Abstract
Jaeger et al. (Math Proc Camb Philos Soc 108(1):35–53, 1990) proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: the evaluation is #P-hard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén (in: ICALP 2010, vol. 6198, pp. 426–437, Springer, Berlin, Heidelberg, 2010) and Husfeldt and Taslaman (in: IPEC 2010, vol. 6478, pp. 192–203, Springer, Berlin, Heidelberg, 2010) in combination with the results of Curticapean (in: ICALP 2015, pp. 380–392, Springer, 2015), extended the #P-hardness results to tight lower bounds under the counting exponential time hypothesis #ETH, with the exception of the line $$y=1$$ , which was left open. We complete the dichotomy theorem for the Tutte polynomial under #ETH by proving that the number of all acyclic subgraphs of a given n-vertex graph cannot be determined in time unless #ETH fails. Another dichotomy theorem we strengthen is the one of Creignou and Hermann (Inf Comput 125(1):1–12, 1996) for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that the #P-hard cases cannot be solved in time unless #ETH fails. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time unless #ETH fails. In order to prove our results, we use the block interpolation idea by Curticapean and transfer it to systems of linear equations that might not directly correspond to interpolation.
Highlights
Counting combinatorial objects is at least as hard as detecting their existence, and often it is harder
Valiant [20] introduced the complexity class #P to study the complexity of counting problems and proved that counting the number of perfect matchings in a given bipartite graph is #P-complete
It has been hypothesized that no O 1.99n/2 -time algorithm for the problem exists, but we do not know whether such an algorithm has implications for the strong exponential time hypothesis, which states that for all ε > 0, there is some k such that the problem of deciding satisfiability of boolean formulas in k-CNF on n variables does not have an algorithm running in time (2 − ε)n
Summary
Counting combinatorial objects is at least as hard as detecting their existence, and often it is harder. Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP polynomial-time algorithm that is given access to an oracle for counting perfect matchings. If #ETH holds, neither of these problems has an algorithm running in time exp(o(n)) even in simple n-vertex graphs of bounded maximum degree. If #ETH holds, there exist constants ε, C > 0 such that no O(exp(εn))-time algorithm can compute the number of all forests in a given simple n-vertex graph with at most C · n edges. If #ETH holds, there exist constants ε > 0 and D ∈ N such that no O(exp(εn))-time algorithm can compute the number of independent sets in bipartite n-vertex graphs of maximum degree at most D. We consider Corollary 4 to be a first step towards understanding the fine-grained complexity of technically much more challenging dichotomies, such as the ones for counting CSPs with complex weights of Cai and Chen [3], or the dichotomy for Holant problems with symmetric signatures over the Boolean domain of Cai, Lu and Cia [4]
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