Sequence independent lifting for 0 1 knapsack problems with disjoint cardinality constraints

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1 Sequence independent lifting for knapsack problems with disjoint cardinality constraints Bo Zeng and Jean-Philippe P. Richard School of Industrial Engineering, Purdue University, 35 N. Grant Street, West Lafayette, IN {beng, September 9, 26 Abstract In this paper, we study the set of integer solutions to a single knapsack constraint and a set of non-overlapping cardinality constraints MCKP. This set is a generaliation of the traditional knapsack polytope and the knapsack polytope with generalied upper bounds. We derive strong valid inequalities for the convex hull of its feasible solutions by lifting the generalied cover inequalities presented in [32]. For problems with a single cardinality constraint, we derive a set of multidimensional superadditive lifting functions and prove that they are maximal and nondominated under some mild conditions. We then show that these functions can also be used to build strong valid inequalities for problems with multiple disjoint cardinality constraints. Introduction In 983, Crowder et al. [8] proposed to use strong inequalities for knapsack sets as cuts for more general mixed integer programs MIPs with multiple constraints. They demonstrated that this approach yields significant computational improvements over pure branch-and-bound algorithms on a set of general MIPs. Gu et al. [2] later implemented the idea in MINTO and confirmed these results. As a consequence, cuts from knapsack relaxations of MIPs were progressively implemented in commercial software. They are now an integral part of major commercial MIP solvers, such as CPLEX and XPRESS. In this paper, we consider an extension of the basic knapsack model KP in which the integer solutions to a single knapsack constraint are also required to satisfy a set of non-overlapping cardinality constraints MCKP. Specifically, for N = {,...,n}, N i N for i =,...,r with N i N k = for i k, we study Ŝ r = x {, }n : â j x j ˆb, j N x j ˆK i, i =,...,r,, j N i where â j R for j N, ˆb R and ˆK i Z + for i =,...,r. Without loss of generality, we assume that ˆK i for i =,...,r. We also define PŜr := convŝr. Because Ŝr is a finite set, the following proposition can easily be verified. Proposition. PŜr is a polytope. The study of PŜr is interesting both theoretically and practically. From a theoretical perspective, PŜr generalies the traditional knapsack problem if N i = for all i, the knapsack problem

2 with generalied upper bounds GUBKP if ˆKi = for all i, and the knapsack problem with disjoint precedence constraints if N i = {j i, k i }, â ji â ki and ˆK i = for i =,...,r. Also, it is a first step in the polyhedral study of unstructured problems with multiple constraints. In fact, we can always relax an MIP into PŜr by selecting any one of its row to be the knapsack constraint of PŜr and by generating a set of r disjoint extended cover inequalities from the problem formulation to constitute the cardinality constraints; see Nemhauser and Wolsey [2] for a discussion of extended cover inequalities. Clearly, strong valid inequalities for PŜr can be used as cuts for the original problem. Furthermore, these cuts are generated from a relaxation that is tighter than the traditional knapsack relaxation PŜ. Therefore, it is reasonable to assume that these cuts will be strong. From a practical perspective, MCKP appears as a subproblem of more complicated integer programs. As an example, GUBKP appears as a relaxation of product pricing, capital budgeting, scheduling and planning, resource allocation, computer and circuit layout design problems; see Armstrong et al. [], Ferreira et al. [9], Sherali and Lee [23], Sinha and Zoltners [24], and Wolsey [29]. The general cardinality-constrained knapsack model, although less studied, also appears as a subproblem of many practical applications, including parallel task allocation, capacity allocation and various combinatorial problems; see Hollermann et al. [6], Toktay and Usoy [25], and Bruglieri et al. [7]. To the best of our knowledge, there are few polyhedral studies of the knapsack problem with cardinality constraints. Exceptions include recent work by Glover and Sherali [, ]. In [], the authors study sets defined by a knapsack constraint and one additional cardinality constraint that covers all variables. Their focus is on deriving all strong second-order cover inequalities. We note that the set they studied is a special case of PS since we do not assume that all the variables belong to the cardinality constraint. In [], the authors design a procedure to generate similar high-order cover inequalities for KP with a single two-sided bounding constraint over all variables and a set of disjoint two-sided bounding inequalities. Our research is different from Glover and Sherali s work in that we concentrate on developing and applying superadditive lifting theory to efficiently generate strong valid inequalities from facet-defining extensions of classical cover inequalities. Although polyhedral studies of PŜ r are scarce, special cases such as KP and GUBKP have been studied extensively in the past. In 975, Balas [4], Hammer et al. [5], Wolsey [27] presented facetdefining inequalities for KP based on the notion of a cover. Balas and Zemel [5] gave bounds on the lifted coefficients of minimal cover inequalities and Zemel [3] proposed a general simultaneous lifting procedure to obtain facets. Balas and Zemel [6] also showed that all the facet-defining inequalities of KP can be obtained by lifting cover inequalities after possibly complementing variables. Padberg [2] introduced, k-configurations inequalities for KP. Weismantel [26] presented a complete linear description of the convex hull of particular knapsack polytopes and proposed a reduction principle that can be used to generate strong valid inequalities for KP. The polyhedral structure of GUBKP has been less studied. Johnson and Padberg [7] described an equivalent transformation from instances of GUBKP with arbitrarily signed coefficients to instances of GUBKP where all the variable coefficients are nonnegative. This implies that when studying PŜr with ˆK i = for i =,...,r, it is not restrictive to assume that a j for j N. Wolsey [29] gave a set of valid inequalities for GUBKP and proved that they are strong by deriving conditions under which they describe the convex hull of the problem. Nemhauser and Vance [9] presented a method based on independent sets to lift cover inequalities and obtain facet-defining inequalities. Sherali and Lee [23] also generated inequalities from covers using lifting. Their lifting procedure is different because it simultaneously lifts all variables within a GUB, which yields a polynomial lifting scheme. Finally, Gu et al. [2] discussed some computational aspects of using lifted GUB cover inequalities in the solution of integer programs. In this paper, we are interested in studying the knapsack with disjoint cardinality constraints. To simplify the presentation, we first complement the variables x j of Ŝr whose coefficients â j are 2

