E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....a mixed integer knapsack set, strong valid inequalities for K can be used as cutting planes for MIP. There are many important polyhedral studies on special cases of the mixed integer knapsack set K. The most studied is probably the 0 1 knapsack set (u = 1 and C = for which seminal works [5, 7, 19, 33] date back to 70 s; see also [16, 28, 32, 37] Crowder, Johnson, and Padberg [13] demonstrate the e#ectiveness of cutting planes from individual 0 1 knapsack constraints in solving 0 1 programming problems. Date: March 2002, December 2002. Alper Atamturk: Department of Industrial Engineering and ....

E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.

....Knapsack Integer Knapsack Mixed Integer Knapsack Figure 1. A hierarchy of relaxations and validity of inequalities. The 0 1 knapsack set is probably the most studied and well understood single constraint relaxation. Seminal works on the 0 1 knapsack polytope date back to 70 s (Balas [7], Hammer et al. 18] Wolsey [37] The main topic of these early papers are facets based on covers and extensions of covers through sequential lifting (Padberg [29] Balas and Zemel [8] describe bounds on the lifting coe#cients and other facets of the 0 1 knapsack set obtained by simultaneous ....

....of the pack P . For a cover C let us consider the restriction KB (N C, N C to zero. Since the sum of the coe#cients a i i C exceeds the knapsack capacity by # 0, all variables x i i C cannot be one simultaneously in a feasible solution to KB (N #) Therefore, the cover inequality [7, 18, 37] (6) is valid for KB (N #) Cover inequality (6) defines a facet of KB (N if and 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 KB (#, P ) obtained by fixing all P to one. Since any x i i P ....

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E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.

....used in the branch and cut algorithm. Let (x ) denote the current solution. Lifted Quadratic Cover Inequalities (Yaman [19] A subset C I such that i2C a i m2InC Tim M is called a quadratic cover. This notion is very close to that of cover for knapsack constraints (see Balas [4], Hammer et al. 12] and Wolsey [18] If C I is a quadratic cover, then the quadratic cover x ij (jCj 1)x jj is valid for PQH . To separate the quadratic cover inequalities, we use the branch and bound algorithm given in Yaman [19] For a given j 2 I , we de ne I 1 = fi 2 I : x ij = ....

Balas, E. (1975): Facets of the Knapsack Polytope. Mathematical Programming 8, 146-164

....paper presents an algorithm which solves large sized Strongly Correlated Knapsack Problems faster than uncorrelated Knapsack Problems usually are solved. In Section 2 it is shown that tight upper bounds may be derived by surrogate relaxation of the original weight constraint with a Balas cut [1]. The relaxed problem becomes a Subset sum problem when optimal surrogate multipliers are applied, and it is proved that an optimal solution to the relaxed problem which apply most possible items also will be an optimal solution to the original problem. Thus in Section 3 we use a 2 optimal ....

..... b 1 and x b = c j=1 w j ) w b . The IP solution which choses the rst b 1 items will be denoted the break solution x . As the rst b 1 items de ne a cover, we may impose the following additional constraint x j b 1; 3) to problem (1) without excluding any optimal solutions [1]. Since this two constrained knapsack problem however is very dicult to solve, we will surrogate relax (1) and (3) using multipliers 1 and S respectively. There is no loss of generality in this assumption, since the rst multiplier always can be chosen as 1 by appropriate scaling. Thus we get the ....

E. Balas (1975), \Facets of the Knapsack Polytope", Mathematical Programming, 8, 146-164.

....that the capacity of each edge is no more than the demand. Let t N t = The subproblem constraint is enforced by a knapsack cover (KC) inequality: d 08 a[ 2.1) Under certain conditions, these inequalities are facet defining, see [33] for example. Previous researchers [2, 17, 33] have considered an uncapacitated form of inequality (2.1) that forces the choice of at least one edge in . We can show that if only these weaker constraints are added to IP1, the IP LP ratio can still be as bad as . These researchers used lifting procedures to strengthen these basic ....

Egon Balas. Facets of the knapsack polytope. MP, 8:146-- 164, 1975.

....and partial multistar inequalities in great detail. In this section we move on to consider multistar inequalities which are not necessarily homogeneous. Consider the following polytope, which we denote by K(q, Q) Definition: K(q, Q) conv (y This is a 0 1 knapsack polytope (see, e.g. [7, 30]) the extreme points of which represent feasible allocations of customers to a single (arbitrary) vehicle. Now suppose also that we allow VRP instances in which one or more customers have qi 0. This does not affect the formulation (1) 4) except that it is necessary to define k(S) 1 ....

