Beating Treewidth for Average-Case Subgraph Isomorphism
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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 is true, proved a lower bound of Ω(n^const· emb(G)) for a certain graph parameter emb(G) = Ω(tw(G)/ 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)/ 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 hypercube, then κ(G) is Θ(tw(G)/√( 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^2κ(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.
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