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.
APSP and APBP can be easily computed in time @math on quantum computers @cite_40 . designed the first quantum algorithm for computing in time @math , and noted that every problem equivalent to APBP admits a nontrivial @math -time algorithm in the quantum realm.
@cite_40
1554473710
We consider the quantum time complexity of the all pairs shortest paths (APSP) problem and some of its variants. The trivial classical algorithm for APSP and most all pairs path problems runs in @math time, while the trivial algorithm in the quantum setting runs in @math time, using Grover search. A major open problem in classical algorithms is to obtain a truly subcubic time algorithm for APSP, i.e. an algorithm running in @math time for constant @math . To approach this problem, many truly subcubic time classical algorithms have been devised for APSP and its variants for structured inputs. Some examples of such problems are APSP in geometrically weighted graphs, graphs with small integer edge weights or a small number of weights incident to each vertex, and the all pairs earliest arrivals problem. In this paper we revisit these problems in the quantum setting and obtain the first nontrivial (i.e. @math time) quantum algorithms for the problems.