Finding the smallest H-subgraph in real weighted graphs and related problems

Let G be a graph with real weights assigned to the vertices (edges). The weight of a subgraph of G is the sum of the weights of its vertices (edges). The MIN H-SUBGRAPH problem is to find a minimum weight subgraph isomorphic to H, if one exists. Our main results are new algorithms for the MIN H-SUBGRAPH problem. The only operations we allow on real numbers are additions and comparisons. Our algorithms are based, in part, on fast matrix multiplication. For vertex-weighted graphs with n vertices we obtain the following results. We present an O(n^t(w,h)) time algorithm for MIN H-SUBGRAPH in case H is a fixed graph with h vertices and ω < 2.376 is the exponent of matrix multiplication. The value of t(ω, h) is determined by solving a small integer program. In particular, the smallest triangle can be found in O(n ^2+1/(4-ω)) ≤ o(n^2.616) time, the smallest K ₄ in O(n^ω+1) time, the smallest K₇ in O(n^4+3/(4-ω)) time. As h grows, t(ω,h) converges to 3h/(6 - ω) < 0.828h. Interestingly, only for h = 4, 5, 8 the running time of our algorithm essentially matches that of the (unweighted) H-subgraph detection problem. Already for triangles, our results improve upon the main result of [VW06]. Using rectangular matrix multiplication, the value of t(ω, h) can be improved; for example, the runtime for triangles becomes O(n ^2.575). We also present an algorithm whose running time is a function of m, the number of edges. In particular, the smallest triangle can be found in O(m^{(18-4ω)/(13-3ω)}) ≤ o(m^1.45) time. For edge-weighted graphs we present an O(m^2-1/k log n) time algorithm that finds the smallest cycle of length 2k or 2k - 1. This running time is identical, up to a logarithmic factor, to the running time of the algorithm of Alon et al. for the unweighted case. Using the color coding method and a recent algorithm of Chan for distance products, we obtain an O(n³/log n) time randomized algorithm for finding the smallest cycle of any fixed length.