Detecting and Counting Small Subgraphs, and Evaluating a Parameterized Tutte Polynomial: Lower Bounds via Toroidal Grids and Cayley Graph Expanders
Abstract
Given a graph property , we consider the problem EdgeSub(), where the input is a pair of a graph G and a positive integer k, and the task is to decide whether G contains a k-edge subgraph that satisfies . Specifically, we study the parameterized complexity of EdgeSub() and of its counting problem \#EdgeSub() with respect to both approximate and exact counting. We obtain a complete picture for minor-closed properties : the decision problem EdgeSub() always admits an FPT algorithm and the counting problem \#EdgeSub() always admits an FPTRAS. For exact counting, we present an exhaustive and explicit criterion on the property which, if satisfied, yields fixed-parameter tractability and otherwise \#W[1]-hardness. Additionally, most of our hardness results come with an almost tight conditional lower bound under the so-called Exponential Time Hypothesis, ruling out algorithms for \#EdgeSub() that run in time f(k)·|G|o(k/ k) for any computable function f. As a main technical result, we gain a complete understanding of the coefficients of toroidal grids and selected Cayley graph expanders in the homomorphism basis of \#EdgeSub(). This allows us to establish hardness of exact counting using the Complexity Monotonicity framework due to Curticapean, Dell and Marx (STOC'17). Our methods can also be applied to a parameterized variant of the Tutte Polynomial TkG of a graph G, to which many known combinatorial interpretations of values of the (classical) Tutte Polynomial can be extended. As an example, TkG(2,1) corresponds to the number of k-forests in the graph G. Our techniques allow us to completely understand the parametrized complexity of computing the evaluation of TkG at every pair of rational coordinates (x,y).