Online purchasing under uncertainty

TL;DR

This paper studies an online decision-making problem where one must select specific graph structures like spanning trees or matchings from randomly revealed costs, aiming to minimize total cost without prior knowledge.

Contribution

It introduces a framework for online purchasing of complex graph structures under uncertainty, extending previous models to include various target structures such as spanning trees and Hamilton cycles.

Findings

Develops algorithms for online selection of graph structures

Analyzes cost bounds for different target structures

Provides theoretical guarantees for online strategies

Abstract

Suppose there is a collection $x_1,x_2,\dots,x_N$ of independent uniform $[0,1]$ random variables, and a hypergraph $\cF$ of \emph{target structures} on the vertex set $\{1,\dots,N\}$. We would like to buy a target structure at small cost, but we do not know all the costs $x_i$ ahead of time. Instead, we inspect the random variables $x_i$ one at a time, and after each inspection, choose to either keep the vertex $i$ at cost $x_i$, or reject vertex $i$ forever. In the present paper, we consider the case where $\{1,\dots,N\}$ is the edge-set of some graph, and the target structures are the spanning trees of a graph, spanning arborescences of a digraph, the paths between a fixed pair of vertices, perfect matchings, Hamilton cycles or the cliques of some fixed size.