Cover and Pack Inequalities for (Mixed) Integer Programming

Transcription

1 Annals of Operations Research 139, 21 38, 2005 c 2005 Springer Science + Business Media, Inc. Manufactured in The Netherlands. Cover and Pack Inequalities for (Mixed) Integer Programming ALPER ATAMTÜRK atamturk@ieor.berkeley.edu Department of Industrial Engineering and Operations Research, University of California, Berkeley, CA , USA Abstract. We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 0-1 knapsack set, the mixed 0-1 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a unified presentation of the inequalities based on covers and packs and highlight the connections among them. The focus of the paper is on recent research on the use of superadditive functions for the analysis of knapsack polyhedra. We also present some new results on integer knapsacks. In particular, we give an integer version of the cover inequalities and describe a necessary and sufficient facet condition for them. This condition generalizes the well-known facet condition of minimality of covers for 0-1 knapsacks. Keywords: integer programming, knapsack polyhedra, lifting, superadditive functions In the last two decades there have been major advances in our capability of solving mixed integer programs (MIPs). Strong cutting planes obtained through polyhedral analysis of fundamental substructures of MIPs contributed to this success substantially by strengthening their linear programming (LP) relaxations. Polyhedral cuts based on problem substructures have already become standard features in leading optimization software packages. MIP solvers in these packages automatically identify knapsack, set packing, fixed-charge flow substructures of problems and generate polyhedral cutting planes specifically defined for these substructures. The use of such structural cuts improves the performance of the solvers dramatically. In this paper we review strong inequalities for single-constraint relaxations of (mixed) integer programs. Since every constraint of a mixed integer program defines a knapsack set, strong valid inequalities for the relevant knapsack relaxations can be used as cutting planes for solving MIPs in a branch-and-cut approach. Furthermore, it is also possible to generate other knapsack relaxations by aggregating constraints into one. Gomory cuts from individual simplex tableau rows may also be viewed in this context. Indeed, generating cutting planes from single-constraint relaxations is currently the most effective general method in state-of-the-art MIP software systems. Moreover, strong cuts for many structured problems can be obtained as knapsack cuts by building appropriate knapsack relaxations for them. Therefore, a good The author is supported, in part, by NSF Grants and

2 22 ATAMTÜRK understanding of fundamental knapsack polyhedra is of utmost importance in integer programming. Here we consider the following fundamental knapsack sets. The 0-1 knapsack set: The mixed 0-1 knapsack set: K B = conv{x {0, 1} N : ax b}. K MB = conv{x {0, 1} N, y R M + : ax + gy b}. The integer knapsack set: K I = conv{x Z N + : ax b, x u}. The mixed integer knapsack set: K MI = conv{x Z N +, y RM + : ax + gy b, x u}. We assume that a, u Q N, g Q M, and b Q with nonzero entries. Without loss of generality, u is an integral vector with positive entries, and, if necessary by complementing variables, a is a vector with positive entries. Throughout, if M = or g > 0, we assume that 1 u i b/a i for all i N. Note that in this case, u i can be reduced to b/a i without changing the set of feasible points if u i > b/a i. Then the knapsack sets are full-dimensional. In figure 1 we illustrate the hierarchy of the knapsack relaxations. The mixed-integer knapsack set is the most general one and, therefore, valid inequalities for it are also valid for the three other knapsack sets. When the upper bound for integer variables is 1, we obtain the mixed 0-1 knapsack set, whereas when there are no continuous variables, we obtain the integer knapsack set. Finally, the 0-1 knapsack set is the simplest and the most specialized one. Valid inequalities for mixed 0-1 knapsack and integer knapsack sets are also valid for the 0-1 knapsack set. The 0-1 knapsack set is probably the most studied and best understood knapsack set. Seminal works on the 0-1 knapsack polytope date back to the seventies (Balas, 1975; Hammer, Johnson, and Peled, 1975; Wolsey, 1975). The main topic of these Figure 1. A hierarchy of relaxations and validity of inequalities.

