Exponential Number Of Inequalities Research Articles

Convexifying multilinear sets with cardinality constraints: Structural properties, nested case and extensions

The problem of minimizing a multilinear function of binary variables is a well-studied NP-hard problem. The set of solutions of the standard linearization of this problem is called the multilinear set. We study a cardinality constrained version of it with upper and lower bounds on the number of nonzero variables. We call the set of solutions of the standard linearization of this problem a multilinear set with cardinality constraints. We characterize a set of conditions on these multilinear terms (called properness) and observe that under these conditions the convex hull description of the set is tractable via an extended formulation. We then give an explicit polyhedral description of the convex hull when the multilinear terms have a nested structure. Our description has an exponential number of inequalities which can be separated in polynomial time. Finally, we generalize these inequalities to obtain valid inequalities for the general case.

On the generation of metric TSP instances with a large integrality gap by branch-and-cut

This paper introduces a computational method for generating metric Travelling Salesman Problem (TSP) instances having a large integrality gap. The method is based on the solution of an integer programming problem, called IH-OPT, that takes as input a fractional solution of the Subtour Elimination Problem (SEP) on a TSP instance and computes a TSP instance having an integrality gap larger than or equal to the integrality gap of the first instance. The decision variables of IH-OPT are the entries of the TSP cost matrix, and the constraints are defined by the intersection of the metric cone with an exponential number of inequalities, one for each possible TSP tour. Given the very large number of constraints, we have implemented a branch-and-cut algorithm for solving IH-OPT. Then, by sampling cost vectors over the metric polytope and by solving the corresponding SEP, we can generate random fractional vertices of the SEP polytope. If we solve the IH-OPT problem for every sampled vertex using our branch-and-cut algorithm, we can select the generated TSP instance (i.e., cost vector), yielding the longest runtime for Concorde, the state-of-the-art TSP solver. Our computational results show that our method is very effective in producing challenging instances. As a by-product, we release the Hard-TSPLIB, a library of 41 small metric TSP instances which have a large integrality gap and are challenging in terms of runtime for Concorde.

Most vital vertices for the shortest s–t path problem: complexity and Branch-and-Cut algorithm

In this paper we study the most vital vertices for the shortest s–t path problem. This problem consists, given a digraph $$D=(V\cup \{s,t\},A)$$ and a threshold $$d\in \mathbb {N}$$ , in finding the minimum number of nodes to delete from D in such a way that there does not exist a s–t path of length less or equal than d. We prove the NP-hardness of this problem and propose an integer linear program using an exponential number of inequalities. We investigate the facial structure of the associated polytope. We prove that this polytope is integer and we derive an efficient Branch-and-Cut algorithm. Finally, we present and discuss some computational results.

The probabilistic pickup-and-delivery travelling salesman problem

Transportation problems are essential in commercial logistics and have been widely studied in the literature during the last decades. Many of them consist in designing routes for vehicles to move commodities between locations. This article approaches a pickup-and-delivery single-vehicle routing problem where there is susceptibility to uncertainty in customer requests. The probability distributions of the requests are assumed to be known, and the objective is to design an a priori route with minimum expected length. The problem has already been approached in the literature, but through a heuristic method. This article proposes the first exact approach to the problem. Two mathematical formulations are proposed: one is a compact model (i.e. defined by a polynomial number of variables and constraints); the other one contains an exponential number of inequalities and is solved within a branch-and-cut framework. Computational results show the upsides as well as the breakdowns of both formulations.

Complete formulations of polytopes related to extensions of assignment matrices

Let k,n denote two positive integers and consider the family of the polytopes defined as the convex hull of pairs of the form (Y,h) where Y is a 0/1-matrix with k rows, n columns, containing exactly one nonzero coefficient per column, and where h stands for the smallest index of a nonzero row of Y.These polytopes and some variants naturally emerge in formulations of different classical combinatorial optimization problems such as minimum makespan scheduling and minimum span frequency assignment.In this paper, we provide complete formulations for these polytopes and show the associated separation problem can be solved in polynomial time. The complete formulations in the original space of variables generally contain an exponential number of inequalities. Alternative extended compact formulations are also presented.

A dynamical constraint-generating algorithm for the Single-Vehicle Cyclic Inventory-Routing Problem

The Single-Vehicle Cyclic Inventory-Routing Problem (SVCIRP) is a variant of the Inventory-Routing Problem (IRP) in which the replenishment decisions of a recurring distribution plan are optimized. In this paper, we investigate the current formulation of the SVCIRP and propose a number of improvements for it. First, we introduce a new binary variable in the model to distinguish tours in which only one retailer is visited. Additionally, we formulate three new sets of valid inequalities. Because one of the sets contains an exponential number of inequalities, we develop a procedure to insert only the violated inequalities. We present an algorithm, based on this procedure, to solve the SVCIRP effectively. Our computational results show that our algorithm outperforms the other exact methods in literature. We obtain 25 new best bounds and we find 13 improved solutions.

