Lower and upper bounds for the linear arrangement problem on interval graphs

We deal here with theLinear Arrangement Problem(LAP) onintervalgraphs, any interval graph being given here together with its representation as theintersectiongraph of some collection of intervals, and so with relatedprecedenceandinclusionrelations. We first propose a lower boundLB, which happens to be tight in the case ofunit intervalgraphs. Next, we introduce the restriction PCLAP of LAP which is obtained by requiring any feasible solution of LAP to be consistent with theprecedencerelation, and prove that PCLAP can be solved in polynomial time. Finally, we show both theoretically and experimentally that PCLAP solutions are a good approximation for LAP onintervalgraphs.

A note on minimum linear arrangement for BC graphs

A linear arrangement is a labeling or a numbering or a linear ordering of the vertices of a graph. In this paper, we solve the minimum linear arrangement problem for bijective connection graphs (for short BC graphs) which include hypercubes, Möbius cubes, crossed cubes, twisted cubes, locally twisted cube, spined cube, [Formula: see text]-cubes, etc., as the subfamilies.

Variable neighborhood search for minimum linear arrangement problem

The minimum linear arrangement problem is widely used and studied in many practical and theoretical applications. It consists of finding an embedding of the nodes of a graph on the line such that the sum of the resulting edge lengths is minimized. This problem is one among the classical NP-hard optimization problems and therefore there has been extensive research on exact and approximative algorithms. In this paper we present an implementation of a variable neighborhood search (VNS) for solving minimum linear arrangement problem. We use Skewed general VNS scheme that appeared to be successful in solving some recent optimization problems on graphs. Based on computational experiments, we argue that our approach is comparable with the state-of-the-art heuristic.