3 COVER AND PACK INEQUALITIES FOR (MIXED) INTEGER PROGRAMMING 23 early papers are facets based on covers and extensions of covers through sequential lifting (Padberg, 1979b). Balas and Zemel (1978) describe bounds on the lifting coefficients and other facets of the 0-1 knapsack set obtained by simultaneous lifting (Zemel, 1978). See Gu, Nemhauser, and Savelsbergh (1998, 1999a) and Zemel (1989) for complexity and computational aspects of sequential lifting of knapsack covers. Padberg (1980) gives a generalization of covers referred to as the (1, k)-configuration for packing problems, including 0-1 knapsacks. Weismantel (1997) describes the socalled weight inequalities based on feasible sets, which we refer to as packs here. Weismantel (1996) studies a special case of the knapsack set with two distinct coefficients. See also Padberg (1979a) for a survey on covering, packing, and knapsack problems. In an influential paper in computational integer programming, Crowder, Johnson, and Padberg (1983) show how to effectively incorporate strong valid inequalities for the 0-1 knapsack set in cut generation procedures for solving 0-1 programming problems. Using strong valid inequalities for 0-1 knapsacks, they have been able to solve much bigger problems than it was possible until that time. Even though the success of strong inequalities from single constraint relaxations of integer programs has been demonstrated in the early eighties, other than a few exceptions, almost all research on cutting planes for knapsacks has been limited to the 0-1 knapsack set. Special cases of the integer knapsack set with divisible coefficients have been investigated. Pochet and Weismantel (1998) describe the convex hull of the integer knapsack set with divisible coefficients. Pochet and Wolsey (1995) give the convex hull of the integer knapsack cover set ( replaced with ) with divisible coefficients and unbounded variables. They also give an extension that includes a continuous variable. See also Magnanti and Mirchandani (1993) for a study on a mixed integer set with divisible coefficients arising in telecommunication problems. Mazur and Hall (2002) study a polyhedron related to the integer knapsack cover set, referred to as the integer capacity cover polyhedron. Ceria et al. (1998) give a generalization of the 0-1 cover inequalities for mixed integer knapsacks. Martin and Weismantel (1997) extend the weight inequalities (Weismantel, 1997) to mixed integer knapsacks. Marchand and Wolsey (1999) give strong inequalities for the 0-1 knapsack set with a continuous variable. Richard, de Farias, and Nemhauser (2003) study mixed 0-1 knapsack set with bounded continuous variables. Atamtürk and Rajan (2002) study continuous and 0-1 knapsack sets with a single integer variable, that arise as a common relaxation of network design problems; see also Brockmüller, Günlük, and Wolsey (1996), van Hoesel et al. (2002) and Magnanti, Mirchandani, and Vachani (1993) for these sets. Atamtürk (2003) studies the facets of the mixed integer knapsack set in a more general setting than treated here, in which upper bounds are allowed for integer as well as continuous variables. Atamtürk (2002), Klabjan and Nemhauser (2002) study the flow set with integer variable upper bounds. Atamtürk and Rajan (2004) describe valid inequalities for the mixed-integer knapsack set lifted from two-integer variable restrictions. Marchand and Wolsey (2001) show that strong inequalities for many structured MIP problems can be obtained by a single application of mixed-integer rounding (Nemhauser

4 24 ATAMTÜRK and Wolsey, 1990) after aggregating constraints and complementing variables appropriately. In a follow-up paper, Louveaux and Wolsey (2003) show how mixed-integer rounding (MIR) and MIR lifting functions can be used to derive strong approximations of the facets for mixed integer sets. Here we will focus on inequalities obtained from two central objects, covers and packs, and their superadditive lifting. A cover is a subset of the variables that exceeds the knapsack capacity when all are set to their upper bounds. On the other hand, a pack is a subset of variables that leave a residual capacity when all are set to their upper bounds. In Section 1 we review the use of superadditive functions in lifting inequalities. In Sections 2 5 we describe the cover and pack inequalities for 0-1 knapsack, mixed 0-1 knapsack, mixed integer knapsack, and integer knapsack sets, respectively. We aim to give a unified presentation of the inequalities for these four knapsack sets and highlight the connections among the inequalities. Although most of the paper is a review of recent developments in the last six to seven years, Section 5 includes some new results on integer knapsack sets. In particular, we define integer cover inequalities and describe a necessary and sufficient facet condition for them. This condition generalizes the well-known facet condition of minimality of covers for 0-1 knapsacks. In Section 6 we conclude with a few open research questions. Throughout the paper, for v R T we let v(s) = i S v i for any S T, where v( ) = 0. We also let v + denote the nonnegative vector with entries max{v i, 0}, i T, whereas v denote the nonpositive vector with entries min{v i, 0}, i T. 1. Superadditive lifting An important and very useful concept in deriving strong valid inequalities is lifting of inequalities. Lifting refers to extending valid inequalities for low dimensional restrictions of polyhedra to ones that are valid in high dimensions. The concept of lifting has been introduced by Gomory (1969) in the context of the group problem. Padberg (1979b) described the sequential lifting procedure for 0-1 programming. Since then lifting has been studied and used extensively (Atamtürk, 2004; Balas and Zemel, 1978; Escudero, Garín, and Péres, 2003; Gu, Nemhauser, and Savelsbergh,1998, 1999a, 1999b, 2000; Johnson and Padberg, 1981; Louveaux and Wolsey, 2003; Marchand and Wolsey, 1999; Nemhauser and Vance, 1994; Padberg, 1973; Richard, de Farias, and Nemhauser, 2003; Sherali and Lee, 1995; Wolsey, 1976, 1977; Zemel, 1978, 1989), particularly, for 0-1 and mixed 0-1 programming problems. In this section we focus on the use of superadditive functions in lifting inequalities for K MI.Amore general description can be found in Atamtürk (2004). Let K MI (L, U) = conv{x R Z+ R, y RM + : a Rx R + gy d, x R u R } be the nonempty restriction of K MI obtained by fixing x i = 0, i L and x i = u i, i U for