3 negative. We obtain the equivalent set S r = {x Z n : a j x j b, 2 j N x j x j K i, i =,...,r 3 j N + i j N i x j j N, 4 x j j N} 5 where b R, K i Z, a j R + for j N, N i + = {j N i : â j } and Ni = {j N i : â j < }. Clearly, from, we have K i = ˆK i N i and N i = N + i N i N with N + i N i =. We define r N := N\ N i. We assume without loss of generality that if j, k N i + or Ni, or N and j < k, i= then a j a k. We also assume that b since otherwise, S r =, a j b for j N since otherwise x j = and N i K i if K i < since otherwise S r =. We denote the convex hull of S r by PS r. Furthermore, when N i = for i =,...,r, we denote the set S r as S r + and its convex hull as PS+ r. We observe that PS r + is an independence system. In Section 2, we describe basic lifting results and discuss general conditions under which lifting is sequence independent. In particular, we show that for PS r, conditions weaker than superadditivity already imply that lifting is sequence independent. In Section 3, we briefly review some results about generalied cover inequalities, a family of inequalities described in [32] that are strong for PS r. In Section 4, we derive a set of superadditive lifting functions for PS + and prove that they are nondominated and maximal. In Section 5, we present a set of superadditive lifting functions for PS and show that the valid inequalities obtained using sequence independent lifting in PS are either coefficientwise stronger than those obtained from superadditive lifting in PS, or cannot be obtained from PS. In Section 6, we extend the results of Section 4 and present a r+-dimensional maximal and non-dominated superadditive lifting function for generalied cover inequalities in PS r +. 2 Lifting for PS r In this section, we give a brief review of how lifting can be used to generate valid inequalities in integer programming. We also generalie sequence independence results and describe how we will use these results to generate inequalities for PS r. In Section 2., we give basic definitions and results about lifting. In Section 2.2, we describe general conditions for which lifting is sequence independent. These superadditive conditions were introduced by Wolsey [28] in the case of integer programs. Because of the specific nature of our problem, we show that it is sufficient to consider a subset of the superadditive conditions to guarantee sequence independent lifting in PS r. This result is important since it permits an easier derivation of stronger lifting coefficients for MCKP in Sections 4, 5 and 6. In Section 2.3, we discuss the use of superadditive functions to obtain strong valid inequalities. 2. Basic Lifting Results We consider G to be the set of integer solutions to a finite set of linear inequalities, i.e. G = {x {, } n : j N A j x j b} 6 where A j R m for j =,...,n, b R m and N = {,...,n}. We denote the convex hull of G as PG. We consider the restriction of PG obtained by fixing some of the variables at, i.e. we define PGN := conv{g {x j = : j N }} where N = {,..., n } N. Assuming that all the variables 3

4 are fixed at is without loss of generality since variables can be complemented. Assume that the seed inequality j N\N α j x j α 7 is valid for PGN. Sequential lifting is the process by which 7 is converted into a valid inequality for PG of the form α j x j α 8 j N by reintroducing the variables x,..., x n in 7 one at a time. Since we can always reorder variables, we assume that x i is the i th variable to be lifted. For sequential lifting, the coefficients α i for i N can be obtained by solving the lifting problems f i Z = min α α j x j α j x j j N\N j<i s.t. A j x j + A j x j b Z 9 j N\N j<i x j {, } j N\N {,..., i } where Z R m and where we define f i Z = + when f i Z is infeasible. When f i A i < + for i N, it is clear that setting α i = f i A i in 8 yields a valid inequality for PG. In particular, it can easily be proven that if 7 is facet-defining for PGN and if PGN is full-dimensional, then setting α i = f i A i for i N yields a facet-defining inequality of PG. The case where f i A i = + for some i N corresponds to situations where the variable x i to be lifted cannot take any value other than the one it is fixed at in the current set. This typically means that x i should be lifted later in the sequence. In the remainder of this paper, we will use f to denote f and refer to it as the exact lifting function of 7. In the following propositions, we give some properties of f i. These properties can easily be derived from the fact that 7 is valid for PG and from the fact that the problem defining f i is a relaxation of that defining f i. Proposition 2. Let i N. Then, f i. Proposition 3. Let i j n and Y Z R m be such that f i Y < + and f i Z < +. Then f i Y f i Z if f i Y f j Y. 2.2 Sequence Independent Lifting The inequalities generated by computing the lifting coefficients α i exactly through the computation of f i A i are strong. However, the amount of computation needed to obtain them is often prohibitive as a different optimiation problem must be solved for every variable that is lifted. Fortunately this computational burden can be significantly reduced if the exact lifting function f is well-structured. Definition. A function g : D R is superadditive if gx + gy gx + y for all x, y, x + y D. In 977, Wolsey [28] showed that for knapsack problems, lifting coefficients can be directly from f when f is superadditive. Gu et al. [4] extended Wolsey s results to mixed-integer programs. Atamtürk [3] later generalied the results to general mixed integer programs. Because of its computational advantages, the superadditive lifting theory has been used in different applications to derive strong inequalities for MIPs. As an example, Marchand and Wolsey [8] used superadditive lifting to derive two families of closed form facet-defining inequalities for knapsack sets with a single continuous variable. The condition that f be superadditive over D = R n is sufficient for sequence independent to hold. However, there are weaker conditions that still imply sequence independent lifting. We give such conditions next. 4

