We study an integrated market selection and production planning problem. There is a set of markets with deterministic demand, and each market has a certain revenue that is obtained if the market’s demand is satisfied throughout a planning horizon. The demand is satisfied with a production scheme that has a lot-sizing structure. The problem is to decide on which markets’ demand to satisfy and plan the production simultaneously. We consider both single and multiobjective settings. The single objective problem maximizes the profit, whereas the multiobjective problem includes the maximization of the revenue and the minimization of the production cost objectives. We develop a decomposition-based exact solution algorithm for the single objective setting and show how it can be used in a proposed three-phase algorithm for the multiobjective setting. The master problem chooses a subset of markets, and the subproblem calculates an optimal production plan to satisfy the selected markets’ demand. We investigate the subproblem from a cooperative game theory perspective to devise cuts and strengthen them based on lifting. We also propose a set of valid inequalities and preprocessing rules to improve the proposed algorithm. We test the efficacy of our solution method over a suite of problem instances and show that our algorithm substantially decreases solution times for all problem instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms – Discrete. Funding: This work was supported by TUBITAK [Grant 1059B191801782].

Abstract We study a bipartite graph (BG) construction problem that arises in digital communication systems. In a digital communication system, information is sent from one place to another over a noisy communication channel using binary symbols (bits). The original information is encoded by adding redundant bits, which are then used to detect and correct errors that may have been introduced during transmission. Harmful structures, such as small cycles, severely deteriorate the error correction capability of a BG. We introduce an integer programming formulation to generate a BG for a given smallest cycle length. We propose a branch‐and‐cut algorithm for its solution and investigate the structural properties of the problem to derive valid inequalities and variable fixing rules. We also introduce heuristics to obtain feasible solutions for the problem. The computational experiments show that our algorithm can generate BGs without small cycles in an acceptable amount of time for practically relevant dimensions.

Graph coloring is the problem of assigning a minimum number of colors to all vertices of a graph such that no two adjacent vertices receive the same color. The selective graph coloring problem is a generalization of the standard graph coloring problem; given a graph with a partition of its vertex set into clusters, the objective is to choose exactly one vertex per cluster so that, among all possible selections, the number of colors necessary to color the vertices in the selection is minimum. This study focuses on a decomposition based exact solution framework for selective coloring in some perfect graph families: in particular, permutation, generalized split, and chordal graphs where the selective coloring problem is known to be NP‐hard. Our method combines integer programming techniques and combinatorial algorithms for the graph classes of interest. We test our method on graphs with different sizes and densities, present computational results and compare them with solving an integer programming formulation of the problem by CPLEX, and a state‐of‐the art algorithm from the literature. Our computational experiments indicate that our decomposition approach significantly improves solution performance in low‐density graphs, and regardless of edge‐density in the class of chordal graphs.

We investigate the maximum induced matching problem (MIM), which is the problem of finding an induced matching having the largest cardinality on an undirected graph. The problem is known to be NP-hard for general graphs. We first propose a vertex-based integer programming formulation for MIM, which is more compact compared to an edge-based formulation found in the literature. We also introduce the maximum weight induced matching problem (MWIM), which generalizes MIM so that vertices and edges have weights. We adapt the edge-based formulation to MWIM, and propose a quadratic programming formulation of MWIM based on our vertex-based formulation. We then linearize our quadratic programming formulation, and devise a Benders decomposition algorithm that exploits a special structure of the linearized formulation. We also propose valid inequalities and formulation tightening procedures to improve the efficiency of our approach. Our computational tests on a large suite of randomly generated graphs show that our vertex-based formulation and decomposition approach significantly improve the solvability of MIM and MWIM, especially on dense graphs. The online appendix and data are available at https://doi.org/10.1287/ijoc.2017.0764 .

We consider graph search problems involving an intruder and mobile searchers. The graph consists of nodes on which the intruder and searchers may be located, and edges on which these entities travel. Associated with each node is a set of nodes that are visible from that node. The goal is to find the minimum number of searchers needed to detect the intruder within a given time limit. We investigate three variants of the graph search problem: (i) a hide‐and‐seek problem, in which a stationary intruder “hides” at an unknown node, (ii) a pursuit‐evasion problem, in which the intruder moves among the nodes to avoid being detected, and (iii) a patrol problem, which is similar to the pursuit‐evasion problem except that searchers patrol the graph in repeated circuits to seek intruders. Our contribution provides exponential‐size set‐covering formulations for these problems, along with a class of branch‐cut‐price algorithms tailored for solving them. These algorithms leverage results from the orienteering literature to solve pricing problems related to searcher routes. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 70(1), 4–18 2017

Abstract– We investigate the problem of finding a maximal matching that has minimum total weight on a given edge‐weighted graph. Although the minimum weight maximal matching problem is NP‐hard in general, polynomial time exact or approximation algorithms on several restricted graph classes are given in the literature. In this article, we propose an exact algorithm for solving several variants of the problem on general graphs. In particular, we develop integer programming (IP) formulations for the problem and devise a decomposition algorithm, which is based on a combination of IP techniques and combinatorial matching algorithms. Our computational tests on a large suite of randomly generated graphs show that our decomposition approach significantly improves the solvability of the problem compared to the underlying IP formulation. © 2013 Wiley Periodicals, Inc. NETWORKS, Vol. 62(4), 273–287 2013