5 COVER AND PACK INEQUALITIES FOR (MIXED) INTEGER PROGRAMMING 25 some partition (L, U, R)ofN; thus, d = b a U u U. Let π R x R + σ y π o (1) be a valid inequality for K MI (L, U) and define the lifting function : R R { } for (1) as (a) = π o max{π R x R + σ y : a R x R + gy d a, x R u R, x R Z R +, y RM + }. We let (a) = if the optimization problem above, referred to as the lifting problem, is infeasible. Since (1) is valid for K MI (L, U), the lifting problem is bounded for all a R for which it is feasible. The lifting function is useful in proving the validity of extensions of (1) to higher dimensions. Observe that is valid for K MI if and only if π R x R + π L x L + π U (u U x U ) + σ y π o (2) π L x L + π U (u U x U ) (a L x L + a U (x U u U )) (3) for all (x, y) K MI.Now, suppose that φ : R R is a superadditive lower bound on, i.e., φ(a 1 ) (a 1 ) and φ(a 1 ) + φ(a 2 ) φ(a 1 + a 2 ) for all a 1, a 2 R, and π i = φ(a i ) i L, and π i = φ( a i ), i U in (2). Then, since φ(a i )x i + φ( a i )(u i x i ) φ(a L x L +a U (x U u U )) (a L x L +a U (x U u U )), i L i U (4) inequality (2) is valid for K MI.Inother words, for any superadditive lower bound φ on, inequality (1) can be extended to a valid inequality π R x R + i L φ(a i )x i + φ( a i )(u i x i ) + σ y π o (5) i U for K MI. Furthermore, if φ(a i ) = (a i ) for all i L, φ( a i ) = ( a i ) for all i U, and inequality (1) defines a k-dimensional face of K MI (L, U), then inequality (5) defines an, at least, k + L + U dimensional face of K MI. 2. The 0-1 knapsack set In this section we consider the simplest of the knapsack sets, the 0-1 knapsack polyhedron K B = conv{x {0, 1} N : ax b}. Covers and packs play a central role in the analysis of knapsack polyhedra. A subset C of the index set N is called a cover if λ = a(c) b > 0, whereas a subset P of N is

6 26 ATAMTÜRK called a pack if µ = b a(p) > 0. We refer to λ as the excess of the cover C and to µ as the residual capacity for the pack P. For a cover C let us consider the restriction K B (N \ C, ), obtained by fixing all x i i N \ C to zero. Since the sum of the coefficients a i i C exceeds the knapsack capacity by λ>0, all variables x i i C cannot be one simultaneously in a feasible solution to K B (N \C, ). Therefore, the cover inequality (Balas, 1975; Hammer, Johnson, and Peled, 1975; Wolsey, 1975) x i C 1 (6) is valid for K B (N \ C, ). Cover inequality (6) defines a facet of K B (N \ C, ) ifand only if C is a minimal cover, that is, a(c \{i}) b for all i C. On the other hand for a pack P, consider the restriction K B (, P), obtained by fixing all x i, i P to one. Since any x i, i N \ P with coefficient a i greater than the residual capacity µ cannot take the value one in a feasible solution to K B (, P), the pack inequality (a i µ) + x i 0 (7) is valid for K B (, P). Sequential lifting and simultaneous lifting of cover inequalities have been studied extensively (Balas, 1975; Balas and Zemel, 1978, 1984; Gu, Nemhauser, and Savelsbergh, 1998, 1999a; Hammer, Johnson, and Peled, 1975; Wolsey, 1975; Zemel, 1989) to extend them to strong inequalities for K B. Here we illustrate the use of superadditive functions for sequence independent lifting of cover inequalities (Gu, Nemhauser, and Savelsbergh, 2000). Suppose the cover inequality (6) defines a facet of K B (N \ C, ), i.e., a i λ for all i C.For convenience, we write the inequality as λx i λ( C 1). (8) Suppose (wlog) C ={1, 2,..., C } and a 1 a 2 a C. Let A i = i k=1 a k for i {1, 2,..., C } and A 0 = 0. Then it is easy to see that the lifting function of (8) { (a) = λ( C 1) max λx i : } a i x i b a for a 0 can be expressed in a closed form as 0 if A 0 a A 1 λ, (a) = iλ if A i λ<a A i+1 λ, if a > b,