5 Theorem 4. Assume that A i b for i =,...,n. Assume also that fa i +f j R A j f for all R N \{i} such that {A i,..., A n }. j R {i} A j b. Then f i x = f x for i =,...,n j R {i} A j and for x Proof. The proof is by induction. The proof is obvious when i =. So, we assume that we have already established that for i k where 2 k n, f i x = f x for x {A i,...,a n }. It remains to prove that the result holds for i = k +. It follows from Proposition 3, that f i A t fa t for i =,...,n. Since x j {, } for j i, we observe from 9 that f i A t = min R {,...,i} {f j R A j + A t j R f ja j }. From the inductive hypothesis, we have j R f ja j = j R fa j since R {,...,i }. From the theorem assumption on f, it can easily be verified that j R fa j f j R A j for R {,..., i }. So, f i A t min R {,...,i} {f j R A j + A t f j R A j} for t = i,...,n. Again, from the theorem assumption on f, we observe that f i A t fa t. We conclude that f i A t = fa t for t = i,...,n. Theorem 4 shows that it is sufficient to prove that f is superadditivity over the set consisting of all possible values of j R A j for R N to ensure sequence independent lifting. In many cases, the number of such conditions increases exponentially with the number of variables to be lifted. As a result, it becomes more convenient to verify that f is superadditive over R or R +. However, the number of conditions to verify for a cardinality constraint does not increase exponentially. This is because all the coefficients of the variables are either, or. By combining this observation with the result of Theorem 4, we derive next simpler conditions for sequence independent lifting in PS r. To simplify the presentation we define for N, N N satisfying N N =, We define PS + r N, N similarly. PS r N, N := conv{x S r : x j =, j N, x j =, j N }. Proposition 5. Let f be the lifting function of a valid inequality for PS r N,. Then, the lifting of variables in N is sequence independent if y y + f I + f f I + h for y, I [, b] {, ± e,..., ± e r } and, [, b] Z r in PS r or for y, I [, b] {, e,..., e r } and, [, b] Z r + in PS + r where e,..., e r are the unit vectors in R r. Note that a lifting function satisfying the conditions of Proposition 5 is not necessarily superadditive over [, b] Z r. However, in the remainder of this paper, we will refer to functions satisfying as superadditive. Verifying that a function is superadditive is often cumbersome, even for one-dimensional functions. Therefore, reducing the set of conditions to verify is a very crucial step in developing strong sequence independent lifting schemes for problems with multiple constraints. For this reason, Proposition 5 will be used extensively in later sections. 2.3 Approximate Superadditive Lifting Very often, the exact lifting function of a seed inequality of interest is not superadditive and so the result of Theorem 4 cannot be used directly to derive valid inequalities. In such as situation, a superadditive lower approximation of the exact lifting function can be used to obtain strong valid inequalities without having to solving lifting problems repeatedly. This idea was first used in Gu et al. [3, 4] to derive efficient lifting procedures for knapsack and single node flow problems. An application of this procedure is also given by Atamtürk [2] for general mixed-integer knapsack sets and by Shebalov and Klabjan [22] for mixed-integer program with variable upper bounds. 5

6 For any given lifting function, Gu et al. [4] give a constructive proof of the existence of a superadditive lower approximation. In practice, there are usually many such approximations. Therefore, evaluating the quality of a proposed approximation is important. To measure the strength of a superadditive approximation, Gu et al. [3, 4] propose two criteria: non-dominance and maximality. Here, we summarie these concepts and present them for higher dimensions. We also introduce a new criterion to describe superadditive approximations of lifting functions defined for multiple constraints that yield coefficients stronger than those obtained from the individual constraints. Thereafter, we refer to the exact lifting function as f and to its superadditive approximation as g. We denote the domain of f as X R m. Definition 2. We say that a valid superadditive approximation vx of ux is non-dominated over X if there is no valid superadditive approximation v x of ux such that vx v x for all x X and vx < v x for some x X. Note that, for lifting, it is not crucial that the superadditive approximation be non-dominated over the domain of f. In fact, it is only important that it is non-dominated in a range X that contains all the coefficients of the variables to be lifted. In PS r for example, we must verify that the exact lifting function is superadditive over [, b] Z r so as to guarantee sequence independent lifting. However, we only care about the strength of the superadditive approximation over X = [, b] {, ± e,..., ± e r } since all the lifting coefficients will be obtained by evaluating the lifting function in this range. This observation motivates the following definitions. Definition 3. Let X X be a set that contains all the coefficients of the variables to be lifted. Given that fx is the exact lifting function over X, we say gx is a non-dominated superadditive approximation of fx if gx is a superadditive approximation of fx over X and is non-dominated over X. Next, we describe the concept of a maximal approximation. We begin with a definition of maximal set. Definition 4. Let n be the number of variables to be lifted. Let X be a set that contains all the coefficients of the variables to be lifted, i.e. A i X for i =,...,n. We say that E = {x X : f i x = fx for all choices of A i X, i =,...,n and for all lifting orders} is the maximal set of the lifting function fx. Definition 5. Given a valid superadditive approximation gx of fx, we say that gx is maximal over X if gx = fx for x E. Clearly, the best possible superadditive approximations of f are maximal and non-dominated. However, non-dominance and maximality are sometimes difficult to achieve and to verify for multipleconstraint problems. In these situations, it is important to guarantee that the approximation obtained for multiple constraints is at least as good as that obtained from a single row relaxation. It is clear that a function φ that is superadditive over X R m can be extended into a superadditive function over X 2 X R s by setting ψ x y = φx for x, y X R s. When applied to lifting, this observation simply means that a valid superadditive lifting function for an integer program with m constraints can be extended into a valid lifting function for a program with m+s constraints if the former model is the relaxation of the latter model obtained by dropping the last s constraints. For this reason, we introduce the following definition where we use E[gx] to denote the trivial extension of a superadditive lifting function gx to a higher dimensional space. Definition 6. Let g and g be two valid lifting functions that are superadditive over X and X 2 respectively. Assume also that X X 2 R s. We say that g is superior to g on X if g x y dominates E[g x]. 6

