n}, seach fo a node v i such that K i = Γ[v i ] v i,..., v n } induces a d-hypeclique o the empty hypegaph without hypeedges. If H[K i ] is not empty, then it is eithe a maximal hypeclique, o K i K j fo some j 1,..., i 1}, povided that i > 1. Theefoe, the numbe of maximal hypecliques in a d-unifom tiangulated hypegaph is bounded by n. They can be enumeated inductively in this way. Since evey maximal hypeclique of a d-unifom hypegaph is a basis (maximal independent set) in its d-complement hypegaph, we get the subsequent coollay: Coollay 5.3. Let a CCLP be given with an associated d-unifom hypegaph H. If H is co-tiangulated, then it has at most n bases and CCLP can be solved in polynomial time. Deaing et al. [18] showed that a maximal tiangulated subgaph of a given gaph G can be found in polynomial time. Futhemoe, Dahlhaus [13] showed that a minimal tiangulated gaph, whose subgaph is G, can be found in polynomial time. Thus, if all cicuit constaints of (CCLP) ae complementaity constaints, then a valid uppe bound on the objective value of (CCLP) can be computed in polynomial time by eplacing its conflict gaph by a maximal cotiangulated subgaph. Likewise, a lowe bound can be obtained by eplacing the conflict gaph by a minimal co-tiangulated supegaph. It emains an open question whethe the esults of Deaing and Dahlhaus can be genealized to tiangulated hypegaphs. 6 Banching Rules The objective of banching ules is to split the solution set of a poblem into smalle pats. Standad MIP-based banching ules ealize this by patitioning the domain of an individual 10

intege vaiable. Howeve, these ules ae not applicable to (CCLP) whose vaiables ae continuous. In this section, we pesent constaint-based banching ules that decompose the solution set by eithe adding new cicuit o cadinality constaints to each banch o modifying the existing ones. We aleady tested these banching ules computationally in [23] fo the special case that all cicuit constaints ae complementaity constaints. The pupose of this section is to genealize the esults in [23] to cicuit constaints and to answe open theoetical questions. Moeove, we will investigate an equivalent chaacteization of the constaint-based banching ules via independent sets. Thoughout this section, we conside H = (V, C) as the local conflict hypegaph, which is assumed to be simple and induced by the vaiables not fixed to zeo. This means, in paticula, that C > 1 fo all C C. 6.1 Stuctue Based Banching We pesent two specialized banching ules, that enfoce the cicuit constaints by taking specific stuctues of the conflict hypegaph into account. The fist one aises fom the hypeclique stuctue and oiginates fom wok of Bienstock [8]. We give a shot eview of a genealized vesion of de Faias et al. [14]. Aftewads, we descibe a second banching ule consideing the neighbohood of a vaiable. In the following let x K 0 k fo (K, k) K be an initial set of cadinality constaints given fo (CCLP), whee the induced hypecliques H[K] = (K, ( K k+1) ) fo (K, k) K togethe cove the hypeedges of H; a paticula example would be K = (C, C 1) : C C}. Hypeclique banching The idea of hypeclique banching, which was intoduced by de Faias et al. [14] unde the tem specialized banching, is based on constaint splitting. Suppose we have given a cadinality constaint x K 0 k, (K, k) K, which is violated by the cuent LP solution x. We choose a nonempty set S K and a numbe 0 s < mink, S }. Then a valid coveing of the solution set is deived fom the disjunction x S 0 s x K\S 0 k s 1, (6.1) see [14] o Section 6.3 fo the poof of coectness. The disjunction