7 COVER AND PACK INEQUALITIES FOR (MIXED) INTEGER PROGRAMMING 27 where i {0, 1,..., C 1}. is a step function and is, unfortunately, not superadditive. Let {i C : a i >λ}={1, 2,...,p}. Then it is easy to check that the continuous function ϕ : R + R + defined as iλ if A i a A i+1 λ, ϕ(a) = iλ + (a A i ) if A i λ a A i, pλ + (a A p ) if A p λ a, where i {0, 1,...,p 1}, for a 0isasuperadditive lower bound on. Therefore, from Section 1, x i + i N\C ϕ(a i ) λ x i C 1 (9) is a valid inequality for K B. Moreover, since ϕ(a) = (a) for A i a A i+1 λ, inequality (9) is facet-defining for K B if A ik a k A ik +1 λ for some i k {0, 1,...,p 1} for all k N \ C. This result gives the characterization of the exact lifting coefficients within one unit due to Balas and Zemel (1978). Gu, Nemhauser, and Savelsbergh (2000) describe a stronger superadditive lower bound than ϕ for. The pack inequalities can also be lifted in a similar way. The lifting function of the pack inequality (7) { } (a) = max (a i µ) + x i : a i x i µ a (10) for a 0 can also be characterized easily because of the simple relation between coefficients of the constraint and the objective. Suppose (wlog) {i N \ P : a i >µ}= {1, 2,...,q} and a 1 a 2 a q. Then let A i = i k=1 a k for i {1, 2,...,q} and A 0 = 0. It can be checked that iµ + a if A i+1 + µ a A i, ψ(a) = iµ + A i if A i a A i + µ, qµ + A q if a A q + µ, where i {0, 1,...,q 1},isasuperadditive lower bound on. Therefore, inequality ψ( a i )(1 x i ) + (a i µ) + x i 0 (11) i P is valid for K B. Since K B (, P) isnot full-dimensional (when q > 0), it is harder to describe facet conditions of the lifted pack inequalities (11). Nevertheless, since (a) = ψ(a) for A i a A i + µ, inequality (11) defines an at least N q 1 dimensional face of K B if A ik a k A ik + µ, for some i k {0, 1,...,q 1} for all k P.

8 28 ATAMTÜRK Weismantel (1997) refers to inequalities a i (1 x i ) + (a i µ) + x i 0 (12) i P as the weight inequalities and describes facet conditions for (12) in a special case with a i = 1 for all i P. Since ψ(a) a for a 0, the lifted pack inequality (11) is at least as strong as the weight inequality (12). 3. The mixed 0-1 knapsack set By introducing continuous variables to the 0-1 knapsack set, we obtain the mixed 0-1 knapsack set. Since most practical problems contain integer variables as well as continuous variables, extending the concepts for 0-1 knapsacks to mixed 0-1 knapsacks is essential to be able to apply polyhedral results to such problems. Here we review the results of Marchand and Wolsey (1999) on the mixed 0-1 knapsack set K MB ={x {0, 1} N, y R M + : ax + gy b}. Let C N be a cover, i.e., λ = a(c) b > 0 and suppose there exists l C such that a l λ. Let us consider the restriction K MB (N \ C, C \{l}) obtained by fixing all x i, i C \{l} to one and all x i, i N \ C to zero. Thus, the restriction is given by a l x l + gy a l λ. The mixed integer rounding inequality (Nemhauser and Wolsey, 1990) λx l + g y 0 (13) is facet-defining for K MB (N \ C, C \{l}). Lifting (13) with x i, i C \{l}, weobtain the mixed 0-1 cover inequality min{a i,λ}(1 x i ) + g y λ. (14) Mixed 0-1 cover inequality (14) is facet-defining for K MB (N \ C, ). In contrast with 0-1 cover inequalities (6), minimality of cover C is not a necessary facet condition for the mixed 0-1 cover inequalities. Observe that inequality (14) generalizes the 0-1 cover inequality (6). If C is a minimal cover, i.e., a i λ for all i C, inequality (14) reduces to x i + g y λ C 1, (15) which is equivalent to (6) when y is zero. Now let P N be a pack, i.e., µ = b a(p) > 0 and suppose that there exists l N \ P such that a l µ. Fixing all x i, i N \ P {l} to zero and all x i, i P to

