Efficiently parallelizable problems on a class of decomposable graphs

Abstract

In the literature, there are quite a few sequential and parallel algorithms to solve problems on decomposable graphs utilizing distinct techniques. Trees, series-parallel graphs, outerplanar graphs, and bandwidth-k graphs all belong to decomposable graphs. Let $T_{\mathrm{d}}(|V|,|E|)$ and $P_{\mathrm{d}}(|V|,|E|)$ denote the time complexity and processor complexity required to construct a parse tree representation $T_G$ for a decomposable graph $G=(V,E)$ on a PRAM model $M_{\mathrm{d}}$. We define a general problem-solving paradigm to solve a wide class of subgraph optimization problems on decomposable graphs in $O(T_{\mathrm{d}}(\left|V\right|,\left|E\right|)+\log |V(T_G)|)$ time using $O(P_{\mathrm{d}}(\left|V\right|,\left|E\right|)+\frac{|V(T_G)|}{\log |V(T_G)|})$ processors on $M_{\mathrm{d}}$.

We also demonstrate the following examples fitting into our paradigm:

• The maximum independent set problem on trees,

• The maximum matching problem on series-parallel graphs, and

• The efficient domination problem on series-parallel graphs.

Our results improve the previously best known results of (1) and (2).