....negative. Using a standard construction [27] this can be reduced to a max fiow min cut problem and therefore solved in polynomial time. Given Theorem 7, the question then arises as to which knapsack inequalities to use. Initially, we experimented with lifted cover (LC) inequalities (see, e.g. [7, 25, 30]. However, we were unable to find any violated KLM inequalities at all for any of our test instances when we used LC inequalities, even though we tried several ways of generating them. Instead, we found that much better results were obtained by the naYve option of merely setting q and F Q. ....

E. Balas, "Facets of the knapsack polytope", Math. Program., vol. 8, pp. 146-64, 1975.

....easy to show. Theorem 3.2 The upper bound constraint x ij 1 defines a facet for PMKCP if and only if O i O i W j where O i = max k2N j Gammafig O k . 9 The polyhedral structure of single knapsack and multiple knapsack problems has been widely studied in the operations research literature [1][8] 9] 10] 4] 2] A well known result, proved in [4] is that all non trivial facets for the single knapsack problem are facets for the multiple knapsack problem. Given the proximity of the MKCP to the MKP, it is then natural to ask a similar question about which facets for the MKP are also facets ....

....P i2C O i W k . The cover is minimal with respect to k if P i2Cnfcg O i W k for all c 2 C. Suppose that C N is a minimal cover with respect to some knapsack k. The inequality X i2C x ik jCj Gamma 1 (3) is called the minimal cover inequality corresponding to C and k. It was shown in [1], 5] and [9] that the minimal cover inequality corresponding to C and k defines a facet for SK(C;O;W k ) 4] shows that all non trivial facets for SK(C;O;W k ) are facets for PMKP . Suppose the minimal cover C is composed of p N distinct colors, c 1 ; c 2 ; c p . Let S(c 1 ) S(c 2 ) ....

Balas, E. (1975) Facets of the knapsack polytope, Mathematical Programming, 8, 146-164.

....# N isa cover if # = # j C a j b 0. 13) 9 In addition, the cover C is said to be minimal if a j # # for all j # C. To each cover C, we can associate a simple valid inequality which states that not all variables x j for j # C can be set to one simultaneously . Proposition 2.1. [8, 59, 88, 106] Let C # N be a cover. The cover inequm445 # j#C x j # C 1 (14) is valid for K. Moreover, if C is minimal, then the inequmKHH (14) defines a facet of conv (KC ) where KC = K# x : x j =0,j# N C . Example 2.2. Consider the 0 1 knapsack set K = x # B6 :5x 1 5x 2 5x 3 5x 4 ....

.... for the 0 1 knapsack set this can be done e#ciently using a dynamic programming approach based on the following recursion formula, #M# k (u) min[#M (u) #M (u a k ) #M (a k ) Using such a lifting approach, facet defining inequalities for the 0 1 knapsack set have been derived [8, 59, 106] and embedded in a branch and bound framework to solve to optimality particular types of 0 1 integer programs [37] In theory, exact sequential lifting can be applied to derive valid inequalities for any kind of mixed integer set S. However, in practice, this approach is only useful to ....

E. Balas, Facets of the knapsack polytope, Mathematical Programming 8, 146 -- 164 (1975).

....variable (called the continuous 0 1 knapsack set) Y = y, s) # B n R 1 : X j#N a j y j # b s where a j 0 for j # N = 1, n , and b # 0. When s = 0, Y reduces to a binary knapsack set for which a large number of facet defining inequalities have been derived [1] [11] 20] 21] For such sets, the concepts of projection, lifting and cover have played an important role. Using separation heuristics, the resulting inequalities have been successfully used as cutting planes in various branch and bound and branch and cut systems [6] for problems containing ....

E. Balas, Facets of the Knapsack Polytope, Mathematical Programming 8, 146-164 (1975)

....knapsack problem, where jM j = 1. In analogy to the definition of MK let SK(N; f; F ) convfx 2 IR N j X i2N f i x i F; x i 2 f0; 1g; i 2 Ng denote the single knapsack polytope. Although a lot of emphasis has been put on studying the facial structure of SK(N; f; F ) see, for example, [B75], W75] HJP75] P75] BZ78] P80] MK and generalizations of it have not yet been studied to the same extent. In a few papers we find investigations in this direction. Crowder, Johnson and Padberg [CJP83] consider general 0 1 linear programs with no apparent structure: Let be given a matrix A 2 ....

