Fine-Grained Dichotomies for the Tutte Plane and Boolean #CSP

Abstract

Jaeger , Vertigan, and Welsh [15] proved a dichotomy for the complexity of evaluating the Tutte polynomial at fixed points: The evaluation is #Phard almost everywhere, and the remaining points admit polynomial-time algorithms. Dell, Husfeldt, and Wahlén [9] and Husfeldt and Taslaman [12], in combination with Curticapean [7], 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 exp o(n) unless #ETH fails.Another dichotomy theorem we strengthen is the one of Creignou and Hermann [6] for counting the number of satisfying assignments to a constraint satisfaction problem instance over the Boolean domain. We prove that all #P-hard cases are also hard under #ETH. The main ingredient is to prove that the number of independent sets in bipartite graphs with n vertices cannot be computed in time exp o(n) unless #ETH fails.In order to prove our results, we use the block interpolation idea by Curticapean [7] and transfer it to systems of linear equations that might not directly correspond to interpolation.

Parameterized Counting of Trees, Forests and Matroid Bases

“…The problem of counting all forests is #P-hard even when restricted to 3-regular bipartite planar graphs [19,9]. We will complement these results by showing that the problems of counting k-trees and k-forests are #W [1] hard when parameterized by k. In both proofs, we reduce from the problem of counting k-matchings. Note that we cannot hope to achieve hardness results on planar graphs, as Eppstein showed that the problem of counting subgraphs of size k is fixed parameter tractable on planar graphs [6].…”

“…During the last years, much work has been done in the field of parameterized counting complexity. Important results are the proof of #W [1]-hardness for counting the number of k-matchings in a simple graph [2], and the dichotomies for counting graph homomorphisms [10,4] and embeddings [3].…”

Achieving quantum supremacy with sparse and noisy commuting quantum computations

“…Recent results relating lower bounds for computing sparse Tutte polynomials to the exponential time hypothesis demonstrate that this is likely close to the optimal depth. Last year it was shown that precise evaluations of Tutte polynomials on sparse graphs cannot be performed in time exp(o(n)) without a violation of the counting equivalent of the exponential time hypothesis [9,15]. That is, if there were a sub-exponential runntime for Tutte polynomials on sparse graphs at all #P-hard points, then key NP-hard problems such as 3SAT could also be solved in subexponential time.…”

“…It remains to be seen whether a more sparse version of IQP sampling can be devised while retaining its classical hardness. Standard tensor network contraction techniques would allow any output probability of the above circuits on a square lattice to be classically computed in time O(2 D √ n ), so achieving a similar hardness result for D = o( √ n) would violate the counting exponential time hypothesis [9,15]. The challenge remains to remove a factor of log n from the depth while maintaining the anticoncentration requirements of [1,11].…”

Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders

“…The classical Tutte polynomial (as well as its specializations like the chromatic, flow or reliability polynomial) have received widespread attention, both from a combinatorial as well as a complexity theoretic perspective [47,2,78,7,41,29,11,9]. The classical Tutte polynomial is of special interest from a complexity theoretic perspective, as the Tutte polynomial encodes a plethora of properties of a graph: prominent examples include the chromatic number, the number of acyclic orientations, and the number of spanning trees; we refer the reader to the work of Jaeger et al [47] for a comprehensive overview.…”