9 COVER AND PACK INEQUALITIES FOR (MIXED) INTEGER PROGRAMMING 29 one, we obtain the restriction K MB (N \ (P {l}), P), given by a l x l + gy µ. Lifting the mixed integer rounding inequality (a l µ)x l + g y 0 (16) with x i, i N \ P {l},weobtain the mixed 0-1 pack inequality (a i µ) + x i + g y 0. (17) In contrast with the 0-1 knapsack set, K MB (, P)isfull-dimensional and the mixed 0-1 pack inequality is facet-defining for K MB (, P). By computing the lifting functions of the mixed 0-1 cover inequality and mixed 0-1 pack inequality, we can extend them to strong valid inequalities for K BM. Marchand and Wolsey show that the lifting function of (14) is precisely ϕ, the superadditive lower bound for used in Section 2 for lifting the 0-1 cover inequalities. Therefore, lifting inequality (14) with x i, i N \ C using ϕ, weobtain the continuous cover inequality (Marchand and Wolsey, 1999) min{a i,λ}(1 x i ) + ϕ(a i )x i + g y λ, (18) i N\C which is facet-defining for K MB. On the other hand, the exact lifting function of (17) is ψ, which is the superadditive lower bound used to lift the 0-1 pack inequalities (7) for 0-1 knapsacks in Section 2. Therefore lifting (17) with x i, i P using ψ, weget this time the so-called reverse continuous cover inequality (Marchand and Wolsey, 1999) ψ( a i )(1 x i ) + (a i µ) + x i + g y 0, (19) i P which is also facet-defining for K MB. Strong inequalities for many structured mixed 0-1 programming problems can be obtained from their appropriate mixed 0-1 knapsack relaxations. For instance, inequalities (18) and (19) are also good approximations of lifted flow cover (Gu, Nemhauser, and Savelsbergh, 1999b) and lifted flow pack (Atamtürk, 2001) inequalities for fixed-charge network flows. Recently (Richard, de Farias, and Nemhauser, 2003) study a generalization of K MB, where they consider upper bounds on the continuous variables as well. Some of the facets of this generalization can be obtained by complementing and aggregating continuous variables and using the results of Marchand and Wolsey (1999). But as shown in Richard, de Farias, and Nemhauser (2003), the mixed 0-1 knapsack set with bounded continuous variables has other facets that can be obtained by lifting inequalities with the bounded continuous variables.

10 30 ATAMTÜRK 4. The mixed integer knapsack set For mixed integer programs in which integer variables are not restricted to 0 and 1, the mixed integer knapsack set K MI ={x Z N +, y RM + : ax + gy b, x u} is the appropriate single-constraint relaxation to consider for generating valid inequalities. In this section we review the results of Atamtürk (2003) on the mixed integer knapsack polyhedron. Let C N beacoverifλ = u ia i b > 0. ForacoverC suppose that there exists l C such that µ = u l a l λ>0. Fixing all x i, i C \{l} to u i and all x i, i N \ C to zero, we obtain the restriction a l x l + gy µ. Defining η = µ/a l and r = µ µ/a l a l,itcan be seen that the MIR inequality (a l r)(u l x l ) + g y (u l η)r λ (20) defines a facet of this restriction. It is shown in Atamtürk (2003) that the lifting function l (a) = (u l η)r λ max{ (a l r)(u l x l ) + g y : a l x l + gy µ, x l Z +, y R + } can be expressed in closed form as (η u l 1)(a l r) if a < λ, k(a l r) if ka l a < ka l + r, l (a) = a (k + 1)r if ka l + r a < (k + 1)a l, a ηr if a µ, (21) where k {η u l 1,η u l,...,η}. The lifting function l is not superadditive on R;however, it is superadditive on R + and on R, separately. Therefore using l to lift (20) with x i, i C \{l}, weobtain the mixed-integer cover inequality l ( a i )(u i x i ) + g y (u l η)r λ. (22) Inequality (22) is facet-defining for K MI (N \ C, ). Observe that for the mixed 0-1 knapsack set, since η = u l = 1, for every l the mixed-integer cover inequality reduces to the same mixed 0-1 cover inequality (14). Next we extend (22) for K MI with x i, i N \ C. Let C + ={i C \{l} : a i (u l η + 1)a l }. Suppose (w.l.o.g.) C + ={1, 2,...,s} and a 1 a 2 a s. Then for a 0, the function γ l defined below is a superadditive lower bound on the lifting

