Counting and Detecting Small Subgraphs via Equations

Abstract

We present a general technique for detecting and counting small subgraphs. It consists of forming special linear combinations of the numbers of occurrences of different induced subgraphs of fixed size in a graph. These combinations can be efficiently computed by rectangular matrix multiplication. Our two main results utilizing the technique are as follows. Let H be a fixed graph with k vertices and an independent set of size s. 1. Detecting if an n-vertex graph contains a (not necessarily induced) subgraph isomorphic to H can be done in time O(n ω((k−s)/2 ,1, (k−s)/2)), where ω(p, q, r) is the exponent of fast arithmetic matrix multiplication of an n p × n q matrix by an n q × n r matrix. 2. When s = 2, counting the number of (not necessarily induced) subgraphs isomorphic to H can be done in the same time, i.e., in time O(n ω((k−2)/2 ,1, (k−2)/2)). It follows in particular that we can count the number of subgraphs isomorphic to any H on four vertices that is not K 4 in time O(n ω), where ω = ω(1, 1, 1) is known to be smaller than 2.373. Similarly, we can count the number of subgraphs isomorphic to any H on five vertices that is not K 5 in time O(n ω(2,1,1)), where ω(2, 1, 1) is known to be smaller than 3.257. Finally, we derive inputsensitive variants of our time upper bounds. They are partially expressed in terms of the number m of edges of the input graph and do not rely on fast matrix multiplication.

Finding Four-Node Subgraphs in Triangle Time

“…They gave an interesting equivalence: if one can compute the number of occurrences of any induced subgraph on 4 nodes in T (n) time, then the number of occurrences of all other 4-node subgraphs on 4 nodes can also be computed in O(n ω + T (n)) time. Kowaluk et al [KLL13] generalized this result for k > 4, to show i of H. If (i, j) is not an edge of H, then for every node u in partition i and node v in partition j, remove (u, v) if it was an edge. If the new graph G contains an induced copy of H, then G contains a noninduced one since G is a supergraph of G .…”

Homomorphisms are a good basis for counting small subgraphs

“…[40,17]). Essentially, one can exhaustively iterate over the image of the minimum vertex-cover in G, which gives rise to n vc(G) choices; the rest of H can then be embedded by dynamic programming.…”

Counting Thin Subgraphs via Packings Faster than Meet-in-the-Middle Time

“…In particular in the context of subgraph counting such techniques can be traced back at least to the triangle-and cycle-counting algorithms of Itai and Rodeh [16], with more recent uses including the algorithms of Kowaluk, Lingas, and Lundell [20] that improve upon algorithms of Nešetřil and Poljak [24] and Vassilevska and Williams [26] for counting small dense subgraphs (k < 10) with a maximum independent set of size 2. Also the counting-in-halves technique of Björklund et al [6] can be seen to solve an (implicit) system of linear equations to recover weighted disjoint packings.…”

“…For now we will be content in simply defining the equations and providing an illustration in Fig. 2. (The eventual algorithmic serendipity of this construction will be revealed only later in (20) and (21).) Let i = 0, 1, .…”

“…Let us use the shorthandp = 1 − p for the complement of p ∈ {0, 1}. Denoting pointwise multiplication of functions by "·", from (20), (7) and (18) it immediately follows that Proof. First evaluate the parity transforms f π p , gπ p , hπ p for all p ∈ {0, 1} and s ≤ (3/2 − γ)q using Lemma 4.…”