Combinatorial Optimization with Rational Objective Functions

Abstract

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[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 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.

Budgeted Matching via the Gasoline Puzzle

“…This can be done in polynomial time by standard techniques whenever the unbudgeted problem (Π ) can be solved in polynomial time [9]. It can even be done in strongly polynomial time by using Megiddo's parametric search technique [5]. This technique can be used because combinatorial algorithms (only using comparisons and additions of weights) exist for (Π ) (see, e.g., [10]).…”

Approximation Algorithms for Reliable Stochastic Combinatorial Optimization

“…Similarly, continuous optimization work with probability (chance) constraints (e.g., [29]) applies for linear and not discrete optimization problems. Additional related work on the combinatorial optimization side includes research on multi-criteria optimization (e.g., [32,1,35,40]) and combinatorial optimization with a ratio of linear objectives [27,33]. Our models can also be seen as instances of concave discrete minimization; however, the existing work in this area requires assumptions that do not hold in our framework, such as restrictive properties on the feasible set, strictly positive range of the objective function, or boundedness/positivity of the objective function gradient [31,3,22,14].…”

Verifying Nash Equilibria in PageRank Games on Undirected Web Graphs

“…Problem Q can be solved directly by Megiddo's method in O(n 2 log 2 n) time [3], and it can be also solved by Newton's Method in time O(n 2 log n) [4]. However we are able to specialize Megiddo's method for our problem to obtain an O(n 2 ) time algorithm.…”

“…However we are able to specialize Megiddo's method for our problem to obtain an O(n 2 ) time algorithm. Our approach initially follows the technique describe in [3]. Since max f (x) = max l l g(x), we can solve problem P by solving problem Q for each l = 1, ..., d v .…”

“…One of the novel ideas is the application of line arrangements to speed up Megiddo's [3] technique for fractional optimization.…”