July 28, 2008
Virginia Panayotova Vassilevska
10:00 AM, 1507 Newell-Simon Hall
Title: Efficient Algorithms for Path Problems in Weighted Graphs
Problems related to computing optimal paths have been abundant in computer science since its emergence as a field. Yet for a large number of such problems we still do not know whether the state-of-the-art algorithms are the best possible. A notable example of this phenomenon is the all pairs shortest paths problem in a directed graph with real edge weights. The best algorithm (modulo small polylogarithmic improvements) for this problem runs in cubic time, a running time known since the 1960s. Our grasp of many such fundamental algorithmic questions is far from optimal, and the major goal of this thesis is to bring some new insights into efficiently solving path problems in graphs.
We focus on several path problems optimizing different measures: shortest paths, maximum bottleneck paths, minimum nondecreasing paths, and various extensions. For the all-pairs versions of these path problems we use an algebraic approach. We obtain improved algorithms using reductions to fast matrix multiplication. For maximum bottleneck paths and minimum nondecreasing paths we are the first to break the cubic barrier, obtaining truly subcubic strongly polynomial algorithms. We also consider a nonalgebraic, combinatorial approach, which is considered more efficient in practice compared to methods based on fast matrix multiplication. We present a data structure which maintains a matrix so that products with given sparse vectors can be computed efficiently. This allows us to obtain good running times for several path problems in unweighted sparse graphs.
This thesis also gives algorithms for some single source path problems. We obtain the first linear time algorithm for the single source minimum nondecreasing paths problem. We give some extensions to this, including algorithms to find shortest minimum nondecreasing paths and cheapest minimum nondecreasing paths. Besides finding optimal paths, we consider the problem of finding optimal cycles. In particular, we focus on the problem of finding in a weighted graph a triangle of maximum weight sum. We obtain the first truly subcubic algorithm for finding a maximum weight triangle in a node-weighted graph. We also present algorithms for the edge-weighted case. These algorithms immediately imply good algorithms for finding maximum weight k-cliques, or arbitrary maximum weight pattern subgraphs of fixed size.
Guy Blelloch, Chair
Uri Zwick, Tel Aviv University