is enfoced by ceating two banches, whee we identify K (S, s)} with the cadinality constaint set of the left banch and K (K \ S, k s 1)} with the cadinality constaint set of the ight one. If s = 0 and S = 1, then the banching disjunction (6.1) is the one that was poposed by Bienstock [8]. Bienstock additionally suggests in [8] to use bound inequalities as a linea epesentation of the cadinality constaints in the LP-elaxation. This means to add j S s and k s 1 to the espective banches if finite uppe bounds u K of x K ae known. j K\S x j It emains to state how to choose S and s. De Faias et al. suggest in [14] to define S as some (not futhe specified) set with S = K /2 and s = k/2. Hee, we adopt a diffeent appoach that was oiginally poposed in the context of SOS1 constaints by Beale and Tomlin [6]: Fo simplicity, we assume that K = 1,..., K }. Futhe, we suppose that the vaiables x j, j K, ae soted inceasingly accoding to pedefined weights w R K with 0 < w 1 < < w K. If no weights ae specified befoehand, one usually takes w j = j fo j K. Ou goal is to split K into S = 1,..., } and K \ S = + 1,..., K } fo some index with 1 K 1. We fist calculate a weighted mean w as w := j K w j x j. j K x j Notice that the denominato is nonzeo by assumption. Then we choose such that w w < w +1. x j 11

Since now S is detemined as S = 1,..., }, it emains to select s: We suppose that all uppe bounds u K ae finite. If the bound inequalities x j j S s and x j j K\S k s 1 ae violated, then the violation of these inequalities is δ 1 := j S x j / s and δ 2 := j K\S x j / k + s + 1, espectively. To obtain balance, we choose s such that δ 1 and δ 2 ae appoximately equal: 1 ( x j s = max 0, 2 u j S j j K\S x ) } j + k 1. If the uppe bounds ae not finite, then we apply neighbohood banching as an altenative, see the next section. Neighbohood banching This banching ule aises fom the neighbohood of a given vaiable x i, i V, in H. The idea is to banch on the disjunction x i = 0 x K\i} 0 k 1 (K, k) K : i K, (6.2) which is implied by x i = 0 x i 0. Note, howeve, that x i = 0 is feasible fo both sides of (6.2). We implemented neighbohood banching in such a way that it does not explicitly add new cadinality constaints to each banch, i.e., the set K (also C) is not enlaged duing the banching pocess. Instead, we establish a fomulation with auxiliay binay vaiables y 0, 1} V. We use the y-vaiables to stoe the banching decisions belonging to the ight hand side of (6.2), but they do not appea in the LP-elaxation. If y i = 1 fo some i V, then this designates that x i can be teated as nonzeo. The local cadinality constaints of each node can be epoduced fom the initial cadinality constaints x K 0 k, (k, K) K, using the model x W (K,y) 0 k j K y j, W (K, y) = K \ j K : y j = 1}, (6.3) fo (K, k) K. Note that (6.3) efes to an initial cadinality constaint if no vaiable y j with j K is fixed to 1. Now (6.2) may be eplaced by the equivalent and moe pactical disjunction x i = 0 y i = 1, i.e., we fix x i = 0 on the left banch and y i = 1 on the ight one. To impove the LP-elaxations of each banch, one can add bound inequalities j W (K,y) fo (K, k) K if finite uppe bounds u W (K,y) of x W (K,y) ae known. x j k j K y j Example 6.1. Fo an illustation of neighbohood banching, conside the conflict hypegaph in Figue 6.1 whose hypeedges epesent the disjunctions x 1 = 0 x 5 = 0, x 1 = 0 x 2 = 0 x 3 = 0 x 4 = 0, x 2 = 0 x 6 = 0, x 3 = 0 x 4 = 0 x 5 = 0 x 6 = 0. We apply neighbohood banching to the vaiable x 1, which leads to the following banching decision (accoding to 6.2): x 1 = 0 x 5 = 0, x 2 = 0 x 3 = 0 x 4 = 0. (6.4) An altenative banching disjunction will be demonstated late in Example 6.4. Obseve that neighbohood banching is equivalent to standad 0/1-banching on the binay vaiables of (MIPPC): Banching on y i, i V, esults in y i = 0 in one banch and an update of the constaints j K y j k fo (K, k) K with i K to j K\i} y j k 1 in the 12