7 3 Deriving Strong Valid Inequalities from Generalied Cover Inequalities In this section, we describe the procedure we will use to generate strong valid inequalities for PS r. A traditional approach to generate strong valid inequalities for KP is to lift minimal cover inequalities that are facet-defining for restrictions of the set; see [5, 6, 3]. However, for MCKP, there may not exist any minimal cover C that yields a facet-defining inequality for PS r N\C, even when PS r is non-trivial. We illustrate this observation in Example. Example. Consider S = {x {, } 5 : 6x + 5x 2 + 4x 3 + 4x 4 + 2x 5 7, x + x 2 + x 3 x 4 2}. Observe first that S S where S = {x {, } 5 : x +x 2 +x 3 x 4 2} since the point,,,, S but,,,, / S. Clearly, C =, 2, 3, 4 is the only minimal cover that can be obtained from the knapsack constraint. It is easily seen that PS N\C, is full-dimensional. Finally, observe that x + x 2 + x 3 + x 4 3 is not facet-defining for PS N\C,. This illustrates the fact that PS might be non-trivial and might not have minimal cover inequalities that can be lifted into facets of PS. Example illustrates that minimal covers from the knapsack constraint do not always produce the best seed inequalities for lifting when studying PS r. Therefore, to derive strong inequalities for PS r, we propose to first fix variables x j with j ˆN r to so as to loosen the cardinality constraints i= and then to identify a minimal cover C that is strong. The minimal cover inequality is then lifted with respect to these variables to obtain a facet-defining inequality for PS r N\{ ˆN C},. Next we describe necessary and sufficient conditions for PS r to be full-dimensional. Throughout this subsection, we define N + i := {p i,..., q i } and N i := {q i +,..., s i } for i =,...,r, and N = {s r +,...,n} where p = and p i = s i + for i 2. We define a k W := max{ j V a j : V W, V = k}, and a k W := min{ j V a j : V W, V = k}. We also define a k W := if W < k and a k W := if W < k. Proposition 6. [32] Let b = max{, i:k i a K i N i }. PS r is full-dimensional iff a N +b b and for each i =,...,r, one of the following conditions holds i K i and a N i + b b, ii K i =, a N + i + a N i + b b and a N i + b b, iii K i, N i K i +, a N i +ba s i K i + b, and max{a N + i, }+b+a s i K i b. In Proposition 7, we present necessary and sufficient conditions under which a minimal cover C of PS r yields an inequality x j C j C that is facet-defining for PS r N\C,. Let C i = C N i, C + i = C N + i and C i = C N i for i =,...,r. We define η i := K i C + i + C i. Note that η i is an integer that represents the amount of slack there is in the i th cardinality constraint when all the members of the cover are set to. Proposition 7. [32] Let C be a minimal cover for PS r N\C, and assume that PS r N\C, is full-dimensional. Then, the minimal cover inequality is facet-defining for PS r N\C, if and only if for each i =,...,r, one of the following conditions is satisfied i C i and η i ; ii C + i, C i = : a C C + i and η i ; b C = C + i and η i ; iii C + i = C i = and η i. N i 7

8 For PS r +, the conditions of Proposition 7 become simpler. Corollary 8. Let C be a minimal cover. The cover inequality is facet-defining for PS r + N\C, if and only if one of the following conditions is satisfied i C i = for i =,...,r, i.e. C N ; ii C = C i for some i {,...,r}, and η i ; iii C C i and η i for i =,...,r. We prove in [32] that if PS r is not completely determined by the cardinality constraints, there always exist ˆN N and C N\ ˆN such that is facet-defining for PS r N\C ˆN, ˆN. For PS r +, a minimal cover from the knapsack constraint that satisfies conditions of Corollary 8 can always be obtained if we choose ˆN =. Observe also that in the case of GUBKP, condition iii corresponds to the notion of minimal GUB cover; see Nemhauser and Vance [9], Sherali and Lee [23], and Wolsey [29]. Therefore, it is possible to obtain a facet-defining inequality for PS r N\C ˆN, by lifting a minimal cover inequality from. It is shown in Zeng and Richard [32] that, except for the case C = C + i with η i = and the case C = C i with η i = for some i {,..., r}, all the lifted inequalities are of the form x j + x j C + EZ 2 j C j EZ where EZ ˆN is the set of exact lifting coefficients that are equal to and where we denote EZ = ˆN\EZ. Note that although 6 has the form of a cover inequality, C EZ may not be a cover for the knapsack constraint. We illustrate this observation in the following example. Example 2. Consider PS = conv{x {, } 6 : 7x + 7x 2 + x 3 + x 4 + x 5 + x 6 5, 3 x x 2 x 3 x 4 x 5 x 6 2}. 4 Note that C = {, 2} is a minimal cover for the full-dimensional polyhedron PS, ˆN where ˆN = {3, 4, 5, 6}. The minimal cover inequality derived from C is x + x 2. 5 It follows from Proposition 7 that 5 is facet-defining for PS N\C ˆN, ˆN. The lifting coefficients for x 3, x 4, x 5, x 6 obtained through exact lifting are eros. Therefore, 5 is a facet-defining inequality for PS. It is easily seen that C is not a cover for the knapsack constraint 3. To streamline the notation, we will denote in the remainder of this paper the set C EZ as C and the set C EZ as C. Using this new definition of C, 2 becomes with x j C. 6 j C C + i C i EZ N i + η i = K i 7 for i {,..., r}. Because of the similarity and difference to the traditional cover inequality, we refer to 2 as a generalied cover inequality; see Zeng and Richard [32]. Note that, for PS + r, a generalied cover is always a minimal cover for the knapsack constraint with C = EZ =. The following corollary presents some properties of generalied cover. Corollary 9. [32] For a generalied cover C, we have a j > for j C. Furthermore, if C is a cover for the knapsack constraint, then it is also a minimal cover for the knapsack constraint. 8

9 In this paper, we show how to lift generalied cover inequalities with respect to the variables x j from for j N\C ˆN from to obtain facet-defining inequality for PS r. Since lifting is difficult to perform exactly, we compute strong approximate lifting coefficients using a superadditive lower approximation of the exact lifting function. We note that the lifting function used here is multidimensional since we need to consider the coefficients of the variables in the knapsack constraint and in the different cardinality constraints. To the best of our knowledge, this is the first time superadditive lower approximations of multidimensional lifting functions are proposed and proven to be strong. In fact, research on approximate superadditive lifting has been limited to one-dimensional lifting functions even when the sets studied have multiple constraints; see [3, 4, 22]. 4 Superadditive Approximation of Lifting Functions in PS + In this section, we study how to lift generalied cover inequalities that satisfy conditions of Corollary 8 for PS +. Because the lifting function of the generalied cover inequality is not superadditive, we approximate it from below to obtain a valid superadditive lifting function. We first construct a superadditive lifting function for the case where K =. Then, for the case where K 2, we introduce a composition method that can be used to create a 2-dimensional superadditive approximation from a single-dimensional one. Such a composition method is very useful since it is difficult in general to build strong multidimensional superadditive lifting functions. For both the cases where K = and K 2, we prove that the proposed approximations are strong by showing that they are non-dominated and maximal. In the remainder of this paper, we refer to the exact lifting function of the generalied cover inequality in PS + as Θ and to its superadditive approximation as θ. 4. Exact Lifting and Superadditive Lifting function Because PS + has a single cardinality constraint with only nonnegative coefficients, we use the notation N +, C + and K to represent N + = N, C + and K respectively. We assume without loss of generality that the variables x j for j C are sorted in non-decreasing order of their coefficients, i.e. a a 2 a C. We also denote A i = j i a j for i =,..., C, A = and λ = A C b. Because the generalied cover C is also a minimal cover for the knapsack constraint, we have λ >. The i th lifting problem associated with a generalied cover inequality of PS + is given by Θ i = min C x j α j x j h j C j i s.t. a j x j + a j x j b j C j i 8 x j + I j x j K h j C + j i x j {, } for j C {,..., i } where, [, b] Z + and I j is the coefficient of x j in the only cardinality constraint. Similar to Proposition 3, we present the following result. Corollary. The function Θ i is non-decreasing over, [, b] Z+. Instead of computing exact solutions to a set of different lifting problems, we derive a superadditive lower approximation of Θ to generate strong valid inequalities for PS +. From Corollary 8, we observe that there are different types of generalied cover inequalities that lead to different lifting functions. We will show next in Sections 4.2 and 4.3 that all the valid superadditive approximations of generalied cover inequalities we propose can be constructed as in Theorem. 9

