Beating Treewidth for Average-Case Subgraph Isomorphism

Abstract

For any fixed graph G, the subgraph isomorphism problem asks whether an n-vertex input graph has a subgraph isomorphic to G. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of the problem, denoted G-SUB, and then solves G-SUB in time O(n tw(G)+1 ) where tw(G) is the treewidth of G. Marx (2010) conjectured that G-SUB requires time Ω(n const•tw(G) ) and, assuming the Exponential Time Hypothesis, proved a lower bound of Ω(n const•emb(G) ) for a certain graph parameter emb(G) = Ω(tw(G)/ log tw(G)). With respect to the size of AC 0 circuits solving G-SUB, Li, Razborov and Rossman (2017) proved an unconditional average-case lower bound of Ω(n κ(G) ) for a different graph parameter κ(G) = Ω(tw(G)/ log tw(G)).Our contributions are as follows. First, we show that emb(G) is at most O(κ(G)) for all graphs G. Next, we show that κ(G) can be asymptotically less than tw(G); for example, if G is a hypercubelog tw(G) . Finally, we construct AC 0 circuits of size O(n κ(G)+const ) that solve G-SUB in the average case, on a variety of product distributions. This improves an O(n 2 κ(G)+const ) upper bound of Li et al., and shows that the average-case complexity of G-SUB is n o(tw(G)) for certain families of graphs G such as hypercubes. ACM Subject ClassificationTheory of computation → Circuit complexity; Theory of computation → Fixed parameter tractability; Mathematics of computing → Graph algorithms Keywords and phrases subgraph isomorphism, average-case complexity, AC 0 , circuit complexity

Current Algorithms for Detecting Subgraphs of Bounded Treewidth are Probably Optimal

“…Additional to the conditional lower bound of O(n o(tw(H)/ log(tw(H))) g(k)) by Marx [54], there is an unconditional lower bound of O(n κ(H) ) for the size of any AC 0 -circuit, for some graph parameter κ(H) = Ω(tw(H)/ log(tw(H))), which holds even when considering the average case [50]. Interestingly, the factor of 1/ log(tw(H)) does not seem to be an artefact of the proof: There is an AC 0 -circuit of size O(n o(tw(H)) g(k)) that solves the problem on certain unbounded-treewidth classes in the average case [62].…”