1 2 3 4 5 6 Figue 6.1: Example of a conflict hypegaph othe. Due to the big-m constaints x j y j, j V, this indiectly means to banch on the disjunction (6.2). Using neighbohood banching instead of hypeclique banching has seveal advantages and disadvantages: On the positive side, neighbohood banching does not explicitly add local cadinality constaints to the banching nodes, and thus keeps the memoy equiements and the time needed fo node-switching elatively low. Moeove, if the cadinality constaints ovelap, then neighbohood banching may take multiple cadinality constaints in the enfocement of the disjunction (6.2) into account; this is not the case fo hypeclique banching, which always consides only a single cadinality constaint in the banching disjunction (6.1). On the othe hand, neighbohood banching can esult in a vey unbalanced banching tee, since it does not make the attempt to decompose the solution set evenly. Unbalanced banching can lead to elatively lage enumeation; this was confimed computationally by Appleget and Wood [2] who compaed specific constaint-based banching ules like SOS banching [6] and Ryan-Foste banching [37] to the standad vaiable-based banching. Implementation of the two banching ules We implemented a selection ule that decides whethe to use neighbohood o hypeclique banching depending on the situation. If all cadinality constaints x K 0 k, (K, k) K, ae satisfied by the cuent LP solution x, then x is feasible fo (CCLP). Othewise, the selection ule chooses a pai (K, k ) K whose cadinality constaint is violated by x and has lagest value j K x j. We fist ty to apply hypeclique banching if the depth of the cuent node is not lage than 20 and if all uppe bounds u K of x K ae finite. As explained above, we choose a set S and a numbe s fom which we deive the disjunction x S 0 s x K \S 0 k s 1, see (6.1). We equie that S and s should satisfy the conditions (i) S > 1 and K \ S > 1, (ii) δ 1 := j S x j / s > ε and δ 2 := j K \S x j / k + s + 1 > ε fo some ε 0. If condition (i) would not be satisfied then we would fix just one vaiable to zeo on at least one side of the banch-and-bound tee and neighbohood banching would be a bette choice if the hypecliques ovelap. The second condition (ii) guaantees that the bound inequalities j S x j/ s and j K \S x j/ k s 1 cut off x. Consequently, the selection ule does not always select the same cadinality constaint fo banching if bound inequalities ae sepaated with node depth fequency 1. Peliminay tests showed that a (emakably high) value ε = 2.0 is an adequate choice fo the instances we consideed. If the conditions fo hypeclique banching ae not satisfied, then we apply neighbohood banching as follows: Choose fom the set K an index i with lagest value x i. Then banching is pefomed on the disjunction x i = 0 y i = 1 as explained above. We call this combination of hypeclique and neighbohood banching balanced banching. 13