10 Theorem. For the lifting function Θ of the generalied cover inequality, there exists a nondominated and maximal superadditive approximation of Θ over [, b] {,..., K} that is of the form θ if h = max{θ a +, θ } if h = θ = h h j 9 sup θ if h = 2,...,K È h {= j= j,j,j=,...,h} j= where θ is a superadditive approximation of Θ and a depends on C and N +. We now describe how to obtain θ and a. The case K = is described in Section 4.2 and the case K 2 is described in Section Building a Superadditive Approximation of Θ when K = First we use Theorem 4 and Proposition 5 to derive sufficient conditions for sequence independent lifting. Corollary 2. Define d + = max{ S : S N + \C +, j S a j b}. For PS + sequence independent over [, b] {, } if y y + Θ + Θ Θ y y + Θ + Θ Θ h h where h = min{, d + } and y,, y + [, b]. with K =, lifting is Note that when d + =, i.e. N + = C +, 2 and 2 reduce to the traditional superadditive conditions for sequence independent lifting in the knapsack polytope. In fact, the traditional knapsack set can be viewed as a special case of PS + where a cardinality constraint of the form x j is imposed for some j N. There are two cases. On the first hand, if j C +, N + \C + = then h = d + =, and therefore 2 reduces to 2. On the other hand, if j / C +, Θ = Θ and therefore 2 reduces again to 2. We conclude that in both cases, the conditions of Corollary 2 reduces to those given by Gu et al. [4]. The nontrivial case is when d + =, i.e. there exists a variable x j such that j N + \C +. From Corollary 8 we see that, in order to yield a facet-defining inequality, the generalied cover inequality must satisfy C + 2. Therefore, we discuss next the following 3 cases: i C + =, ii C + = and iii C + = 2. Because the case C + = is the most interesting, we consider it first Case i: C + = Assume that C + = {l}. Note that since any single variable of PS + cannot form a cover, we must have C C +. We first derive the exact lifting function. Theorem 3. The exact lifting function Θ I of the generalied cover inequality is { if A λ i if A i λ < A i+ λ i =,..., C Θ I = if a l λ i if A i + a l λ < A i + a l λ i =,...,l i if A i λ < A i+ λ i = l,..., C. if I = if I =

11 Note that we use Θ to denote this lifting function rather than Θ since we will use this function in this section and Section 6. We observe that Θ = Θ for Al λ. Note also that Θ = Θ for all if a l = a. In the following corollary, we derive additional relations between Θ and Θ. Corollary 4. Θ Θ [, b]. Furthermore, if a l λ Θ if a = l λ < < a l Θ a l 23 + if al A l λ Θ, if Al λ < b. Next, we derive a valid superadditive approximation θ for the lifting function Θ. To derive this approximation, we first observe that Θ is the lifting function of a cover inequality. Therefore it can be approximated with the superadditive function proposed by Gu et al. [4]. We then build θ from θ using the relations presented in Corollary 4. The result of this construction is presented in Theorem 5. Theorem 5. Let ρ i = max{, a i+ a λ} for i =,..., C. The function θ = I { θ if I = if = i if A i λ + ρ i < A i+ λ i A i λ + ρ i /ρ if A i λ < A i λ + ρ i max{θ a l if < a l +, θ } if al b if I = for i =,..., C is a valid superadditive approximation of Θ x I. Proof. The proof is in two steps. We first show that θ I Θ I for, I [, b] {, } and then prove that the lifting function θ I satisfies conditions 2 and 2 of Corollary 2 with d + =. Since θ is the valid superadditive approximation of Θ proposed by Gu et al. [4], θ Θ for [, b]. Now, for [, al, we deduce from Corollary 4 that Θ Θ θ = θ. Observe now that because ai a l for i C such that i < l, Θ a l + Θ for a l A l λ. Similarly, because a j a l for j C such that j > l, we have Θ a l + Θ for A l λ < b. Therefore, we conclude from Corollary 4 that for [a l, b] Θ =max{θ al +, Θ } max{θ al +, θ } = θ. We now verify that θ satisfies the conditions of Corollary 2. The fact that condition 2 is satisfied follows from Gu et al. [4]. To verify condition 2, we must show that y y + θ + θ θ for y,, y + [, b]. For y [, b], [, a l and y + [, b], we obtain that y y y + y + θ + θ = θ + θ θ θ 24

