Tight local approximation results for max-min linear programs

TL;DR

This paper establishes tight bounds for local approximation algorithms in bipartite max-min linear programs, showing the best possible approximation ratio achievable with local information and introducing a new graph unfolding technique.

Contribution

The paper provides tight approximation bounds for local algorithms in bipartite max-min LPs and introduces the graph unfolding technique for designing such algorithms.

Findings

Achieves approximation ratio of Δ_I (1 - 1/Δ_K) + ε with local algorithms.

Proves no local algorithm can surpass the ratio Δ_I (1 - 1/Δ_K).

Introduces the graph unfolding technique for local approximation algorithms.

Abstract

In a bipartite max-min LP, we are given a bipartite graph $\myG = (V \cup I \cup K, E)$, where each agent $v \in V$ is adjacent to exactly one constraint $i \in I$ and exactly one objective $k \in K$. Each agent $v$ controls a variable $x_v$. For each $i \in I$ we have a nonnegative linear constraint on the variables of adjacent agents. For each $k \in K$ we have a nonnegative linear objective function of the variables of adjacent agents. The task is to maximise the minimum of the objective functions. We study local algorithms where each agent $v$ must choose $x_v$ based on input within its constant-radius neighbourhood in $\myG$. We show that for every $\epsilon>0$ there exists a local algorithm achieving the approximation ratio ${\Delta_I (1 - 1/\Delta_K)} + \epsilon$. We also show that this result is the best possible -- no local algorithm can achieve the approximation ratio ${\Delta_I (1 - 1/\Delta_K)}$. Here $\Delta_I$ is the maximum degree of a vertex $i \in I$, and $\Delta_K$ is the maximum degree of a vertex $k \in K$. As a methodological contribution, we introduce the technique of graph unfolding for the design of local approximation algorithms.