Logic Programming with Max-Clique and its Application to Graph Coloring (Tool Description)

This paper presents pl-cliquer, a Prolog interface to the cliquer tool for the maximum clique problem. Using pl-cliquer facilitates a programming style that allows logic programs to integrate with other tools such as: Boolean satisfiability solvers, finite domain constraint solvers, and graph isomorphism tools. We illustrate this programming style to solve the Graph Coloring problem, applying a symmetry break that derives from finding a maximum clique in the input graph. We present an experimentation of the resulting Graph Coloring solver on two benchmarks, one from the graph coloring community and the other from the examination timetabling community. The implementation of pl-cliquer consists of two components: A lightweight C interface, connecting cliquer\u27s C library and Prolog, and a Prolog module which loads the library. The complete tool is available as a SWI-Prolog module

Mixed integer-linear formulations of cumulative scheduling constraints - A comparative study

This paper introduces two MILP models for the cumulative scheduling constraint and associated pre-processing filters. We compare standard solver performance for these models on three sets of problems and for two of them, where tasks have unitary resource consumption, we also compare them with two models based on a geometric placement constraint. In the experiments, the solver performance of one of the cumulative models, is clearly the best and is also shown to scale very well for a large scale industrial transportation scheduling problem

A Tutorial on Clique Problems in Communications and Signal Processing

Since its first use by Euler on the problem of the seven bridges of K\"onigsberg, graph theory has shown excellent abilities in solving and unveiling the properties of multiple discrete optimization problems. The study of the structure of some integer programs reveals equivalence with graph theory problems making a large body of the literature readily available for solving and characterizing the complexity of these problems. This tutorial presents a framework for utilizing a particular graph theory problem, known as the clique problem, for solving communications and signal processing problems. In particular, the paper aims to illustrate the structural properties of integer programs that can be formulated as clique problems through multiple examples in communications and signal processing. To that end, the first part of the tutorial provides various optimal and heuristic solutions for the maximum clique, maximum weight clique, and $k$-clique problems. The tutorial, further, illustrates the use of the clique formulation through numerous contemporary examples in communications and signal processing, mainly in maximum access for non-orthogonal multiple access networks, throughput maximization using index and instantly decodable network coding, collision-free radio frequency identification networks, and resource allocation in cloud-radio access networks. Finally, the tutorial sheds light on the recent advances of such applications, and provides technical insights on ways of dealing with mixed discrete-continuous optimization problems

A Computational Comparison of Optimization Methods for the Golomb Ruler Problem

The Golomb ruler problem is defined as follows: Given a positive integer n, locate n marks on a ruler such that the distance between any two distinct pair of marks are different from each other and the total length of the ruler is minimized. The Golomb ruler problem has applications in information theory, astronomy and communications, and it can be seen as a challenge for combinatorial optimization algorithms. Although constructing high quality rulers is well-studied, proving optimality is a far more challenging task. In this paper, we provide a computational comparison of different optimization paradigms, each using a different model (linear integer, constraint programming and quadratic integer) to certify that a given Golomb ruler is optimal. We propose several enhancements to improve the computational performance of each method by exploring bound tightening, valid inequalities, cutting planes and branching strategies. We conclude that a certain quadratic integer programming model solved through a Benders decomposition and strengthened by two types of valid inequalities performs the best in terms of solution time for small-sized Golomb ruler problem instances. On the other hand, a constraint programming model improved by range reduction and a particular branching strategy could have more potential to solve larger size instances due to its promising parallelization features