Discrete Optimization最新文献

This paper presents a highly effective reinforcement learning enhancement of multi-neighborhood tabu search for the max-mean dispersion problem. The reinforcement learning component uses the Q-learning mechanism that incorporates the accumulated feedback information collected from the actions performed during the search to guide the generation of diversified solutions. The tabu search component employs 1-flip and reduced 2-flip neighborhoods to collaboratively perform the neighborhood exploration for attaining high-quality local optima. A learning automata method is integrated in tabu search to adaptively determine the probability of selecting each neighborhood. Computational experiments on 80 challenging benchmark instances demonstrate that the proposed algorithm is favorably competitive with the state-of-the-art algorithms in the literature, by finding new lower bounds for 3 instances and matching the best known results for the other instances. Key elements and properties are also analyzed to disclose the source of the benefits of our integration of learning mechanisms and tabu search.

With each separable optimization problem over a given set of vectors is associated its multichoice counterpart which involves choosing $n$ rather than one solutions from the set so as to maximize the given separable function over the sum of the chosen solutions. Such problems have been studied in various contexts under various names, such as load balancing in machine scheduling, congestion routing, minimum shared and vulnerable edge problems, and shifted optimization. Separable multichoice optimization has a very broad expressive power and can be hard already for explicitly given sets of binary points. In this article we consider the problem over monotone systems, also called independence systems. Typically such a system has exponential size, and we assume that it is presented implicitly by a linear optimization oracle. Our main results for separable multichoice optimization are the following. First, the problem over any monotone system with any separable concave function can be approximated in polynomial time with a constant approximation ratio which is independent of $n$. Second, the problem over any monotone system with an arbitrary separable function can be approximated in polynomial time with an approximation ratio of $1/\left(O(logn)\right)$.

This work considers the graph partitioning problem known as maximum $k$-cut. It focuses on investigating features of a branch-and-bound method to obtain global solutions. An exhaustive experimental study is carried out for the two main components of a branch-and-bound algorithm: Computing bounds and branching strategies. In particular, we propose the use of a variable neighborhood search metaheuristic to compute good feasible solutions, the $k$-chotomic strategy to split the problem, and a branching rule based on edge weights to select variables. Moreover, we analyze a linear relaxation strengthened by semidefinite-based constraints, a cutting plane algorithm, and node selection strategies. Computational results show that the resulting method outperforms the state-of-the-art approach and discovers the solution of several instances, especially for problems with $k\ge 5$.

The quadratic stable set problem (QSSP) is a natural extension of the well-known maximum stable set problem. The QSSP is NP-hard and can be formulated as a binary quadratic program, which makes it an interesting case study to be tackled from different optimization paradigms. In this paper, we propose a novel representation for the QSSP through binary decision diagrams (BDDs) and adapt a hybrid optimization approach which integrates BDDs and mixed-integer programming (MIP) for solving the QSSP. The exact framework highlights the modeling flexibility offered through decision diagrams to handle nonlinear problems. In addition, the hybrid approach leverages two different representations by exploring, in a complementary way, the solution space with BDD and MIP technologies. Machine learning then becomes a valuable component within the method to guide the search mechanisms. In the numerical experiments, the hybrid approach shows to be superior, by at least one order of magnitude, than two leading commercial MIP solvers with quadratic programming capabilities and a semidefinite-based branch-and-bound solver.

We consider the Bipartite Boolean Quadratic Programming Problem (BQP01), which generalizes the well-known Boolean quadratic programming problem (QP01). The model has applications in graph theory, matrix factorization and bioinformatics, among others. The primary focus of this paper is on studying the structure of the Bipartite Boolean Quadric Polytope (BQP${}^{m,n}$) resulting from a linearization of a quadratic integer programming formulation of BQP01.

We present some basic properties and partial relaxations of BQP${}^{m,n}$, as well as some families of facets and valid inequalities. We find facet-defining inequalities including a family of odd-cycle inequalities. We discuss various approaches to obtain a valid inequality and facets from those of the related Boolean quadric polytope. The key strategy is based on rounding coefficients, and it is applied to the families of clique and cut inequalities in BQP${}^{m,n}$.

The min knapsack problem appears as a major component in the structure of capacitated covering problems. Its polyhedral relaxations have been extensively studied, leading to strong relaxations for networking, scheduling and facility location problems.

