The Complexity of Counting in Sparse, Regular, and Planar Graphs

Abstract

We show that a number of graph-theoretic counting problems remain ${\cal NP}$-hard, indeed $#{\cal P}$-complete, in very restricted classes of graphs. In particular, we prove that the problems of counting matchings, vertex covers, independent sets, and extremal variants of these all remain hard when restricted to planar bipartite graphs of bounded degree or regular graphs of constant degree. We obtain corollaries about counting cliques in restricted classes of graphs and counting satisfying assignments to restricted classes of monotone 2-CNF formulae. To achieve these results, a new interpolation-based reduction technique which preserves properties such as constant degree is introduced.