Efficiently implementing maximum independent set algorithms on circle graphs

Abstract

Circle graphs are an important class of graph with applications in a number of areas, including compiler optimization, VLSI design and computational geometry. In this article, we provide an experimental evaluation of the two most efficient algorithms for computing the maximum independent set of a circle graph. We provide details of our implementations of the algorithms of Apostolico et al. and Valiente [2003], together with time, space, and other measurements. We describe optimizations to the algorithm of Apostolico et al. that significantly reduce both its running time and its memory consumption. We also correct an error in this algorithm. In addition, we show how to restructure and simplify Valiente's algorithm, allowing us to remove redundant computations from the algorithm resulting in much improved performance. Moreover, in the case of both algorithms, when the density of the circle graph's associated interval representation is increased beyond a certain point the efficacy of the optimizations we apply increases dramatically and as a function of the density. We provide experimental results over dense and sparse random circle graphs, as well as for circle graphs that arise when performing register allocation of software pipelined loops in a compiler.

An output sensitive algorithm for computing a maximum independent set of a circle graph

“…In these experiments, a variation of the algorithm described in this paper was used. This variation appears to have favourable constant factors and operates in time O(n min{d, α log n}), it was compared to an optimized implementation of the previous best algorithm, whose experimental performance was documented in [8]. The output sensitive algorithm was observed to offer much improved performance (by a factor of between 3 and 7).…”

“…Valiente [11] solved the problem in Θ(nd) time and only Θ(n) space. Nash et al [8] experimentally studied the relative performance of optimized implementations of both of the preceding algorithms, showing that when suitably implemented, Valiente's algorithm performs better. As we note in Section 6 a variation of the algorithm described in this paper has been experimentally observed to significantly out-perform the best performing previous algorithm.…”

Counting Hexagonal Patches and Independent Sets in Circle Graphs

“…Recognizing them and constructing a chord representation can be done in polynomial time [4,15], and the current fastest algorithm uses time O(n 2 ), where n is the number of vertices [22]. A number of problems that are NP-hard on general graphs are easy on circle graphs, such as in particular finding maximum independent sets [16,23,24,19].…”

Counting Hexagonal Patches and Independent Sets in Circle Graphs

“…Recognizing them and constructing a chord representation can be done in polynomial time [5,18], and the current fastest algorithm uses time O(m 2 ), where m is the number of vertices [26]. A number of problems that are NP-hard on general graphs are easy on circle graphs, such as in particular finding maximum independent sets [19,23,27,28]. The first algorithm for the optimization problem by Gavril [19] has time complexity O(m 3 ), where m is the number of vertices of the circle graph, which is the number of edges of the corresponding chord model graph.…”