A valid inequality ${\alpha }^{T}x\ge {\alpha }_0$ with $\alpha \ge 0$ for a min knapsack instance is said to have pitch $\le \pi$($\pi$ a positive integer) if the $\pi$ smallest strictly positive ${\alpha }_j$ sum to at least ${\alpha }_0$. An inequality with coefficients and right-hand side in $\{0,1,\dots ,\pi \}$ has pitch $\le \pi$. The notion of pitch has been used for measuring the complexity of valid inequalities for the min knapsack polytope. Separating inequalities of pitch-1 is already NP-Hard. In this paper, we show an algorithm for efficiently separating inequalities with coefficients in $\{0,1,\dots ,\pi \}$ for any fixed $\pi$ up to an arbitrarily small additive error. As a special case, this allows for approximate separation of inequalities with pitch at most 2. We moreover investigate the integrality gap of minimum knapsack instances when bounded pitch inequalities (possibly in conjunction with other inequalities) are added. Among other results, we show that the CG closure of minimum knapsack has unbounded integrality gap even after a constant number of rounds.

Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra’s algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between $\mathrm{FPT}$ and $\mathrm{XP}$ algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining $\mathrm{FPT}$ algorithms with runtime $f\left(k\right)$poly$\left(n\right)$. We focus on:

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  Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used.

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  Optimality program: after giving an $\mathrm{FPT}$ algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups.

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  Minding the poly$\left(n\right)$: reducing $f\left(k\right)$ often has the unintended consequence of increasing poly$\left(n\right)$; so we highlight the common trade-offs and show how to get the best of both worlds.

Let $E^d$ denote the $d$-dimensional Euclidean space. The $r$-ball body generated by a given set in $E^d$ is the intersection of balls of radius $r$ centered at the points of the given set. In this paper we prove the following Blaschke–Santaló-type inequalities for $r$-ball bodies: for all $1\le k\le d$ and for any set of given volume in $E^d$ the $k$th intrinsic volume of the $r$-ball body generated by the set becomes maximal if the set is a ball. As an application we investigate the Gromov–Klee–Wagon problem for congruent balls in $E^d$, which is a question on proving or disproving that if the centers of a family of $N$ congruent balls in $E^d$ are contracted, then the volume of the intersection does not decrease. In particular, we investigate this problem for uniform contractions, which are contractions where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers, that is, when the pairwise distances of the two sets are separated by some positive real number. Bezdek and Naszódi (2018), proved that the intrinsic volumes of the intersection of $N$ congruent balls in $E^d$, $d>1$ increase under any uniform contraction of the center points when $N\ge {\left(1+\sqrt{2}\right)}^d$. We give a short proof of this result using the Blaschke–Santaló-type inequalities of $r$-ball bodies and improve it for $d\ge 42$.

We consider three known bounds for the quadratic assignment problem (QAP): an eigenvalue, a convex quadratic programming (CQP), and a semidefinite programming (SDP) bound. Since the last two bounds were not compared directly before, we prove that the SDP bound is stronger than the CQP bound. We then apply these to improve known bounds on a discrete energy minimization problem, reformulated as a QAP, which aims to minimize the potential energy between repulsive particles on a toric grid. Thus we are able to prove optimality for several configurations of particles and grid sizes, complementing earlier results by Bouman et al. (2013). The semidefinite programs in question are too large to solve without pre-processing, and we use a symmetry reduction method by Permenter and Parrilo (2020) to make computation of the SDP bounds possible.

Circuits play a fundamental role in the theory of linear programming due to their intimate connection to algorithms of combinatorial optimization and the efficiency of the simplex method. We are interested in better understanding the properties of circuit walks in integral polyhedra. In this paper, we introduce a hierarchy for integral polyhedra based on different types of behavior exhibited by their circuit walks. Many problems in combinatorial optimization fall into the most interesting categories of this hierarchy — steps of circuit walks only stop at integer points, at vertices, or follow actual edges. We classify several classical families of polyhedra within the hierarchy, including $0/1$-polytopes, polyhedra defined by totally unimodular matrices, and more specifically matroid polytopes, transportation polytopes, and partition polytopes. Finally, we prove three characterizations of the simple polytopes that appear in the bottom level of the hierarchy where all circuit walks are edge walks, showing that such polytopes constitute a generalization of simplices and parallelotopes.