6.2 Independent Set Banching In this section, we intoduce a banch-and-bound method that we call independent set banching. Conside the feasible solution set Q of the oiginal poblem, which can be descibed via the union B B Q(B) ove the basis set B (see Coollay 2.4). The key idea of independent set banching is to ecusively cove Q by subsets, which themselves can be descibed via unions ove bases. This is done by ecusive patitioning of B. The banch-and-bound tee is constucted in the following way: The oot node of the tee coesponds to the feasible solution set Q of the oiginal poblem. Fo banching on the oot node, we select a patition B 1 B 2 of B (with B 1, B 2 ). Then we ceate two subpoblems with feasible solution set Q, 1, 2}, defined simila to Notation 2.1: Q := x X : Ax b, x 0}, X := X(B). B B Recusive application of the poposed banching stategy defines a seach tee that enumeates all possible solutions to (CCLP) on its leaf nodes: Lemma 6.2. If B = B 1 B 2, then X = X 1 X 2 and Q = Q 1 Q 2. Futhemoe, if B 1 B and B 2 B, then X 1 X and X 2 X. Poof. The statements follow diectly fom Lemma 2.3 and Coollay 2.4. Thee exists an uppe bound on the maximum numbe of banching nodes that the algoithm geneates: Lemma 6.3. Let B =. The tee of subsets geneated by ecusive patitioning of B has 2 B 1 nodes. Poof. We use induction on B : The assetion is tivial fo B = 1. Next, assume that B is patitioned into B 1 B 2. Let t, t 1, and t 2 be the numbe of nodes in the subtee associated to the sets B, B 1, and B 2, espectively. Due to the induction hypothesis, we get that t 1 = 2 B 1 1 and t 2 = 2 B 2 1. Consequently, t = t 1 + t 2 + 1 = (2 B 1 1) + (2 B 2 1) + 1 = 2 B 1, whee the last equality follows fom B = B 1 + B 2. Example 6.4. Fo a demonstation of independent set banching, we etun to Example 6.1 and the conflict hypegaph visualized in Figue 6.1. The bases ae given by B 1 = 3, 5, 6}, B 3 = 2, 3, 4, 5}, B 5 = 1, 2, 4}, B 2 = 4, 5, 6}, B 4 = 1, 2, 3}, B 6 = 1, 3, 4, 6}. We apply independent set banching by using the patition of B = B 1,..., B 6 } into B 1 = B 1, B 2, B 3 } and B 2 = B 4, B 5, B 6 }. We obtain that x 1 = 0, x 3 = 0 x 4 = 0 x 6 = 0, x 5 = 0, x 2 = 0 x 3 = 0 x 4 = 0 (6.5) is a feasible disjunction, since each of the set 1} and 3, 4, 6} is not completely contained in one of the bases in B 1 and each of the set 5} and 2, 3, 4} is not completely contained in one of the bases in B 2. We pesent an efficient way to compute the disjunction (6.5) in Section 6.3. Banching on the disjunction (6.5) esults in a stonge (i.e., moe tending to disjoint) coveage of the solution set than banching on the neighbohood of the vaiable x 1 : Fo neighbohood banching, the condition x 3 = 0 x 4 = 0 x 6 = 0 is not added to the left hand side of the disjunction (6.4) in Example 6.1. 14

Fom Lemma 6.3, we immediately get a second poof of Coollay 5.1, which states that (CCLP) can be solved in polynomial time if thee exists an algoithm that enumeates the elements of B in polynomial time in the poblem size, as it is the case fo CCLPs with cotiangulated conflict hypegaphs (see Coollay 5.3). Fo geneal conflict hypegaphs, the independent set banching ule is usually not pactical, but needed to deive the banching ule of the next section. 6.3 Dependent Set Banching Let C denote the cicuit set of the cuent node and B the coesponding basis set. Dependent set banching is based on selecting two sets C 1, C 2 2 V and to identify C 1 with the cicuit set of the left banch and C 2 with the cicuit set of the ight banch. The disjunction that we enfoce has the fom x C 0 C 1 C C 1 \ C x C 0 C 1 C C 2 \ C. (6.6) We will investigate chaacteizations of C 1 and C 2 which esult in a feasible banching ule. Moeove, we will answe the question, in which case the dependence systems induced by C 1 and C 2 ae inclusion-wise maximal, i.e., enlaging one of the sets C 1 o C 2 by a cicuit