The probability of the existence of a feasible flow in a stochastic transportation network

AbstractStochastic transportation networks arise in various real world applications, for which the probability of the existence of a feasible flow is regarded as an important performance measure. Although the necessary and sufficient condition for the existence of a feasible flow represented by an exponential number of inequalities is a well‐known result in the literature, the computation of the probability of all such inequalities being satisfied jointly is a daunting challenge. The state‐of‐the‐art approach of Prékopa and Boros, Operat Res 39 (1991) 119–129 approximates this probability by giving its lower and upper bounds using a two‐part procedure. The first part eliminates all redundant inequalities and the second gives the lower and upper bounds of the probability by solving two well‐defined linear programs with the inputs obtained from the first part. Unfortunately, the first part may still leave many non‐redundant inequalities. In this case, it would be very time consuming to compute the inputs for the second part even for small‐sized networks. In this paper, we first present a model that can be used to eliminate all redundant inequalities and give the corresponding computational results for the same numerical examples used in Prékopa and Boros, Operat Res 39 (1991) 119–129. We also show how to improve the lower and upper bounds of the probability using the multitree and hypermultitree, respectively. Furthermore, we propose an exact solution approach based on the state space decomposition to compute the probability. We derive a feasible state from a state space and then decompose the space into several disjoint subspaces iteratively. The probability is equal to the sum of the probabilities in these subspaces. We use the 8‐node and 15‐node network examples in Prékopa and Boros, Operat Res 39 (1991) 119–129 and the Sioux‐Falls network with 24 nodes to show that the space decomposition algorithm can obtain the exact probability of these classical examples efficiently. © 2016 Wiley Periodicals, Inc. Naval Research Logistics 63: 479–491, 2016

The Matching Problem Has No Fully Polynomial Size Linear Programming Relaxation Schemes

Recently, Rothvos established that every linear program (LP) expressing the matching polytope has an exponential number of inequalities (formally, the matching polytope has exponential extension complexity). We generalize this result by deriving strong bounds on the LP inapproximability of the matching problem: for fixed $0 , every $(1 - \varepsilon / n)$ -approximating LP requires an exponential number of inequalities, where $n$ is the number of vertices. This is sharp, given the well-known $\rho $ -approximation of size $O\binom {n}{1/(1 - \rho )}$ provided by the odd-sets of size up to $1/(1 - \rho )$ . Thus, matching is the first problem in $P$ , which does not admit a fully polynomial-size LP relaxation scheme (the LP equivalent of an Fully Polynomial-Time Approximation Scheme), which provides a sharp separation from the polynomial-size LP relaxation scheme obtained, e.g., through constant-sized odd-sets mentioned above. Analyzing the size of LP formulations is equivalent to examining the nonnegative rank of matrices. We study the nonnegative rank through an information–theoretic approach; while it reuses key ideas from Rothvos, the main lower bounding technique is different: we employ the information–theoretic notion of Wyner’s common information used for studying LP formulations. This allows us to analyze the nonnegative rank of perturbations of slack matrices, e.g., the approximations of the matching polytope. It turns out that the high extension complexity for the matching problem stems from the same source of hardness as in the case of the correlation polytope: a direct sum structure.

Solving the Prize-collecting Rural Postman Problem

Abstract In this work we present an algorithm for solving the Prize-collecting Rural Postman Problem. This problem was recently defined and is a generalization of other arc routing problems like, for instance, the Rural Postman Problem. The main difference is that there are no required edges. Instead, there is a profit function on the edges that must be taken into account only the first time that an edge is traversed. The problem is modeled as a linear integer program where the system has an exponential number of inequalities. We propose a solution algorithm where iteratively we solve relaxed models with a small number of inequalities, that provide upper bounds, and we propose exact separation procedures for generating violated cuts when possible. We also propose a simple heuristic to generate feasible solutions that provide lower bounds at each iteration. We use a two-phase method with different solvers at each phase. Despite the difficulty of the problem, the numerical results from a series of computational experiments with various types of instances assess the good behavior of the algorithm. In particular, 75% of the instances were optimally solved with the LP relaxation of the model. The remaining instances were optimally solved on a second phase, most of them in small computation times.