....and a il : a i ; if l = k, i 2 V; 0; otherwise. Let (N; M; f; F ) be an instance of the multiple knapsack problem. Suppose, S is a minimal cover with respect to some k 2 M . Then, the minimal cover inequality X i2S x ik jSj Gamma 1 defines a facet for the polytope SK(S; f; F k ) [B75], HJP75] W75] By applying Lemma 2.2 we can conclude that this minimal cover inequality defines a facet for MK(S Theta M; f; F ) Similarly, let N 0 [fzg; N 0 N; jN 0 j = n 0 and z 2 N nN 0 be a (1; d) configuration with respect to some knapsack k. The (1; d) configuration ....

E. Balas, "Facets of the Knapsack Polytope", Mathematical Programming 8, 146 - 164 (1975).

....from University of S ao Paulo, Brazil. 1 This problem is NP hard (cf. K72] and has been extensively studied in terms of approximation algorithms (see, for instance, IK75] in terms of branch and bound methods (see, for example, MT91] and from a polyhedral point of view (see, for instance, [B75], HJP75] P75] W75] The generalized assignment problem is a generalization of the multiple knapsack problem where every item i may have a particular weight f ik for each knapsack k. The corresponding polyhedron was investigated in [GR90] Our motivation for studying the multiple knapsack ....

....are the minimal cover inequality and the (1,d) configuration inequality that we want to present now. Suppose that S N is a minimal cover with respect to some knapsack k 2 M . The inequality X i2S x ik jSj Gamma 1 6 is called minimal cover inequality corresponding to S and k. In [B75], W75] and [HJP75] it was shown that the minimal cover inequality corresponding to S and k defines a facet of SK(S; f; F k ) and, thus, of MK(S Theta M; f; F ) Another well known class of individual inequalities consists of the (1,d) configuration inequalities. Suppose that N 0 [fzg N is ....

E. Balas, "Facets of the Knapsack Polytope", Mathematical Programming 8, 146 - 164 (1975).

....problem; the other is the discovery of beautiful concepts and results associated with minimal covers, 1; k) configurations or the lifting and complementing of variables. Most of the polyhedral studies presented so far involve two basic and general objects: minimal covers (see for instance, [B75], HJP75] W75] and (1; k) configurations (cf. P80] Let N be a subset of items, let b denote the knapsack capacity and suppose, every item i 2 N has a weight W (i) 0. A set S N is a cover if P i2S W (i) b holds. The cover is minimal, if in addition P i2Snfsg W (i) b for all s 2 ....

....j 0 j 00 Gamma 2r b Gamma j 0 Gamma j 00 = b Gamma r Gamma r = jT 1 j j 0 Gamma r jT 1 j j 0 Gamma . This completes the proof. The inequalities defined in Proposition 2. 2 (i) have been presented in [Le93] and can be viewed as special lifted cover inequalities if N r 1 6= [B75], HJP75] W75] If N r 1 = but N j 6= for some j r 1, these inequalities are lifted (1; k) Gammaconfiguration inequalities ( P80] For the inequalities defined in (2.2) ii) the vector P i2T 1 e i e i 0 is a root. Moreover, the coefficient of item i 0 2 N j 0 is defined as the ....

E. Balas, "Facets of the Knapsack Polytope", Mathematical Programming 8, 146 - 164 (1975).

....is a cover if # = # j#C a j b 0. 13) 10 In addition, the cover C is said to be minimal if a j # # for all j # C. To each cover C, we can associate a simple valid inequality which states that not all variables x j for j # C can be set to one simultaneously . Proposition 2.1. [8, 59, 88, 106] Let C # N be a cover. The cover inequality # j#C x j # C 1 (14) is valid for K. Moreover, if C is minimal, then the inequality (14) defines a facet of conv (KC ) where KC = K # x : x j = 0, j # N C . Example 2.2. Consider the 0 1 knapsack set K = x # B 6 : 5x 1 5x 2 ....