11 COVER AND PACK INEQUALITIES FOR (MIXED) INTEGER PROGRAMMING 31 function of (22) u ih (u l η + 1)(a l r) if m ih a m ih + δ i, (u ih (u l η + 1) + k)(a l r) if m ih + δ i + ka l a m ih + δ i + ka l + r, u ih (u l η + 1)(a l r) if m ih + δ i + ka l + r a m ih γ l (a) = + a m ih δ i (k + 1)r + δ i + (k + 1)a l, ( usus (u l η + 1) + p ) (a l r) if m C + + pa l a m C + + pa l + r, u sus (u l η + 1)(a l r) ifm C + + pa l + r a m C + + (p + 1)a l, + a m C + (p + 1)r where δ i = a i (u l η+1)a l for i C +, u ih = i 1 k=1 u k +h, m ih = m (i 1)ui 1 +ha i for h {0, 1,...,u i } and i {1, 2,...,s} with m 0u0 = 0, m C + = m sus, k {0, 1,...,u l η}, and p Z +. Therefore, using γ l to lift (27) with x i, i N \ C,weobtain the valid inequality l ( a i )(u i x i ) + g y (u l η)r λ (23) γ l (a i )x i + i N\C for K MI. Inequality (23) is facet-defining for K MI if a i m C + + ηa l for all i N \ C (Atamtürk, 2003). Remark 1. Observe that if η = u l, then a l r = λ and (a) = max{a,λ} for a < 0. In this case, inequality (23) takes the simpler form γ l (a i )x i min{a i,λ}(u i x i ) + g y λ. (24) i N\C Now if λ a i for all i C and γ l (a i ) = 0 for all i N \ C, then (24) reduces to inequality (u i x i ) α + g y/(a l r), (25) given in Ceria et al. (1998), where α = λ/a l =u l η + 1; otherwise, inequality (24) is stronger than (25). On the other hand, let P N be a pack if µ = b i P u ia i > 0. For a pack P suppose that there exists l N \ P such that λ = u l a l µ>0. Fixing all x i, i P to u i and all x i, i N \ (P {l}) tozero we obtain again the restriction a l x l + gy µ. After rewriting the corresponding MIR inequality this time as (a l r)x l + g y µ ηr, (26) we lift it with x i, i N \ P,toobtain the mixed-integer pack inequality l (a i )x i + g y µ ηr. (27)

12 32 ATAMTÜRK Inequality (27) defines a facet of K MI (P, ). Observe that in the case of mixed 0-1 knapsack set, since µ = r and η = 1, for every l the mixed integer pack inequality (27) reduces to the same mixed 0-1 pack inequality (17). Next we extend (27) with x i i P. Let P + ={i N \ P : a i ηa l }. Suppose (w.l.o.g.) P + ={1, 2,...,t} and a 1 a 2 a t. Then a superadditive lower bound for the lifting function for (27) is given by u ih ηr + a if m ih δ i a m ih, (u ih η + k)r + a if m ih δ i (k + 1)a l + r a m ih δ i ka l, u ih ηr + m ih δ i if m ih δ i (k + 1)a l ω l (a) = (k + 1)(a l r) a m ih δ i (k + 1)a l + r, u sus ηr + pr + a if m P + (p + 1)a l + r a m P + pa l, m P + + u su s ηr if m P + (p + 1)a l a m P + (p + 1)a l + r, (p + 1)(a l r) where δ i = a i ηa l for i P +, u ih = i 1 k=1 u k + h, m ih = m (i 1)ui 1 ha i for h {0, 1,...,u i }, i {1, 2,...,t} with m 0u0 = 0, and m P + = m tu t, k {0, 1,...,η 1} and p Z +. Now using ω l to lift the mixed-integer pack inequality (27) with x i, i P, we obtain the valid inequality l (a i )x i + ω l ( a i )(u i x i ) + g y µ ηr (28) i P for K MI. Inequality (28) is facet-defining for K MI if a i m P + + u l η + 1 for all i P (Atamtürk, 2003). Remark 2. Ifη = 1, then r = µ and (a) = (a µ) + for a > 0. In this case, inequality (28) reduces to ω l ( a i )(u i x i ) + (a i µ) + x i + g y 0. (29) i P If ω( a i ) = a i for all i N \ P, then (29) reduces to the weight inequality (Martin and Weismantel, 1997) a i x i + (a i µ) + x i + g y b µ; (30) i P otherwise, (29) is stronger than the weight inequality (30).

