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On the facets of the mixed–integer knapsack polyhedron
 MATH. PROGRAM., SER. B 98: 145–175 (2003)
, 2003
"... We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities cont ..."
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Cited by 36 (11 self)
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We study the mixed–integer knapsack polyhedron, that is, the convex hull of the mixed–integer set defined by an arbitrary linear inequality and the bounds on the variables. We describe facet–defining inequalities of this polyhedron that can be obtained through sequential lifting of inequalities containing a single integer variable. These inequalities strengthen and/or generalize known inequalities for several special cases. We report computational results on using the inequalities as cutting planes for mixed–integer programming.
Integrality Gaps of Linear and Semidefinite Programming Relaxations for Knapsack
"... Recent years have seen an explosion of interest in lift and project methods, such as those proposed by Lovász and Schrijver [40], Sherali and Adams [49], Balas, Ceria and Cornuejols [6], Lasserre [36, 37] and others. These methods are systematic procedures for constructing a sequence of increasingly ..."
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Cited by 20 (0 self)
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Recent years have seen an explosion of interest in lift and project methods, such as those proposed by Lovász and Schrijver [40], Sherali and Adams [49], Balas, Ceria and Cornuejols [6], Lasserre [36, 37] and others. These methods are systematic procedures for constructing a sequence of increasingly tight mathematical programming relaxations for 01 optimization problems. One major line of research in this area has focused on understanding the strengths and limitations of these procedures. Of particular interest to our community is the question of how the integrality gaps for interesting combinatorial optimization problems evolve through a series of rounds of one of these procedures. On the one hand, if the integrality gap of successive relaxations drops sufficiently fast, there is the potential for an improved approximation algorithm. On the other hand, if the integrality gap for a problem persists, this can be viewed as a lower bound in a certain restricted model of computation. In this paper, we study the integrality gap in these hierarchies for the knapsack problem. We have two main results. First, we show that an integrality gap of 2 − ɛ persists up to a linear number of rounds of SheraliAdams. This is interesting, since it is well known that knapsack has a fully polynomial time approximation scheme [30, 39]. Second, we show that Lasserre’s hierarchy closes the gap quickly. Specifically, after t 2 rounds of Lasserre, the integrality gap decreases to t/(t − 1). Thus, we provide a second example of an integrality gap separation between Lasserre and Sherali Adams. The only other such gap we are aware of is in the recent work of Fernandez de la Vega and Mathieu [19] (respectively of Charikar, Makarychev and Makarychev [12]) showing that the integrality gap for MAXCUT remains 2 − ɛ even after ω(1) (respectively n γ) rounds of SheraliAdams. On the other hand, it is known that 2 rounds of Lasserre yields a relaxation as least as strong as the GoemansWilliamson SDP, which has an integrality gap of 0.878.
A Semidefinite Programming Approach to the Quadratic Knapsack Problem
, 2000
"... In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibi ..."
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Cited by 19 (1 self)
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In order to gain insight into the quality of semidefinite relaxations of constrained quadratic 0/1 programming problems we study the quadratic knapsack problem. We investigate several basic semidefinite relaxations of this problem and compare their strength in theory and in practice. Various possibilities to improve these basic relaxations by cutting planes are discussed. The cutting planes either arise from quadratic representations of linear inequalities or from linear inequalities in the quadratic model. In particular, a large family of combinatorial cuts is introduced for the linear formulation of the knapsack problem in quadratic space. Computational results on a small class of practical problems illustrate the quality of these relaxations and cutting planes.
Separation algorithms for 01 knapsack polytopes
, 2008
"... Valid inequalities for 01 knapsack polytopes often prove useful when tackling hard 01 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the soca ..."
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Cited by 17 (0 self)
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Valid inequalities for 01 knapsack polytopes often prove useful when tackling hard 01 Linear Programming problems. To use such inequalities effectively, one needs separation algorithms for them, i.e., routines for detecting when they are violated. We show that the separation problems for the socalled extended cover and weight inequalities can be solved exactly in O(nb) time and O((n + amax)b) time, respectively, where n is the number of items, b is the knapsack capacity and amax is the largest item weight. We also present fast and effective separation heuristics for the extended cover and lifted cover inequalities. Finally, we present a new exact separation algorithm for the 01 knapsack polytope itself, which is faster than existing methods. Extensive computational results are also given.