.... for the 0 1 knapsack set this can be done e#ciently using a dynamic programming approach based on the following recursion formula, #M# k (u) min[#M (u) #M (u a k ) #M (a k ) Using such a lifting approach, facet defining inequalities for the 0 1 knapsack set have been derived [8, 59, 106] and embedded in a branch and bound framework to solve to optimality particular types of 0 1 integer programs [37] In theory, exact sequential lifting can be applied to derive valid inequalities for any kind of mixed integer set S. However, in practice, this approach is only useful to ....

E. Balas, Facets of the knapsack polytope, Mathematical Programming 8, 146 -- 164 (1975).

....as a 0 1 knapsack problem with a single integer variable representing the capacity of the knapsack. The closely related knapsack problem with a single continuous capacity variable is studied by Marchand and Wolsey [18] They employ valid inequalities of the standard 0 1 knapsack problem (see Balas [1], Hammer et al. 13] Wolsey [21] to obtain valid inequalities for the extended model by projection and lifting. The edge capacity polytope itself has also been studied by Brockm uller et al. 7, 8] who derive valid and facet de ning inequalities for the edge capacity polytope. Magnanti et al. ....

E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146-164, 1975.

....cuts themselves. 4.4.4 Lifted Knapsack Cover Inequalities Constraint (3e) states that we have to pack arcs a # A that take positive time to complete (namely, t a ) into a working day of length T . Hence, solutions satisfying (3e) can be interpreted 18 knapsack sets (see Padberg [29] Balas [3], Wolsey [38] Hammer, Johnson, and Peled [20] Crowder, Johnson and Padberg [12] Weismantel [37] and Wolsey [40] Hence, lifted cover inequalities derived from (3e) are valid for the MSP. We use a separation algorithm as described by Van Roy and Wolsey [35] and Gu, Nemhauser and Savelsbergh ....

E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.

....the complexity of deciding whether a given inequality is a simple LCI, and show that this can also be done in O(n 2 ) time. Let C = fj 1 ; j 2 ; j r g be a minimal cover with a j 1 a j 2 : a jr , 0 = 0, h = P h i=1 a j i for h = 1; r, and = r Gamma b. Theorem 1 (Balas [1975]) Every facet defining simple LCI satisfies the following conditions: 3 1. If h a j h 1 Gamma , then ff j = h. 2. If h 1 Gamma 1 a j h 1 Gamma 1, then ff j 2 fh; h 1g. Although Theorem 1 nearly determines all the lifting coefficients, it does not give sufficient conditions on ....

E. Balas (1975). Facets of the Knapsack Polytope. Math. Prog. 8, 146-164.

....X i2W x ik jW j Gamma 1 is valid for the polytope MK(N Theta M; f; F ) and defines a facet for MK(W Theta M; f; F ) A subset W with the above properties is called a minimal cover with respect to module k. The corresponding inequality is called minimal cover inequality and was discussed in [B75], HJP75] W75] By applying the above theorem we can conclude that the minimal cover inequality defines a facet of the polytope C(W Theta M; f; F; E) For other classes of inequalities that are valid or facet defining for MK(N Theta M; f; F ) we refer the reader to [P80] GR90a] GR90b] and ....

E. Balas, "Facets of the Knapsack Polytope", Mathematical Programming 8, 146 - 164 (1975).

....polyhedron P : convfx 2 f0; 1g n : n X i=1 a i x i bg: We brie y introduce these classes in the following. For convenience we use the notation N = f1; ng and, for T N , a(T ) P i2T a i . A typical example of valid inequalities for P are cover inequalities. 4 De nition 2. 5 [13, 33, 2][cover inequalities] Let S N with P i2S a i b. S is called a cover and the inequality X i2S x i jSj 1 (7) is the cover inequality with respect to S. De nition 2.6 [32] weight inequalities] Let T N with a(T ) b and set r : b a(T ) The weight inequality with respect to T is de ....

E. BALAS. Facets of the knapsack polytope. Mathematical Programming 8:146-164, 1975.

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E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.

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E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.

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E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.

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E. Balas. Facets of the knapsack polytope. Mathematical Programming, 8:146--164, 1975.

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Balas, E. 1975. Facets of the knapsack polytope. Mathematical Programming 8, 146- 164.

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E. Balas (1975) Facets of the knapsack polytope. Math. Program. 8, 146-- 164.

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Balas, E.: Facets of the knapsack polytope. Mathematical Programming 8, 146--164, 1975.

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