By now, we have a good understanding of how NP-hard problems become easier on graphs of bounded treewidth and bounded cliquewidth: for various problems, matching upper bounds and conditional lower bounds describe exactly how the running time has to depend on treewidth or cliquewidth. In particular, Fomin et al. (2009, 2010) have shown a significant difference between these two parameters: assuming the Exponential-Time Hypothesis (ETH), the optimal algorithms for problems such as MAX CUT and EDGE DOMINATING SET have running time 2~(O(t))n~(O(1)) when parameterized by treewidth, but nO(t) when parameterized by cliquewidth. In this paper, we show that a similar phenomenon occurs also for counting problems. Specifically, we prove that, assuming the counting version of the Strong Exponential-Time Hypothesis (#SETH), the problem of counting perfect matchings (1) has no (2 -ε)~kn~(O(1)) time algorithm for any ε > 0 on graphs of treewidth k (but it is known to be solvable in time 2~kn~(O(1)) if a tree decomposition of width k is given), and (2) has no O(n~((1 - ε)k)) time algorithm for any ε > 0 on graphs of cliquewidth k (but it can be solved in time O(n~(k+1)) if a k-expression is given). A celebrated result of Fisher, Kasteleyn, and Temperley from the 1960s shows that counting perfect matchings in planar graphs is polynomial-time solvable. This was later extended by Gallucio and Loebl (1999), Tesler (2000) and Regge and Zechina (2000) who gave 4~k ·n~(O(1)) time algorithms for graphs of genus k. We show that the dependence on the genus k has to be exponential: assuming #ETH, the counting version of ETH, there is no 2~(o(k)) · n~(O(1)) time algorithm for the problem on graphs of genus k.
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