Combinatorial Optimization with Rational Objective Functions

Let A be the problem of minimizing c1, x1, +... + cnxn subject to certain constraints on x = 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[pn] comparisons and O[qn] additions, then B is solvable in time O[pnqn + pn]. 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 withing 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.