For a pattern graph H on k nodes, we consider the problems of finding and counting the number of (not necessarily induced) copies of H in a given large graph G on n nodes, as well as finding minimum weight copies in both nodeweighted and edge-weighted graphs. Our results include:• The number of copies of an H with an independent set of size s can be computed exactly in O * (2 s n k−s+3 ) time. A minimum weight copy of such an H (with arbitrary real weights on nodes and edges) can be found inThe O * notation omits poly(k) factors.) These algorithms rely on fast algorithms for computing the permanent of a k × n matrix, over rings and semirings.• The number of copies of any H having minimum (or maximum) node-weight (with arbitrary real weights on nodes) can be found in O(n ωk/3 + n 2k/3+o(1) ) time, where ω < 2.4 is the matrix multiplication exponent and k is divisible by 3. Similar results hold for other values of k. Also, the number of copies having exactly a prescribed weight can be found within this time. These algorithms extend the technique of Czumaj and Lingas (SODA 2007) and give a new (algorithmic) application of multiparty communication complexity.• Finding an edge-weighted triangle of weight exactly 0 in general graphs requires Ω(n 2.5−ε ) time for all ε > 0, unless the 3SUM problem on N numbers can be solved in O(N 2−ε ) time. This suggests that the edge-weighted problem is much harder than its node-weighted version.

In the all-pairs bottleneck paths (APBP) problem (a.k.a. allpairs maximum capacity paths), one is given a directed graph with real non-negative capacities on its edges and is asked to determine, for all pairs of vertices s and t, the capacity of a single path for which a maximum amount of flow can be routed from s to t. The APBP problem was first studied in operations research, shortly after the introduction of maximum flows and all-pairs shortest paths.We present the first truly sub-cubic algorithm for APBP in general dense graphs. In particular, we give a procedure for computing the (max, min)-product of two arbitrary matrices over R ∪ {∞, −∞} in O(n 2+ω/3 ) ≤ O(n 2.792 ) time, where n is the number of vertices and ω is the exponent for matrix multiplication over rings. Using this procedure, an explicit maximum bottleneck path for any pair of nodes can be extracted in time linear in the length of the path.

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We present the first truly sub-cubic algorithms for finding a maximum node-weighted triangle in directed and undirected graphs with arbitrary real weights. The first is an O(B · n 3+ω 2 ) = O(B · n 2.688 ) deterministic algorithm, where n is the number of nodes, ω is the matrix multiplication exponent, and B is the number of bits of precision. The second is a strongly polynomial randomized algorithm that runs in O(n 3+ω 2 log n) expected worst-case time. To achieve this, we show how to efficiently sample a weighted triangle uniformly at random, out of just those triangles whose total weight falls in some prescribed interval (W1, W2) for arbitrary weights W1 and W2. Previous approaches to the problem resulted in time bounds with either an exponential dependence on B, or a runtime of the form Ω(n 3 /(log n) c ). The algorithms are easily extended to finding a maximum node-weighted induced subgraph on 3k nodes iñ) =Õ(n 2.688k ) time. We give applications to a variety of problems, including a stable matching problem between buyers and sellers in computational economics, and discuss the possibility of extending our approach to a truly sub-cubic algorithm for computing all-pairs shortest paths on directed graphs with arbitrary weights.

Abstract. 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(ω,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 K4 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) Hsubgraph 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(n 3 / log n) time randomized algorithm for finding the smallest cycle of any fixed length.