Algorithms for shortest paths.
Algorithms for shortest paths.
The complexity of dynamic languages and dynamic optimization problems
STOC '81 Proceedings of the thirteenth annual ACM symposium on Theory of computing
Optimal Embedding into Star Metrics
WADS '09 Proceedings of the 11th International Symposium on Algorithms and Data Structures
ISAAC'05 Proceedings of the 16th international conference on Algorithms and Computation
Vyacheslav Tanaev: contributions to scheduling and related areas
Journal of Scheduling
Polynomial-Time approximation schemes for shortest path with alternatives
ESA'12 Proceedings of the 20th Annual European conference on Algorithms
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Let A be the problem of minimizing c1x1+...+cnxn subject to certain constraints on x&equil;(x1,...,xn), and let B be the problem of minimizing (a0+a1x1+...+anxn)/(b0+b1x1+...+bnxn) subject to the same constraints, assuming the denominator is always positive. It is shown that if A is solvable within O[p(n)] comparisons and O[q(n)] additions, then B is solvable in time O[p(n)(q(n)+p(n))]. This applies to most of the “network” algorithms. Consequently, minimum ratio cycles, minimum ratio spanning trees, minimum ratio (simple) paths, maximum ratio weighted matchings, etc., can be computed within polynomial-time in the number of variables. This improves a result of E. L. Lawler, namely, that a minimum ratio cycle can be computed within a time bound which is polynomial in the number of bits required to specify an instance of the problem. A recent result on minimum ratio spanning trees by R. Chandrasekaran is also improved by the general arguments presented in this paper. Algorithms of time-complexity O(¦E¦ • ¦V¦2•log¦V¦) for a minimum ratio cycle and O(¦E¦ • log2¦V¦ • log log ¦V¦) for a minimum ratio spanning tree are developed.