esults in an infeasible disjunction (6.6). Note, howeve, that even if the dependence systems ae inclusionwise maximal, the feasible solution sets of the esulting banching nodes ae not necessaily disjoint, since 0 is feasible fo both if it is feasible fo Ax b. In geneal, a disjoint decomposition of the feasible solution set is not possible by any banching ule. Thoughout this section, we always associate B with the basis set of the hypegaph H := (V, C ), fo 1, 2}. Futhemoe, we denote C := D V : C D, C C } to be the dependence system induced by C and B := I V : I B, B B } to be the independence system induced by B. One chaacteization of C 1 and C 2 that esults in a feasible banching ule is the following: Lemma 6.5. Let two sets C 1, C 2 2 V be given such that the coesponding basis sets satisfy B = B 1 B 2. Then (6.6) is feasible fo (CCLP). If additionally B 1, B 2 B, then C 1, C 2 C. Poof. Obseve fom Lemma 2.3 that X := x R n : x C 0 C 1 fo evey C C } can be witten as X = B B X(B) fo 1, 2}. By Lemma 6.2, it holds that X = X 1 X 2. Thus, the disjunction (6.6) is valid fo X. If additionally B 1, B 2 B, then Lemma 6.2 states that X 1, X 2 X, and consequently, C 1, C 2 C. It is essential fo pactice to have a chaacteization of a banching ule that is independent of basis sets and only elies on the stuctue of the cicuit set C. In ode to get a banching ule that esults in a stong (i.e., nealy disjoint) coveage of the solution set, we futhe need to chaacteize in which case B = B 1 B 2, i.e., B 1 and B 2 patition B. We need the following lemma: Lemma 6.6. Let two sets C 1, C 2 2 V be given such that the coesponding basis sets satisfy B = B 1 B 2. Then the following applies: (i) Fo evey C V, C C 2 if and only if B B : C B} B 1 and C C 1 if and only if B B : C B} B 2. (ii) B 1 = B B : C B} and B 2 = B B : C B}. C C 2 C C 1 15

Poof. We begin with the poof of (i). Since C 2 = 2 V \ B 2, we have that C C 2 C B 2 B B 2 : C B} = B B : C B} B 2 =. Because B = B 1 B 2, we obtain the equivalence of C C 2 to B B : C B} B 1. The second pat is analogous. To see (ii), we fist obseve by (i) that B 1 C C 2 B B : C B}. Fo the evese inclusion, let B B 1. Then B B 2, and by definition, thee exists some C C 2 with C B. This poves that B C C 2 B B : C B}, and hence B 1 C C 2 B B : C B}. The statement fo B 2 can be shown analogously. Theoem 6.7. Let two sets C 1, C 2 2 V sets satisfy B = B 1 B 2 if and only if with C 1, C 2 C be given. Then the coesponding basis C 1 C 1 V : ( C 2 C 2 ) ( C C) C C 1 C 2 } =: G 1, (6.7) C 2 C 2 V : ( C 1 C 1 ) ( C C) C C 1 C 2 } =: G 2. (6.8) Futhemoe, we have that B = B 1 G 2 ae dependence systems). B 2 if and only if C 1 = G 1 and C 2 = G 2 (note that G 1 and Poof. We stat with the poof of the fowad diection of the fist equivalence: It is enough to show that (6.7) (6.8) hold if B = B 1 B 2 ; note that C = 2 V \ B fo 1, 2} such that (6.7) (6.8) is moe difficult to meet if B 1 B 2 =. Let C 1 C 1 and C 2 C 2. Lemma 6.6 (i) implies that B B : C 1 B} B 2 and B B : C 2 B} B 1. Since B 1 and B 2 ae disjoint by assumption, we theefoe get B B : C 1 B} B B : C 2 B} =. Consequently, thee exists no basis B B with C 1 C 2 B. This shows that thee must exist some cicuit C C with C C 1 C 2 and (6.7) (6.8) follow. We pove the backwad diection of the fist equivalence by contadiction: Fist obseve that B 1 B and B 2 B, since C C 1 and C C 2. We assume that B 1 B 2 B. Then one can find some B B with B B 1, B 2. By definition, thee exist cicuits C 1 C 1 and C 2 C 2 with C 1 B and C 2 B, but thee exists no set C C with C B. Since C C 1 C 