Finding, Minimizing, And Counting Weighted Subgraphs

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 it given large graph G on n nodes, its well as finding minimum weight copies in both node-weighted and edge-weighted graphs. Our results include:The number of copies of an H with ail independent set of size s can be computed exactly in O*(2(s)n(k-s+3)) time. A minimum weight COPY of such ail H (with arbitrary real weights oil nodes and edges) can be found in O(4(s+o(s))n(k-s+3)) little. (The O* notation omits poly(k) factors.) These algorithms rely oil fast algorithms for computing the permanent of a k x 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(omega k/3) + n(2k/3+o(1))) time, where omega < 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 it 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 Omega(n(2.5-epsilon)) time for all epsilon > 0, unless the 3SUM problem on N numbers can be solved in O(N2-epsilon) time. This suggests that the edge-weighted problem is much harder than its node-weighted version.