Purchasing a C_4 online

TL;DR

This paper studies an online decision-making problem for purchasing a specific subgraph, a 4-cycle, in a complete graph with randomly assigned edge costs, aiming to minimize total cost under uncertainty and sequential inspection constraints.

Contribution

It introduces a novel online algorithm for selecting a 4-cycle in a complete graph with unknown uniform edge costs, balancing exploration and exploitation.

Findings

Developed an online strategy for purchasing a C_4 cycle

Achieved bounds on expected purchase costs

Provided analysis of decision thresholds in the online setting

Abstract

Let $G$ be a graph with edge set $(e_1,e_2,...e_N)$. We independently associate to each edge $e_i$ of $G$ a cost ${x}_i$ that is drawn from a Uniform [0, 1] distribution. Suppose $\mathcal{F}$ is a set of targeted structures that consists of subgraphs of $G$. We would like to buy a subset of $\mathcal{F}$ at small cost, however we do not know a priori the values of the random variables ${x}_1,...,{x}_N$. Instead, we inspect the random variables $x_i$ one at a time. As soon as we inspect the random variable associated with the cost of an edge we have to decide whether we want to buy that edge or reject it for ever. In the present paper we consider the case where $G$ is the complete graph on $n$ vertices and $\mathcal{F}$ is the set of all $C_4$ -cycles on 4 vertices- out of which we want to buy one.