Complexity Dichotomies of Counting Problems

Complexity Dichotomies for Counting Problems

Dichotomies in Equilibrium Computation and Membership of PLC Markets in FIXP

Piecewise-linear, concave (PLC) utility functions play an important role in work done at the intersection of economics and algorithms. We prove that the problem of computing an equilibrium in Arrow-Debreu markets with PLC utilities and PLC production sets is in the class FIXP. Recently it was shown that these problems are also FIXP-hard (Garg et al., arXiv:1411.5060), hence settling the long-st...

Counting Homomorphisms to Trees Modulo a Prime

Many important graph theoretic notions can be encoded as counting graph homomorphism problems, such as partition functions in statistical physics, in particular independent sets and colourings. In this article we study the complexity of #pHomsToH, the problem of counting graph homomorphisms from an input graph to a graph H modulo a prime number p. Dyer and Greenhill proved a dichotomy stating t...

The challenges of unbounded treewidth in parameterised subgraph counting problems

Many of the existing tractability results for parameterised problems which involve finding or counting subgraphs with particular properties rely on bounding the treewidth of the minimal subgraphs having the desired property. In this paper, we give a number of hardness results – for decision, approximate counting and exact counting – in the case that this condition on the minimal subgraphs havin...

Approximately Counting H-Colorings is $\#\mathrm{BIS}$-Hard

We examine the computational complexity of approximately counting the list H-colourings of a graph. We discover a natural graph-theoretic trichotomy based on the structure of the graph H . If H is an irreflexive bipartite graph or a reflexive complete graph then counting list H-colourings is trivially in polynomial time. Otherwise, if H is an irreflexive bipartite permutation graph or a reflexi...