Tight Local Approximation Results for Max-Min Linear Programs

In a bipartite max-min LP, we are given a bipartite graph G = (V ∪ I ∪ K,E), where each agent v ∈ V is adjacent to exactly one constraint i ∈ I and exactly one objective k ∈ K. Each agent v controls a variable xv. For each i ∈ I we have a nonnegative linear constraint on the variables of adjacent agents. For each k ∈ 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 xv based on input within its constant-radius neighbourhood in G. We show that for every ǫ > 0 there exists a local algorithm achieving the approximation ratio ∆I(1− 1/∆K)+ǫ. We also show that this result is the best possible – no local algorithm can achieve the approximation ratio ∆I(1− 1/∆K). Here ∆I is the maximum degree of a vertex i ∈ I , and ∆K is the maximum degree of a vertex k ∈ K. As a methodological contribution, we introduce the technique of graph unfolding for the design of local approximation algorithms.