Zwick's @math -approximation algorithm for the All Pairs Shortest Path (APSP) problem runs in time @math , where @math is the exponent of matrix multiplication and @math denotes the largest weight. This can be used to approximate several graph characteristics including the diameter, radius, median, minimum-weight triangle, and minimum-weight cycle in the same time bound. Since Zwick's algorithm uses the scaling technique, it has a factor @math in the running time. In this paper, we study whether APSP and related problems admit approximation schemes avoiding the scaling technique. That is, the number of arithmetic operations should be independent of @math ; this is called strongly polynomial. Our main results are as follows. - We design approximation schemes in strongly polynomial time @math for APSP on undirected graphs as well as for the graph characteristics diameter, radius, median, minimum-weight triangle, and minimum-weight cycle on directed or undirected graphs. - For APSP on directed graphs we design an approximation scheme in strongly polynomial time @math . This is significantly faster than the best exact algorithm. - We explain why our approximation scheme for APSP on directed graphs has a worse exponent than @math : Any improvement over our exponent @math would improve the best known algorithm for Min-Max Product In fact, we prove that approximating directed APSP and exactly computing the Min-Max Product are equivalent.
It is also worth mentioning that there are efficient algorithms for products in other algebraic structures, e.g., dominance product, @math -product, @math -product (see, e.g., @cite_31 ).
@cite_31
2268320119
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 (by Floyd and Warshall). 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 combinatorial data structure that maintains a matrix so that products with given sparse vectors can be computed efficiently. This allows us to obtain good running times for 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 an algorithm to find cheapest minimum nondecreasing paths. Besides finding optimal paths, we consider the related 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.