Tight Local Approximation Results for Max-Min Linear Programs

In a bipartite max-min LP, we are given a bipartite graph G = (V boolean OR I boolean OR K, E), where each agent v is an element of V is adjacent to exactly one constraint i is an element of I and exactly one objective k is an element of K. Each agent v controls a variable x(v). For each i is an element of I we have a nonnegative linear constraint on the variables of adjacent agents. For each k is an element of 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 based on input within its constant-radius neighbourhood in G. 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 is an element of I, and Delta(K) is the maximum degree of a vertex k is an element of K. As a methodological contribution, we introduce the technique of graph unfolding for the design of local approximation algorithms.