Orienting (hyper)graphs under explorable stochastic uncertainty

TL;DR

This paper addresses the problem of efficiently identifying minimum-weight nodes in hypergraphs with uncertain weights, proposing competitive algorithms with proven bounds for various graph classes.

Contribution

It introduces polynomial-time algorithms with competitive ratios for hypergraph orientation under uncertainty, extending to special cases like bipartite graphs and single hyperedges.

Findings

Developed a $f(eta)$-competitive algorithm with ratio between 1.618+ε and 2.

Proved no similar approach can surpass 1.5-competitiveness.

Provided 4/3-competitive algorithms for bipartite graphs and single hyperedge hypergraphs.

Abstract

Given a hypergraph with uncertain node weights following known probability distributions, we study the problem of querying as few nodes as possible until the identity of a node with minimum weight can be determined for each hyperedge. Querying a node has a cost and reveals the precise weight of the node, drawn from the given probability distribution. Using competitive analysis, we compare the expected query cost of an algorithm with the expected cost of an optimal query set for the given instance. For the general case, we give a polynomial-time $f(\alpha)$-competitive algorithm, where $f(\alpha)\in [1.618+\epsilon,2]$ depends on the approximation ratio $\alpha$ for an underlying vertex cover problem. We also show that no algorithm using a similar approach can be better than $1.5$-competitive. Furthermore, we give polynomial-time $4/3$-competitive algorithms for bipartite graphs with arbitrary query costs and for hypergraphs with a single hyperedge and uniform query costs, with matching lower bounds.