13 COVER AND PACK INEQUALITIES FOR (MIXED) INTEGER PROGRAMMING Integer knapsack set In this section we consider the integer knapsack set K I ={x Z N + : ax b, x u}. Let C N be a cover if λ = u ia i b > 0 and consider the restriction K I (N \C, ) obtained by fixing all x i, i N \ C to zero. Extending the combinatorial argument on covers for the 0-1 knapsack set, since not all variables x i, i C can be at their upper bound simultaneously, inequality x i u(c) 1 (31) is valid for K I (N \ C, ). However, (31) is generally a weak inequality and may not even define a proper face of K I (N \C, )evenifc is a minimal cover, that is, \{k} u ia i b for all k C. Another way to write inequality (31) is (u i x i ) 1, (32) which states that at least one x i must be less than its upper bound. However, observe that if a i <λfor all i C, the left hand side of (32) is greater than 1 for all feasible solutions. This suggests the strengthening (u i x i ) λ/ā, (33) where ā = max a i,given in Ceria et al. (1998). Consider the example below. Example 1. Suppose the integer knapsack is given as K I = conv{x Z 2 + :5x 1 + 9x 2 45, x 1 6, x 2 4}. Let C ={1, 2} be an integer cover with λ = = 21. As it can be seen in figure 2, inequality (32) (6 x 1 ) + (4 x 2 ) 1orx 1 + x 2 9 does not support K I.However, the strengthened inequality (33) (6 x 1 ) + (4 x 2 ) 21/9 =3orx 1 + x 2 7 defines a facet of K I for this example.

14 34 ATAMTÜRK Figure 2. Facets of the integer knapsack set. For 0-1 knapsacks, whenever C is a minimal cover, the cover inequality (6) defines afacet of K B (N \ C, ). Furthermore, it is also easy to see that, if C is a minimal cover, the bounds on the variables and the cover inequality give a complete description of K B (N \ C, ). As seen in figure 2, K I (N \C, ) has other non trivial facets besides the one defined by x 1 + x 2 7. Also it should be clear that, in general, K I (N \ C, ) may not have any facet of the form x i π o. Therefore, below we give a generalization of inequalities (33), whose coefficients are not restricted to zero and negative one. Proposition 1. Let C N beacover.for any ρ>0, the integer cover inequality min{a i,λ}/ρ (u i x i ) λ/ρ (34) is valid for K I. Proof. Rewriting a i x i b as a i(u i x i ) λ and strengthening the coefficients gives min{a i,λ}(u i x i ) λ.dividing this inequality by ρ and rounding up the coefficients to integers, we obtain (34). It is of interest to know when inequalities (34) define facets of K I (N \C, ). Below we give a necessary and sufficient condition for an important subclass of the integer cover inequalities to define facets of K I (N \ C, ). For a cover C and i,l C, let κ il = min{a i,λ}/a l and consider the integer cover inequality κ il (u i x i ) λ/a l. (35) Theorem 2. Let C N be a cover and l C be such that µ = u l a l λ 0. The integer cover inequality (35) is facet-defining for K I (N \ C, ) ifand only if a i min{λ, κ il a l r} for all i C \{l}, where r = µ µ/a l a l.

