Combinatorial optimization with rational objective functions

Abstract

Let A be the problem of minimizing c1x1+...+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[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.