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

We show that a number of graph-theoretic counting problems remain NP-hard, indeed #P-complete, in very restricted classes of graphs. In particular, it is shown 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. To achieve these results, a new interpolation-based reduction technique which preserves properties such as constant degree is introduced. In addition, the problem of approximately counting minimum cardinality vertex covers is shown to remain NP-hard even when restricted to graphs of maximal degree 3. Previously, restricted-case complexity results for counting problems were elusive; we believe our techniques may help obtain similar results for many other counting problems.