Probabilistic analysis of algorithms for cost constrained minimum weighted combinatorial objects

TL;DR

This paper analyzes algorithms for cost-constrained minimum spanning trees and assignment problems with probabilistic edge weights, providing asymptotic optimality results using Lagrangian duality under specific distributional assumptions.

Contribution

It introduces polynomial-time algorithms for these problems that are asymptotically optimal, extending analysis to multiple constraints for spanning trees.

Findings

Algorithms achieve asymptotic optimality under specified distributional conditions.

Extension of analysis to multiple constraints for spanning trees.

Provides theoretical foundations for probabilistic cost-constrained combinatorial optimization.

Abstract

We consider cost constrained versions of the minimum spanning tree problem and the assignment problem. We assume edge weights are independent copies of a continuous random variable $Z$ that satisfies $F(x)=\Pr(Z\leq x)\approx x^\alpha$ as $x\to0$, where $\alpha\geq 1$. Also, there are $r=O(1)$ budget constraints with edge costs chosen from the same distribution. We use Lagrangean duality to construct polynomial time algorithms that produce asymptotically optimal solutions. For the spanning tree problem, we allow $r>1$, but for the assignment problem we can only analyse the case $r=1$.