Parameterized Algorithms for Boxicity

In this paper we initiate an algorithmic study of BOXICITY, a combinatorially well studied graph invariant, from the viewpoint of parameterized algorithms. The boxicity of an arbitrary graph G with the vertex set V (G) and the edge set E(C), denoted by box(G), is the minimum number of interval graphs on the same set of vertices such that the intersection of the edge sets of the interval graphs is E(C). In the BOXICITY problem we are given a graph G together with a positive integer k, and asked whether the box(G) is at most k. The problem is notoriously hard and is known to be NP-complete even to determine whether the boxicity of a graph is at most two. This rules out any possibility of having an algorithm with running time vertical bar V (G)vertical bar(O(f(k))), where f is an arbitrary function depending on k alone. Thus we look for other structural parameters like "vertex cover number" and "max leaf number" and see its effect on the problem complexity. In particular, we give an algorithm that given a vertex cover of size k finds box(G) in time 2(O(2kk2))vertical bar V(G)vertical bar. We also give a faster additive one approximation algorithm for finding box(G) that given a graph with vertex cover of size k runs in time 2(O(k2 log k))vertical bar V(G)vertical bar. Our next result is an additive two approximation algorithm for BOXICITY when parameterized by the "max leaf number running in time 2(O(k3 log k))vertical bar V(G)vertical bar(O(1)). Our results are based on structural relationships between boxicity and the corresponding parameter and could be of independent interest.