Recoverable Robust Knapsacks: the Discrete Scenario Case
, 2010
"... Admission control problems have been studied extensively in the past. In a typical setting, resources like bandwidth have to be distributed to the different customers according to their demands maximizing the profit of the company. Yet, in realworld applications those demands are deviating and in o ..."
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Cited by 12 (3 self)
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Admission control problems have been studied extensively in the past. In a typical setting, resources like bandwidth have to be distributed to the different customers according to their demands maximizing the profit of the company. Yet, in realworld applications those demands are deviating and in order to satisfy their service requirements often a robust approach is chosen wasting benefits for the company. Our model overcomes this problem by allowing a limited recovery of a previously fixed assignment as soon as the data are known by violating at most k service promises and serving up to ℓ new customers. Applying this approaches to the call admission problem on a single link of a telecommunication network leads to a recoverable robust version of the knapsack problem. In this paper, we show that for a fixed number of discrete scenarios this recoverable robust knapsack problem is weakly NPcomplete and any such instance can be solved in pseudopolynomial time by a dynamic program. If the number of discrete scenarios is part of the input, the problem is strongly NPcomplete and in special cases not approximable in polynomial time, unless P = NP. Next to its complexity status we were interested in obtaining strong polyhedral descriptions for this problem. We thus generalized the wellknown concept of covers to gain valid inequalities for the recoverable robust knapsack polytope. Besides the canonical extension of covers we introduce a second kind of extension exploiting the scenariobased problem structure and producing stronger valid inequalities. Furthermore, we present two extensive computational studies to (i) investigate the influence of parameters k and ℓ to the objective and (ii) evaluate the effectiveness of our new class of valid inequalities.
Quadratic Knapsack Relaxations Using Cutting Planes and Semidefinite Programming
 INTEGER PROGRAMMING AND COMBINATORIAL OPTIMIZATION, LECTURE NOTES IN COMPUTER SCIENCE
, 1995
"... We investigate dominance relations between basic semidefinite relaxations and classes of cuts. We show that simple semidefinite relaxations are tighter than corresponding linear relaxations even in case of linear cost functions. Numerical results are presented illustrating the quality of these relax ..."
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Cited by 11 (5 self)
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We investigate dominance relations between basic semidefinite relaxations and classes of cuts. We show that simple semidefinite relaxations are tighter than corresponding linear relaxations even in case of linear cost functions. Numerical results are presented illustrating the quality of these relaxations.
Cover and Pack Inequalities for (Mixed) Integer Programming
"... We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 01 knapsack set, the mixed 01 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a common presentation of the inequalities based ..."
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Cited by 10 (3 self)
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We review strong inequalities for fundamental knapsack relaxations of (mixed) integer programs. These relaxations are the 01 knapsack set, the mixed 01 knapsack set, the integer knapsack set, and the mixed integer knapsack set. Our aim is to give a common 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
The MCFseparator – detecting and exploiting multicommodity flows in MIPs
 Mathematical Programming C
, 2010
"... Given a general mixed integer program (MIP), we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multicommodity flow formulations. We identify the underlying graph and generate cutting planes based on cuts in the detected ..."
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Cited by 7 (0 self)
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Given a general mixed integer program (MIP), we automatically detect block structures in the constraint matrix together with the coupling by capacity constraints arising from multicommodity flow formulations. We identify the underlying graph and generate cutting planes based on cuts in the detected network. Our implementation adds a separator to the branchandcut libraries of Scip and Cplex. We make use of the complemented mixed integer rounding framework (cMIR) but provide a special purpose aggregation heuristic that exploits the network structure. Our separation scheme speedsup the computation for a large set of MIPs coming from network design problems by a factor of two on average.