2 B fo all C C, we obtain that C 1 G 1 and C 2 G 2, which is a contadiction to (6.7) (6.8). This shows that the assumption B 1 B 2 B was wong and the claim follows. We now poceed to the poof of the second equivalence. Define B 1 := C G 2 B B : C B} and B 2 := C G 1 B B : C B}. We fist show that B 1 B 2 = if B 1 B 2 = B: By contadiction, suppose that B 1 B 2 and B 1 B 2 = B hold. Since by (6.7) (6.8), C 1 G 1 and C 2 G 2, it follows with Lemma 6.6 (ii) that B 1 B 1 and B 2 B 2. Consequently, B 1 B 2 B 1 B 2, and hence, B = B 1 B 2 = B 1 B 2 B. Because B 1 B 2, we thus obtain B 1 B 2 o B 2 B 1. Assume without loss of geneality that the fist is tue, and let B B 1 B 2. By definition of B 2, thee exists some C 1 G 1 with C 1 B. Since (6.7) holds, it follows that fo evey C 2 C 2 thee exists some C C with C C 1 C 2. This implies that C 2 B fo evey C 2 C 2 ; othewise if C 2 B, then C C 1 C 2 B and this would be a contadiction to C C. By Lemma 6.6 (ii), the fact that thee exists some B B 1 such that C 2 B fo evey C 2 C 2 is a contadiction to B 1 B 2 = B. Theefoe, we obtain that B 1 B 2 = and consequently that B 1 B 2 = B (in paticula, this shows that B 1 = B 1 and B 2 = B 2 ). We will use this finding to pove the fowad diection of the second equivalence of the theoem: Supposing that B 1 B 2 = B, we obtain the inclusions C 1 G 1 and C 2 G 2 by the fist equivalence of the theoem (note that G 1 and G 2 ae dependence systems). We now show that convesely G 1 2 V \ B 1 C 1. The poof that G 2 2 V \ B 2 C 2 is then analogous. Let C 1 G 1. By definition of B 2, we have that B B : C 1 B} B 2. Because B 1 B 2 = B, 16

it follows fom the backwad diection in Lemma 6.6 (i) that C 1 2 V \ B 1 and we obtain the inclusion G 1 2 V \ B 1. The second inclusion 2 V \ B 1 C 1 tivially follows fom B 1 = B 1. We show the backwad implication of the second equivalence by contaposition: Let B 1 B 2 = B, but B 1 B 2. Then (6.7) (6.8) hold by the fist equivalence of the theoem. Moeove, since B 1 B 1 and B 2 B 2 (see above), it follows that B 1 B 2. Let B B 1 B 2. By definition of B 1 and B 2, thee exists some C 1 G 1 with C 1 B and some C 2 G 2 with C 2 B. Then C 1 C 2 B, and since B B, thee exists no cicuit C C with C C 1 C 2 B. By (6.7) (6.8), it follows that C 1 C 1 and C 2 C 2. Thus, both inclusions C 1 G 1 and C 2 G 2 ae pope. On the othe hand, if C 1 = G 1 and C 2 = G 2, then B 1 B 2 = B by the fist equivalence of the theoem, and the negated agument of above shows that B 1 B 2 =. Remak 6.8. Theoem 6.7 gives a complete chaacteization of all cicuit constaints that may be added to the banching nodes, neglecting the influence of the linea inequality system Ax b. This is due the fact that the inclusion estictions (6.7) and (6.8) can always be satisfied with equality fo an appopiate choice of C 1 and C 2. As a consequence of Theoem 6.7, we obtain that the banching disjunction of hypeclique banching (see Section 6.1) is coect. Coollay 6.9. Let the cadinality constaint x K 0 k be valid fo (CCLP). Then fo evey nonempty subset S K and numbe 0 s < mink, S }, the disjunction x S 0 s x K\S 0 k s 1 holds. Poof. Let C be the cicuit set of (CCLP) and B the coesponding basis set. By assumption, we have that ( K ) k+1 C. Given C1 ( S ) s+1 we have fo evey C2 ( K\S) that C := C1 C 2 ( K k+1). This shows that (6.7) is satisfied by C 1 := ( S ) s+1 C and C2 := ( K\S k s) C. The poof that (6.8) holds is analogous. By Theoem 6.7, the basis sets B 1 and B 2 coesponding to C 1 and C 2 satisfy B = B 1 B 2 and the assetion follows fom Lemma 6.5. Theoem 6.7 can additionally be