Finding all k-cliques in k-partite graphs, an application in textile engineering

Abstract

In many practical cases one has to choose an arrangement of different objects so that they are compatible. Whenever the compatibility of the objects can be checked by a pair-wise comparison the problem can be modelled using the graph–theoretic notion of cliques. We consider a special case of the problem where the objects can be grouped so that exactly one object in every group has to be chosen. This object has to be compatible to every other object selected from the other groups. The problem was motivated by a braiding application from textile technology. The task is to route a set of thread-spools (bobbins) on a machine from their origins to their destinations so that collisions are avoided. We present a new model and algorithm in order to solve this important practical problem. Scope and purpose An important contribution of Operations Research is to develop computer-based tools for assistance in decision making. In this paper we show how techniques from combinatorial optimization can assist textile engineers in finding a control of a newly developed braiding machine. We have developed new models and have employed shortest path algorithms (Ahuja RK, Magnanti TL, Orlin JB. Network flows, theory, algorithms, and applications, Englewood Cliffs, NJ: Prentice-Hall, 1993. p. 93–165), and a branch and bound algorithm for the maximum clique problem (Johnson DS, Trick MA, (editors). Cliques, coloring, and satisfiability, DIMACS Series in discrete mathematics and theoretical computer science, Providence, RI: American Mathematical Society, 1996.) to compute controls. The determination of controls is a highly complex task, which can take days or even weeks if performed manually. Hence, computer-assistance is a necessity in practice. Indeed, the tools we have developed are employed in practice and have increased productivity enormously.