Discrete Optimization最新文献

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.

In this paper, we explore some basic questions on the complexity of training neural networks with ReLU activation function. We show that it is NP-hard to train a two-hidden layer feedforward ReLU neural network. If dimension of the input data and the network topology is fixed, then we show that there exists a polynomial time algorithm for the same training problem. We also show that if sufficient over-parameterization is provided in the first hidden layer of ReLU neural network, then there is a polynomial time algorithm which finds weights such that output of the over-parameterized ReLU neural network matches with the output of the given data.

We propose a new combinatorial optimization problem that we call the submodular reassignment problem. We are given $k$ submodular functions over the same ground set, and we want to find a set that minimizes the sum of the distances to the sets of minimizers of all functions. The problem is motivated by a two-stage stochastic optimization problem with recourse summarized as follows. We are given two tasks to be processed and want to assign a set of workers to maximize the sum of profits. However, we do not know the value functions exactly, but only know a finite number of possible scenarios. Our goal is to determine the first-stage allocation of workers to minimize the expected number of reallocated workers after a scenario is realized at the second stage. This problem can be modeled by the submodular reassignment problem. We prove that the submodular reassignment problem can be solved in strongly polynomial time via submodular function minimization. We further provide a maximum-flow formulation of the problem that enables us to solve the problem without using a general submodular function minimization algorithm, and more efficiently both in theory and in practice. In our algorithm, we make use of Birkhoff’s representation theorem for distributive lattices.