used to complement and impove neighbohood banching: Given some node i V, neighbohood banching is pefomed on the disjunction x i = 0 x C\i} 0 C 2 C C : i C, (6.9) which follows fom (6.2) using K = (C, C 1) : C C}. Fo the new appoach, we select a nonempty subset M := C \ i} : C C, i C} and define C 1, C 2 2 V such that C 1 = C 1 V : ( C 2 M) ( C C) C C 1 C 2 }, C 2 = C 2 V : ( C 1 C 1 ) ( C C) C C 1 C 2 }. (6.10) Note that the smalle M is, the lage C 1 becomes. On the othe hand, the lage M is, the smalle C 1 becomes and the lage C 2 becomes. The sets C 1 and C 2 have the following popeties: Coollay 6.10. Fo i V, let M C \ i} : C C, i C}, M =, and let C 1, C 2 2 V satisfy (6.10). Then (i) C 1 C and C 2 C, (ii) C 1 = G 1 and C 2 = G 2, and (iii) C 1 and C 2 define a feasible banching disjunction (6.6). 17

Poof. Recall fom the assumption stated at the beginning of Section 6 that C > 1 fo all C C and that H is simple, i.e., C C fo distinct C, C C. Fo the poof of (i), note that C 1, C 2 C by definition of C 1 and C 2. To show that these inclusions ae stict, we obseve that i} C 1 by definition of M and that M C 2 by definition of C 1. On the othe hand, we have that i} C, since C > 1 fo all C C, and we have that M C =, since C contains only inclusion-wise minimal elements. We poceed to the poof of (ii): The equality C 2 = G 2 tivially holds by definition. To show C 1 = G 1, we fist obseve that C 1 C 1 V : ( C 2 C 2 ) ( C C) C C 1 C 2 } (6.11) by definition of C 2. Because M C 2, we get the convese inclusion of (6.11) by definition of C 1, and Statement (ii) is shown. Finally, Statement (iii) esults fom Theoem 6.7 and Lemma 6.5 using (ii). We show an application of Coollay 6.10 in the following example: Example 6.11. Conside the conflict hypegaph visualized in Figue 6.1, whose hypeedge set is C = 1, 5}, 2, 6}, 1, 2, 3, 4}, 3, 4, 5, 6}}. In Example 6.4 it was shown that x 1 = 0, x 3 = 0 x 4 = 0 x 6 = 0, x 5 = 0, x 2 = 0 x 3 = 0 x 4 = 0 (6.12) is a feasible disjunction. The two subpoblems that we obtain by enfocing (6.12) have the hypeedge sets C 1 = C 1}, 3, 4, 6}} and C 2 = C 5}, 2, 3, 4}}. To deive (6.12) efficiently without computing all bases, we use Coollay 6.10: If we choose i = 1 and M = 5}, 2, 3, 4}}, then fo instance, C 1 := 3, 4, 6} C 1, since the union of C 1 and C 2 = 5} M coves C = 3, 4, 5, 6} C and the union of C 1 and C 2 = 2, 3, 4} M coves C = 2, 6} C. By evaluating all possible combinations, one can show that C 1 = G 1 and C 2 = G 2. We have implemented the idea of dependent set banching fo the case that all cicuit constaints ae complementaity constaints, see [23]. Computational esults confimed that thee exist choices of M which esult in an efficient banching ule. 7 Pojecting Inequalities Recall that S denotes the feasible solution set of (MIPPC) and Q the feasible solution set of (CCLP). In this section, we deive a pojection fomula to tansfom an inequality a i x i + g i y i β i V i V with a, g R V and β R that is valid fo conv(s) into an inequality that is valid fo conv(q). It is easy to see that if g 0, then i V (a i + g i /u i ) x i β is a valid inequality fo conv(q). Ou investigations will cove the moe geneal case that some of the g i, i V, may be negative. Moeove, we will deive conditions fo which the pojected inequality defines a facet. Thoughout this section, we assume that the uppe bounds u R n + of x ae finite and nonzeo. In this case, conv(q) and conv(s) ae polyheda, see Lemma 3.1. Futhemoe, we assume that x K 0 k fo (K, k) K := (K, k ) : R} is an initial set of cadinality constaints given fo (CCLP). 18