On the Parameterized Parallel Complexity and the Vertex Cover Problem

Abstract

Efficiently parallelizable parameterized problems have been classified as being either in the class FPP (fixed-parameter parallelizable) or the class PNC (parameterized analog of NC), which contains FPP as a subclass. In this paper, we propose a more restrictive class of parallelizable parameterized problems called fixed-parameter parallel-tractable (FPPT). For a problem to be in FPPT, it should possess an efficient parallel algorithm not only from a theoretical standpoint but in practice as well. The primary distinction between FPPT and FPP is the parallel processor utilization, which is bounded by a polynomial function in the case of FPPT. We initiate the study of FPPT with the well-known k-vertex cover problem. In particular, we present a parallel algorithm that outperforms the best known parallel algorithm for this problem: using $\mathcal {O}(m)$ instead of $\mathcal {O}(n^2)$ parallel processors, the running time improves from $4\log n + \mathcal {O}(k^k)$ to $\mathcal {O}(k\cdot \log ^3 n)$, where m is the number of edges, n is the number of vertices of the input graph, and k is an upper bound of the size of the sought vertex cover. We also note that a few P-complete problems fall into FPPT including the monotone circuit value problem (MCV) when the underlying graphs are bounded by a constant Euler genus.

This work was partially supported by the German Research Foundation (DFG) within the Collaborative Research Center “On-The-Fly Computing” (SFB 901) and the International Graduate School “Dynamic Intelligent Systems”.