Taming computational complexity: efficient and parallel SimRank optimizations on undirected graphs

SimRank has been considered as one of the promising link-based ranking algorithms to evaluate similarities of web documents in many modern search engines. In this paper, we investigate the optimization problem of SimRank similarity computation on undirected web graphs. We first present a novel algorithm to estimate the SimRank between vertices in O(n3+ Kn2) time, where n is the number of vertices, and K is the number of iterations. In comparison, the most efficient implementation of SimRank algorithm in [1] takes O(K n3 ) time in the worst case. To efficiently handle large-scale computations, we also propose a parallel implementation of the SimRank algorithm on multiple processors. The experimental evaluations on both synthetic and real-life data sets demonstrate the better computational time and parallel efficiency of our proposed techniques

Holographic Algorithm with Matchgates Is Universal for Planar #\#CSP Over Boolean Domain

We prove a complexity classification theorem that classifies all counting constraint satisfaction problems ($\#$CSP) over Boolean variables into exactly three categories: (1) Polynomial-time tractable; (2) $\#$P-hard for general instances, but solvable in polynomial-time over planar graphs; and (3) $\#$P-hard over planar graphs. The classification applies to all sets of local, not necessarily symmetric, constraint functions on Boolean variables that take complex values. It is shown that Valiant's holographic algorithm with matchgates is a universal strategy for all problems in category (2).Comment: 94 page