A local algorithm is a distributed algorithm where each node must operate
solely based on the information that was available at system startup within a
constant-size neighbourhood of the node. We study the applicability of local
algorithms to max-min LPs where the objective is to maximise mink∑vckvxv subject to ∑vaivxv≤1 for each i and xv≥0
for each v. Here ckv≥0, aiv≥0, and the support sets Vi={v:aiv>0}, Vk={v:ckv>0}, Iv={i:aiv>0}
and Kv={k:ckv>0} have bounded size. In the distributed setting,
each agent v is responsible for choosing the value of xv, and the
communication network is a hypergraph H where the sets Vk and
Vi constitute the hyperedges. We present inapproximability results for a
wide range of structural assumptions; for example, even if ∣Vi∣ and ∣Vk∣
are bounded by some constants larger than 2, there is no local approximation
scheme. To contrast the negative results, we present a local approximation
algorithm which achieves good approximation ratios if we can bound the relative
growth of the vertex neighbourhoods in H.Comment: 16 pages, 2 figure