12 since it is easily verified that θ u θ u for u [, b]. For y [, b], [al, b] and y + [, b], we have y y θ + θ =max{θ + θ al y +, θ + θ } y + max{θ al y + y + +, θ } = θ since y + a l. We next prove in Theorem 7 that the approximation θ we propose is non-dominated and maximal. In the proof, we make use of the following result of Gu et al. [4]. Theorem 6. Function θ is a non-dominated approximation of Θ that satisfies θ y +θ θ y+ for y,, y + [, b]. Furthermore, it is maximal over [, b] {}. Theorem 7. The function θ I is a superadditive approximation of Θ I that is non-dominated and maximal over [, b] {, }. Proof. We first prove that θ I is non-dominated. When al = a, Θ = Θ and θ = θ. It follows from Theorem 6 that θ is non-dominated. It is therefore sufficient to consider the case where a l < a. Assume by contradiction that θ is dominated, i.e. there exists θ : [, b] {, } R such that θ I θ I for, I [, b] {, } and θ I > θ I for some, I [, b] {, }. It follows from Theorem 6 that I =. It is easily verified that θ θ if < a l = θ a l + if al A l λ θ 25 if A l λ < b using an argument similar to that of Theorem 5. We now consider three cases. If [, a l, then θ < θ Θ. Because Θ = when [, al λ], we conclude that a l λ, a l with Θ = and θ = θ <. Now consider = Al λ. Clearly, A l λ, A l λ]. It follows that θ l 2, l ]. On the first hand, if θ = l, then θ = Θ l = l. Furthermore, l + θ = θ + θ θ Θ = l. It follows that θ = θ =, which is the desired contradiction. On the other hand, if l 2 < θ < l, then Al λ < A l λ + ρ l with ρ l = a l a λ >. It follows that > a λ. Furthermore, θ + θ > θ + θ l = Θ θ which is the desired contradiction. = a + λ ρ + l a l ρ l + ρ = l 2 If a l, A l λ], then θ = θ a l + and Θ = Θ a l +. Also, because θ < θ Θ, it follows that a l A j λ, A j λ+ρ j for some j {,...,l} such 2

13 that ρ j >. Define now = A j +a l λ. From Theorem 3 and Theorem 5, Θ = θ = j. Clearly, a j ρ j, a j a λ, a j. From 24, we obtain j θ θ + θ > θ + θ = θ a l + + θ = j, which is the desired contradiction. 3 If A l λ, b], then θ = θ and Θ = Θ. This is a contradiction to the fact that θ is non-dominated over [, b]. Second, we prove that θ I is a maximal approximation of Θ I over [, b] {, }. Let E [, b] {, } be the maximal set of Θ I. We show that if θ I < Θ I for some, I [, b] {, }, then, I / E. When I =, the proof reduces to that of Theorem 6 given in [4]. Assume therefore that I =. In the case [, a l, we must have a l λ, a l since θ u = Θ u when u al λ. Let = A l λ. Then Θ = l. Note also that > A l λ and so Θ = l. It follows that Θ 2 = min{θ, Θ Θ } = min{, } =. We conclude that, / E. 2 If [a l, A l λ], we must have a l A j λ, A j λ+ρ j for some j {,..., l} with ρ j >. Consider = A j + a l λ. Then, Θ = Θ = j and > a j ρ j = a λ. We conclude that Θ. Furthermore, during the sequential lifting procedure, we have Θ 2 = min{θ, Θ Θ } = j, showing that, / E. 3 If A l λ, b], then θ = θ and Θ = Θ. Therefore, the proof reduces to that that of Theorem 6. In the remainder of this paper, we define θ := M for < with M C. This is not restrictive since a j for j N. Using this definition, we see that the result of Theorem 5 fits the mold presented in Theorem when a := a l Case ii: C + = In this case, it is easily seen from 8 that Θ I = Θ for, I [, b] {, }. Therefore, it is sufficient to verify 2 to guarantee that lifting is sequence independent. Theorem 8 naturally follows from Theorem 6. Theorem 8. Function θ I := θ is a valid superadditive approximation of Θ I that is nondominated and maximal over, I [, b] {, }. Here also, we observe that the approximation θ we propose can be obtained from Theorem by setting a := a. 3

14 4.2.3 Case iii: C + = 2 In this case, we see that C = C + because of Corollary 8. Let C = {, 2} with a a 2 and b = a +a 2 λ. We derive the exact lifting function of the corresponding generalied cover inequality next. Theorem 9. The lifting function of the generalied cover inequality is { if a λ if I = if a Θ = λ a + a 2 λ I if b if I =. 26 Observe that Θ is identical to Θ with C = {, 2}. Because Θ is simple, a non-dominated and maximal valid superadditive approximation can easily be derived. Theorem 2. The function θ θ = { I θ if < a 2 if a 2 a + a 2 λ if I = if I = 27 is a valid superadditive approximation of Θ that is non-dominated and maximal over, I [, b] {, }. It is easy to verify that θ = max{θ a2 +, θ }. Therefore, θ takes the form described in Theorem with a := a Building a Superadditive Approximation of Θ when K 2 Similar to the derivation of the superadditive approximation when K =, we present first conditions that follow from Theorem 4 and Proposition 5 that ensure sequence independent lifting. Again, we define d + := max{ S : S N + \C +, j S a j b} and denote K + := min{k, d + }. Corollary 2. For PS + with K 2, lifting is sequence independent over [, b] {, } if y y + Θ + Θ Θ 28 y y + Θ + Θ Θ h {,..., K + } 29 h h y y + Θ + Θ Θ h {,...,K + } 3 h h + where y,, y + [, b]. Because the conditions of Corollary 2 are most stringent when K + = K, we will show that the approximation we propose satisfy conditions 28-3 when K + = K. The derivation of an approximation is more difficult in this case since the function proposed must satisfy superadditive conditions over [, b] {, K + } rather than [, b] {, }. We observe however that because θ for h 2 is not used in the derivation of lifting coefficients, it is not necessary that the approximation over [, b] {2,..., K} be strong. We take advantage of this observation in developing θ. We first build a strong approximation of Θ over [, b] {, }. We then extend it into a valid approximation over [, b] {2,...,K}. In this case, there are four cases: i C C + and C + = K, ii C C + and C + K, iii C = C + and C + = K +, and iv C = C + and C + K. 4

