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 known that in general not every scaling-based algorithm can be made strongly polynomial, see, e.g., Hochbaum's work on the allocation problem @cite_2 .
@cite_2
1989262026 2162960148 2022049946 1856442002
We demonstrate the impossibility of strongly polynomial algorithms for the allocation problem, in the comparison model and in the algebraic tree computation model, that follow from lower bound results. Consequently, there are no strongly polynomial algorithms for nonlinear (concave) separable optimization over a totally unimodular constraint matrix. This is in contrast to the case when the objective is linear. We present scaling-based algorithms that use a greedy algorithm as a subroutine. The algorithms are polynomial for the allocation problem and its extensions and are also optimal for the sample allocation problem and the generalized upper bounds allocation problem, in that the complexity meets the lower bound derived from the comparison model. For other extensions of the allocation problem the scaling-based algorithms presented here are the fastest known. These algorithms are also polynomial time algorithms for solving with e accuracy the allocation problem and its extension in continuous variables. We present a deterministic strongly polynomial algorithm that computes the permanent of a nonnegative n × n matrix to within a multiplicative factor of en. To this end we develop the first strongly polynomial-time algorithm for matrix scaling an important nonlinear optimization problem with many applications. Our work suggests a simple new (slow) polynomial time decision algorithm for bipartite perfect matching, conceptually different from classical approaches. ∗Hebrew University. Work supported in part by a grant of the Binational Israel-US Science Foundation. †Hebrew University ‡Hebrew University. Work partially supported by grant 032-7736 from the Israel Academy of Sciences. Part of this work was done during a visit to the Institute for Advanced Study, under the support of a Sloan Foundation grant 96-6-2. Combinatorial strongly polynomial algorithms for minimizing submodular functions have been developed by Iwata, Fleischer, and Fujishige (IFF) and by Schrijver. The IFF algorithm employs a scaling scheme for submodular functions, whereas Schrijver's algorithm achieves strongly polynomial bound with the aid of distance labeling. Subsequently, Fleischer and Iwata have described a push relabel version of Schrijver's algorithm to improve its time complexity. This paper combines the scaling scheme with the push relabel framework to yield a faster combinatorial algorithm for submodular function minimization. The resulting algorithm improves over the previously best known bound by essentially a linear factor in the size of the underlying ground set. The eigenvalues of a matrix polynomial can be determined classically by solving a generalized eigenproblem for a linearized matrix pencil, for instance by writing the matrix polynomial in companion form. We introduce a general scaling technique, based on tropical algebra, which applies in particular to this companion form. This scaling, which is inspired by an earlier work of Akian, Bapat, and Gaubert, relies on the computation of “tropical roots”. We give explicit bounds, in a typical case, indicating that these roots provide accurate estimates of the order of magnitude of the different eigenvalues, and we show by experiments that this scaling improves the accuracy (measured by normwise backward error) of the computations, particularly in situations in which the data have various orders of magnitude. In the case of quadratic polynomial matrices, we recover in this way a scaling due to Fan, Lin, and Van Dooren, which coincides with the tropical scaling when the two tropical roots are equal. If not, the eigenvalues generally split in two groups, and the tropical method leads to making one specific scaling for each of the groups.