Probabilistic analysis for metric optimization problems has mostly been conducted on random Euclidean instances, but little is known about metric instances drawn from distributions other than the Euclidean. This motivates our study of random metric instances for optimization problems obtained as follows: Every edge of a complete graph gets a weight drawn independently at random. The distance between two nodes is then the length of a shortest path (with respect to the weights drawn) that connects these nodes.We prove structural properties of the random shortest path metrics generated in this way. Our main structural contribution is the construction of a good clustering. Then we apply these findings to analyze the approximation ratios of heuristics for matching, the traveling salesman problem (TSP), and the k-median problem, as well as the running-time of the 2-opt heuristic for the TSP. The bounds that we obtain are considerably better than the respective worst-case bounds. This suggests that random shortest path metrics are easy instances, similar to random Euclidean instances, albeit for completely different structural reasons.

Abstract. The parallel complexity class NC 1 has many equivalent models such as polynomial size formulae and bounded width branching programs. Caussinus et al.[CMTV98] considered arithmetizations of two of these classes, #NC 1 and #BWBP. We further this study to include arithmetization of other classes. In particular, we show that counting paths in branching programs over visibly pushdown automata is in FLogDCFL, while counting proof-trees in logarithmic width formulae has the same power as #NC 1 . We also consider polynomial-degree restrictions of SC i , denoted sSC i , and show that the Boolean class sSC 1 is sandwiched between NC 1 and L, whereas sSC 0 equals NC 1 . On the other hand, the arithmetic class #sSC 0 contains #BWBP and is contained in FL, and #sSC 1 contains #NC 1 and is in SC 2 . We also investigate some closure properties of the newly defined arithmetic classes.

Given an input graph G and an integer k, the k-PATH problem asks whether there exists a path of length k in G. The counting version of the problem, #k-PATH asks to find the number of paths of length k in G. Recently, there has been a lot of work on finding and counting k-sized paths in an input graph. The current fastest (randomized) algorithm for k-PATH has been given by Williams and it runs in time O * (2 k ) [IPL, 2009 ]. The randomized algorithm for finding a k-path in the input graph was recently generalized by Koutis and Williams for testing whether there exists a subgraph in the input graph which is isomorphic to a given k-vertex tree [ICALP, 2009 ]. Björklund, Husfeldt, Kaski, and Koivisto [ESA, 2009 ] gave a deterministic algorithm for #k-PATH running in time and space O * ( n k/2 ) on an input graph with n vertices and gave a polynomial space algorithm running in time O * (3 k/2 n k/2 ). In this paper we study a natural generalization of both k-PATH and k-TREE problems, namely, the SUBGRAPH ISOMORPHISM problem. In the SUBGRAPH ISOMORPHISM problem we are given two graphs F and G on k and n vertices respectively as an input, and the question is whether there exists a subgraph of G isomorphic to F . We show that if the treewidth of F is at most t, then there is a randomized algorithm for the SUBGRAPH ISOMORPHISM problem running in time O * (2 k n 2t ).To do so, we associate a new multivariate Homomorphism polynomial of degree at most k with the SUBGRAPH ISOMORPHISM problem and construct an arithmetic circuit of size at most n O(t) for this polynomial. Using this polynomial, we also give a deterministic algorithm to count the number of homomorphisms from F to G that takes n O(t) time and uses polynomial space. For the counting version of the SUBGRAPH ISOMORPHISM problem, where the objective is to count the number of distinct subgraphs of G that are isomorphic to F , we give a deterministic algorithm running in time and space O * ( n k/2 n 2p ) or n k/2 n O(t log k) . We also give an algorithm running in time O * (2 k n k/2 n 5p ) and taking space polynomial in n. Here p and t denote the pathwidth and the treewidth of F , respectively. Thus our work not only improves on known results on SUBGRAPH ISOMORPHISM but it also extends and generalize most of the known results on k-PATH and k-TREE.

We prove that the regularity of binomial edge ideals of graphs obtained by gluing two graphs at a free vertex is the sum of the regularity of individual graphs. As a consequence, we generalize certain results of Zafar and Zahid. We obtain an improved lower bound for the regularity of trees. Further, we characterize trees which attain the lower bound. We prove an upper bound for the regularity of certain subclass of blockgraphs. As a consequence we obtain sharp upper and lower bounds for a class of trees called lobsters.

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