15 4.3. Case i: C C + and C + = K Let C + = {l,...,l K } with l < l 2 < < l K. Because a i a j for i, j C such that i < j, we have a l a lk. We denote Âi = i j= a l j and Âh i = h j=hi+ a l j for h =,...,K and for i =,...,h. When i =, we define Âh = for h =,..., K. We next give in Theorem 22 the exact lifting function of the generalied cover inequality. It has a form that is similar to that presented in Corollary 4. Theorem 22. The exact lifting function of the generalied cover inequality is Θ if h = h if Âh λ Θ = h if  h λ < < h Âh  h if h =,...,K. max i=,...,h {Θ i + i} if  h b 3 Proof. Since it is easy to verify the value of Θ for [, b] and Θ for [,  h and h =,..., K, we only derive the value of Θ for [  h, b] and h =,...,K. Fix [Âh, b] and h =,..., K. Define T := {j C : j l i, i =,...,h} and assume that T = {k,..., k C h } with k < < k C h. Let s be the only index such that Âh λ + s j= a k j < Âh λ + s j= a k j. It is easy to verify that the solution x defined as { if j x {l,..., l j = h } {k,..., k s } if j {k s+,...,k C h } is optimal for Θ and that Θ = h + s. First we prove that max i=,...,h {Θ Âh i + i} Θ h. For i {,...,h}, we define x i as the solution obtained by setting the i largest elements of {l,..., l h } to, i.e. The solution x i satisfies j C a j x i j = j C x i j = { x j if j C\{l hi+,..., l h } if j {l hi+,..., l h }. a j x j + h t=hi+ a lt b + h t=hi+ a lt = b + Âh i and also x i j = x j + j C j C h K h + i K. t=hi+ It follows that x i is a feasible solution for the problem Θ Âh i with objective value Θ h i, i.e. Θ Âh i Θ h i. We conclude that maxi=,...,h {Θ Âh i + i} Θ h. Second we prove that max i=,...,h {Θ Âh i +i} Θ h. Define M = {j {l,...,l h } : j k s +}. Consider the solution ˆx defined as { x ˆx j = j if j C\M if j M. The solution ˆx satisfies a jˆx j = a j x j + a j b + Âh M j C j C j M 5

16 and also ˆx j = x j + K h + M K. j C j C j M It follows that ˆx is a feasible solution to the problem Θ Âh M with objective value Θ h M = h M + s. We now prove that ˆx is an optimal solution to Θ Âh M. Because  h λ + s j= a k j, Âh λ + s j= a k j ], we have that Âh M Âh λ + s j= h M a kj Âh M = j= a lj λ + s a kj = A ks λ. The last equality holds because {l,..., l h M } {k,..., k s } = {,..., k s }. Similarly, we can show that Âh M > A k sλ. This follows from 22 that Θ Âh M = Θ Âh M = ks = h M +s. It implies that ˆx is an optimal solution for Θ Âh M and therefore Θ Âh M = Θ h M. As a consequence, we have max i=,...,h {Θ Âh i + i} Θ h. Note that a closed form expression for Θ can easily be derived. However, expressing Θ h as in Theorem 22 has advantages when deriving a valid superadditive approximation; see proof of Proposition 23. Observe that the lifting function Θ on [, b] {, } is identical to Θ after replacing a l with a l. This suggests that a good approximation of Θ can be obtained by setting θ equal to θ I for h = I {, } after replacing al by a l. The function θ over [, b] {2,..., K} is then built from the value of θ over [, b] {, }. The above construction yields a valid approximation of Θ that is presented next. j= Proposition 23. The function θ { θ if < a l θ = max{θ a l +, θ } if al b h h j sup θ È h {= j= j:j,j=,...,h} j= if h = if h = if h = 2,...,K 32 is a valid superadditive approximation of Θ over, [, b] {,,..., K}. Proof. The proof is in two steps. We first show that θ is valid and then show that θ satisfies the conditions of Corollary 2. The proof that θ is valid is by induction over h. For h = and h =, the fact that θ Θ for [, b] was proven in Theorem 5. Assume that we have already proven that θ Θ for [, b] and h =,...,t. We want to show that this result still holds for h = t +. For [, b], we define and R := R 2 := sup t+ θ È t+ {= j= j:j,j=,...,t,t+ [,a l } j= sup t+ θ È t+ {= j= j:j a l,j=,...,t+} j= j j Clearly, θ t+ max{r, R 2 }. We now prove that R Θ t+ and R2 Θ t+. For R, we observe 6

17 first that θ t+ = θ t+ since t+ [, a l. We have R sup {= È t+ j= j:j,j=,...,t,t+ [,a l } t j È sup { θ t {= j= j: j,j=,...,t} j= = θ Θ Θ t t t + θ t { } j= j + θ t + t+ where the first inequality holds because of the superadditivity of θ over [, b] {, }. For R2, we have R 2 t+ È sup { θ t+ {= j= j:j a l,j=,...,t+} j= t+ sup {= È t+ j= j:j a l,j=,...,t+} = È sup t+ {= Â t+ t+ θ + t + Θ j= j:j a l,j=,...,t+} {θ j a l + } j= j t + a l t + al {θ Â t+ t+ + t + } + t + Θ } + t + }. t + where the third inequality follows from the fact that θ is non-decreasing and Ât+ t +a l, and the last inequality follows from Theorem 22. We now prove that θ satisfies the conditions of Corollary 2. Condition 28 has already been established in Theorem 5. Condition 3 is satisfied because of the way θ is defined for h 2. We therefore only need to prove condition 29. Let y,, y + [, b] and h {2,..., K}. We have y y θ + θ = θ h + sup {= È h j= j:j,j=,...,h} { = È sup { h {= j= j: j,j=,...,h} h θ j= j h È j sup { θ h {= j= j:j,j=,...,h} j= {y+= È h h θ j= h j sup { θ j= j: j,j=,...,h} j= j } y + θ } + θ h + y } = θ } y + where the first inequality holds because of the superadditivity of θ for, [, b] {, }. Clearly, deriving the value of θ for h 2 from its value for h = and h = significantly reduces the amount of work to be done to derive a superadditive approximation. Furthermore, we show next in Theorem 24 that the approximation derived in this manner is strong. The proof of Theorem 24 follows that of Theorem 7. The only difference is that the coefficient a l must be replaced with a l in the definition of Θ and θ. Theorem 24. The function θ is a non-dominated and maximal superadditive approximation of Θ over [, b] {, }. h 7

