Greedy Graph Colouring is a Misleading Heuristic

State of the art maximum clique algorithms use a greedy graph colouring as a bound. We show that greedy graph colouring can be misleading, which has implications for parallel branch and bound

Finding maximum k-cliques faster using lazy global domination

No abstract available

A Partitioning Algorithm for Maximum Common Subgraph Problems

We introduce a new branch and bound algorithm for the maximum common subgraph and maximum common connected subgraph problems which is based around vertex labelling and partitioning. Our method in some ways resembles a traditional constraint programming approach, but uses a novel compact domain store and supporting inference algorithms which dramatically reduce the memory and computation requirements during search, and allow better dual viewpoint ordering heuristics to be calculated cheaply. Experiments show a speedup of more than an order of magnitude over the state of the art, and demonstrate that we can operate on much larger graphs without running out of memory

A Constraint Programming Approach to the Hospitals / Residents Problem

An instance I of the Hospitals / Residents problem (HR) involves a set of residents (graduating medical students) and a set of hospitals, where each hospital has a given capacity. The residents have preferences for the hospitals, as do hospitals for residents. A solution of I is a stable matching, which is an assignment of residents to hospitals that respects the capacity conditions and preference lists in a precise way. In this paper we present constraint encodings for HR that give rise to important structural properties. We also present a computational study using both randomly-generated and real-world instances. Our study suggests that Constraint Programming is indeed an applicable technology for solving this problem, in terms of both theory and practice

Diamond-free Degree Sequences

We introduce a new problem, CSPLib problem number 50, to generate all degree sequences that have a corresponding diamond-free graph with secondary properties. This problem arises naturally from a problem in mathematics to do with balanced incomplete block designs; we devote a section of this paper to this. The problem itself is challenging with respect to computational effort arising from the large number of symmetries within the models. We introduce two models for this problem. The second model is an improvement on the first, and this improvement largely consists of breaking the problem into two stages, the first stage producing graphical degree sequences that satisfy arithmetic constraints and the second part testing that there exists a graph with that degree sequence that is diamond-free.Comment: 8 pages, 2 figures, 2 algorithms, 2 models, 1 tabl

Exact Algorithms for Maximum Clique: a computational study

We investigate a number of recently reported exact algorithms for the maximum clique problem (MCQ, MCR, MCS, BBMC). The program code used is presented and critiqued showing how small changes in implementation can have a drastic effect on performance. The computational study demonstrates how problem features and hardware platforms influence algorithm behaviour. The minimum width order (smallest-last) is investigated, and MCS is broken into its consituent parts and we discover that one of these parts degrades performance. It is shown that the standard procedure used for rescaling published results is unsafe.Comment: 40 pages, 14 figures, 10 tables, 12 short java program listings, code afailable to download at http://www.dcs.gla.ac.uk/~pat/maxClique/distribution

A Parallel Branch and Bound Algorithm for the Maximum Labelled Clique Problem

The maximum labelled clique problem is a variant of the maximum clique problem where edges in the graph are given labels, and we are not allowed to use more than a certain number of distinct labels in a solution. We introduce a new branch-and-bound algorithm for the problem, and explain how it may be parallelised. We evaluate an implementation on a set of benchmark instances, and show that it is consistently faster than previously published results, sometimes by four or five orders of magnitude.Comment: Author-final version. Accepted to Optimization Letter