33 research outputs found
Homomorphisms are a good basis for counting small subgraphs
We introduce graph motif parameters, a class of graph parameters that depend
only on the frequencies of constant-size induced subgraphs. Classical works by
Lov\'asz show that many interesting quantities have this form, including, for
fixed graphs , the number of -copies (induced or not) in an input graph
, and the number of homomorphisms from to .
Using the framework of graph motif parameters, we obtain faster algorithms
for counting subgraph copies of fixed graphs in host graphs : For graphs
on edges, we show how to count subgraph copies of in time
by a surprisingly simple algorithm. This
improves upon previously known running times, such as time
for -edge matchings or time for -cycles.
Furthermore, we prove a general complexity dichotomy for evaluating graph
motif parameters: Given a class of such parameters, we consider
the problem of evaluating on input graphs , parameterized
by the number of induced subgraphs that depends upon. For every recursively
enumerable class , we prove the above problem to be either FPT or
#W[1]-hard, with an explicit dichotomy criterion. This allows us to recover
known dichotomies for counting subgraphs, induced subgraphs, and homomorphisms
in a uniform and simplified way, together with improved lower bounds.
Finally, we extend graph motif parameters to colored subgraphs and prove a
complexity trichotomy: For vertex-colored graphs and , where is from
a fixed class , we want to count color-preserving -copies in
. We show that this problem is either polynomial-time solvable or FPT or
#W[1]-hard, and that the FPT cases indeed need FPT time under reasonable
assumptions.Comment: An extended abstract of this paper appears at STOC 201