Tight local approximation results for max-min linear programs
Abstract
In a bipartite max-min LP, we are given a bipartite graph = (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 . We show that for every ε>0 there exists a local algorithm achieving the approximation ratio I (1 - 1/K) + ε. 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.