15 COVER AND PACK INEQUALITIES FOR (MIXED) INTEGER PROGRAMMING 35 Proof (Sufficiency). Suppose the condition of the theorem is satisfied. Consider the points x 1, x 2,...,x C defined as x l l = µ/a l, x l i = u i, i C \{l}, and x l i = 0 i N \ C. x i i = u i 1, x i k = u k, k C \{i} for i C \{l} with λ κ il a l r, and x i k = 0 k N \ C. x i i = u i 1, x i l = µ/a l +κ il, x i k = u k, k C \{i,l} for i C \{l} with λ>κ il a l r, and x i k = 0 k N \ C. By definition of µ,ifx i = u i for all i C \{l}, and x i = 0 for all i {l} N \ C, the knapsack constraint has a slack of µ. Therefore, x l is a feasible point with a slack of r in the knapsack constraint. Since κ ll = 1 and λ/a l = (u l a l µ)/a l =u l µ/a l, inequality (35) is tight at x l. For i C\{l}, suppose λ κ il a l r. Since a i λ, the point x i is feasible; and since a i λ implies that κ il = λ/a l, inequality (35) is satisfied at equality for x i. Otherwise, κ il a l r λ = u l a l µ or equivalently (µ r)/a l + κ il = µ/a l +κ il u l. Since we also have a i + r κ il a l, the point x i is feasible. Since κ ll = 1, cover inequality (35) is tight for x i. This shows that the described points are on the face of K I defined by (35). They are easily seen to be affinely independent. (Necessity) Ifa i < min{λ, κ il a l r}, then since κ il = a i /a l and r < a l,wehave (κ il 1)a l a i <κ il a l r. Inthis case, it is easy to check that the coefficients of inequality (36) satisfy l ( a i ) > κ il (a l r), thus (36) dominates κ il (a l r)(u i x i ) (a l r) λ/a l =(u l µ/a l 1)r λ, which is a rewriting of (35). Remark 3. The condition of Theorem 2 generalizes the facet condition of minimality of covers for 0-1 knapsacks to integer knapsacks. To see this, observe that for the 0-1 knapsack set, we have µ = a l λ<a l and r = µ. Since µ 0 implies that k il = 1 for all i C, Theorem 2 reduces to stating that x i C 1isfacet-defining for K B (N \ C, )ifand only if a i λ for all i C, that is, C is a minimal cover. Example 1 (cont.) Let C ={1, 2} be a cover as before with λ = = 21. For l = 1, we have µ = = 9 and r = 4. Then since λ/a 1 = 21/5 =5, κ 11 = 1 and κ 21 = 9/5 =2, the corresponding integer cover inequality is (6 x 1 ) + 2(4 x 2 ) 5orx 1 + 2x 2 9. On the other hand, for l = 2, we have µ = = 15 and r = 6. In this case, since λ/a 2 = 21/9 =3, κ 12 = 5/9 =1 and κ 22 = 1, the integer cover inequality is (6 x 1 ) + (4 x 2 ) 3orx 1 + x 2 7.

16 36 ATAMTÜRK Both of these inequalities satisfy the condition of Theorem 2, and hence define facets of K I. This is also illustrated in figure 2. The cover inequalities (35) can be strengthened by using the results on mixed integer knapsacks in Section 4, as shown in the necessity part of the proof of Theorem 2. For a cover C and l C such that µ = u l a l λ 0, let η = µ/a l +1 and r = µ µ/a l a l. Then setting the continuous variables y to zero in inequality (22), we obtain the lifted integer cover inequality l ( a i )(u i x i ) (u l η)r λ (36) γ l (a i )x i + i N\C for K I. Similarly for a pack P N and l N \ P such that λ = u i a i µ>0, we have the lifted integer pack inequality l (a i )x i + ω l ( a i )(u i x i ) µ ηr (37) i P for K I, where η and r are defined as before. 6. Final remarks Although the 0-1 knapsack set has been studied extensively since the seventies, research on integer and mixed integer knapsack sets is still limited. The study of integer polyhedra appears to be significantly harder than the study of 0-1 polytopes. One of the difficulties is that combinatorial arguments for 0-1 variables or extensions of these arguments do not lead to sufficiently strong results in the presence of integer variables. This point is illustrated in Section 5 for integer knapsacks. A possible explanation for this difficulty is that for 0-1 problems, feasible points are among the vertices of the LP relaxation, whereas for integer problems, feasible points generally lie deep in their LP relaxations. Another challenge is that integer polyhedra are much richer than 0-1 polytopes. For instance, even though the convex hull of a 0-1 polytope in two dimensions is trivial, the same is not true for an integer polyhedron. However, it is possible to list all facets of an integer knapsack polyhedron in two variables in polynomial time in the size of the data (Atamturk and Rajan, 2004). In Section 5 we give the facet condition for integer cover inequality (34) when ρ = a i for some i N. Itisanopen question what other values for ρ would yield facets. To gain a better understanding of integer knapsack polyhedra, it may be worthwhile to explore the connections between valid inequalities and primal methods for integer programming (Aardal, Weismantel, and Wolsey, 2002). Along this line are the inequalities of Weismantel (1996) based on the elements of the Hilbert basis of the cone of exchange vectors for a special 0-1 knapsack. It is also of interest to see in what ways the knowledge on cyclic group and master knapsack facets (Aráoz et al., 2003; Dash and Günlük, 2004) can facilitate the study of more general integer polyhedra.

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