18 We observe again that θ has the form described in Theorem if we set a := a l. We next illustrate the strength of lifted generalied cover inequalities. The example is obtained from Gu et al. [4] page 22 by adding an cardinality constraint. Example 3. Consider PS = conv{x {, } 5 : 8x + 7x 2 + 6x 3 + 4x 4 + 6x 5 22, x 3 + x 4 + x 5 2}. Clearly, C = {, 2, 3, 4} is a generalied cover with λ = 4 j= a j 22 = = 3 and a l = 6. Figure shows both the exact lifting function Θ and the superadditive approximation θ we propose. Assume that we reintroduce x 5 through sequence independent lifting. Let α 5 be its lifting coefficient. Using the traditional superadditive lifting function from [4], we would obtain α 5 = θ 6 = 2. Using our two-dimensional superadditive approximation, we obtain α 5 = θ 6 =. Furthermore, the resulting inequality x + x 2 + x 3 + x 4 + x 5 3 is facet-defining for PS. a Θ and θ b Θ and θ Figure : Exact lifting function and superadditive lifting function of Example Case ii: C + C and C + K Since C + K, the lifting function Θ h is identical to Θ for, [, b] {, }. Therefore, we can easily build a superadditive approximation of Θ based on θ. Theorem 25. The function θ h = θ for, [, b] {,...,K} is a valid superadditive approximation of Θ that is non-dominated and maximal over [, b] {, }. Proof. From Corollary, we conclude that for [, b], Θ K Θ K Θ Θ = Θ. It follows that Θ h Θ θ = θ for [, b] and h {,...,K}. Furthermore, for, 2 [, b] and h, h 2 {,..., K} such that + 2 [, b] and h + h 2 K, we have 2 θ + θ = θ + θ 2 θ = θ, h h 2 h + h 2 showing that the conditions of Corollary 2 are satisfied. The non-dominance and maximality of θ follows directly from the fact that Θ = Θ = Θ and the fact that θ is a non-dominated and maximal approximation of Θ ; see Theorem 6. Again, θ has the form described in Theorem if we set a := a Case iii: C = C + and C = K + In this case, C = C + = {,...,K + }. Using the similar argument of Theorem 22, we derive an expression for the exact lifting function Θ in Corollary 26 that better describes the relations between Θ for h and Θ. This result will be used to derive a superadditive approximation of Θ in Theorem 27. 8

19 Corollary 26. For, [, b] {,...,K}, Θ h if A h+ λ Θ = h + if A h+ λ < < A h h+ Â h+ max i=,...,h+ {Θ i + i} if A h+ b where Âh i = h j=hi+ a j for h =,...,K + and i h. Theorem 27. The function θ if h = θ replacing al with a 2 if h = θ = h h j sup θ } if h {2,..., K} È { h {= j= j,j,j=,...,,h} j= if h = if h =,...,K is a valid superadditive approximation of Θ over [, b] {,,..., K}. Proof. The proof that θ satisfies the conditions of Corollary 2 is identical to that of Theorem 23. We now prove that Θ θ for, [, b] {,..., K}. We consider the following three cases. h =. Clearly, Θ = Θ for [, b] and so θ Θ. 2 h =. For [a +a 2, b], it follows from the fact that a a K+ that Θ = max{θ a a 2 + 2, Θ a 2 +, Θ } = Θ. Furthermore, since for [a + a 2, b], it follows from 25, that θ = θ, we conclude that θ Θ. Therefore, it is sufficient to show that θ Θ for [, a + a 2. Because Θ is a step function and because θ is non-decreasing, it is sufficient to verify that θ a +a 2λ Θ a+a 2λ and θ a+a 2ǫ Θ a+a 2ǫ for all sufficiently small positive ǫ to prove that θ Θ for [, a +a 2. For = a +a 2 λ, it follows from 5 and 35 that θ = max{θ, θ a2 + } = max{, } = = Θ. For ẑ = a + a 2 ǫ, we have that ẑ ẑ ẑ ẑ θ = max{θ, θ a2 + } = max{2, 2} = 2 = Θ since a λ < ẑ a 2 a + a 2 λ. 3 h 2. The proof is by induction and is similar to that of Proposition 23. Assume that the result holds for h =,..., t. We define R := sup t+ θ È t+ {= j= j:j,j=,...,t,t+ [,a2} j= j 37 and R 2 := sup t+ θ È t+ {= j= j:j a2,j=,...,t+} j= j 38 9

20 and observe that θ t+ max{r, R 2 }. Similar to Proposition 23, it can be proven that R Θ t. For R2, we have t+ R 2 È j a 2 sup { θ + } t+ {= j= j:j a2,j=,...,t+} j= t+ j= sup j t + a 2 + t + } È {θ t+ {= j= j:j a2,j=,...,t+} = È sup t+ {= Â t+2 t+ θ + t + Θ j= j:j a2,j=,...,t+} {θ t + a2 + t + } Â t+2 t+ + t + Θ. t + where the third inequality follows from the fact that θ is non-decreasing and Ât+2 t+ t+a 2 and the last inequality holds because of Corollary 26. The following theorem show that the proposed superadditive approximation is strong. Theorem 28. The function θ is a non-dominated and maximal superadditive approximation of Θ over [, b] {, }. Proof. The proof of non-dominance over [, b] {, } is similar to that of Theorem 24. Next, we show θ is maximal. Let E to denote the maximal set of Θ. We show that if θ I < Θ I for, I [, b] {, }, then, I / E. Since θ = θ for [, b] and θ was shown to be maximal in Theorem 6, we may assume that I =. a 2 a λ. From Theorem 3 and Theorem 5, we know that θ = θ = Θ = Θ for [, b]. From 36, we derive that θ = Θ for [, a2, θ = Θ for [a 2, a +a 2 λ] and Θ = Θ for a +a 2 λ, b]. Therefore, it is sufficient to consider [, a 2 such that Θ 2. Define now = a + a 2 λ. Clearly, > a λ. Therefore, Θ = Θ =. We conclude that Θ 2 = min{θ, Θ Θ } = min{, } = < Θ. This shows that, / E. 2 a 2 > a λ. From Theorem 5, Theorem 27 and the fact that a 2 > a λ, we know that θ = θ = θ for [, b]. Because θ < Θ, we either have [, a 2 or A j λ, A j λ + ρ j where j 2 and ρ j >. If [, a 2, define = a + a 2 λ. Then, > a λ and Θ = Θ =. We obtain Θ 2 = min{θ, Θ Θ } = < Θ. This shows that, / E. If A j λ, A j λ + ρ j, we define = A j+ λ. Then, > a λ. Also, we have Θ = Θ = j and Θ =. We conclude that Θ 2 = min{θ, Θ Θ } = j < Θ, which shows that, / E. Again, we observe that θ has the form presented in Theorem if a := a 2. 2