4 research outputs found
Approximating max-min linear programs with local algorithms
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 subject to for each and
for each . Here , , and the support sets , ,
and have bounded size. In the distributed setting,
each agent is responsible for choosing the value of , and the
communication network is a hypergraph where the sets and
constitute the hyperedges. We present inapproximability results for a
wide range of structural assumptions; for example, even if and
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 .Comment: 16 pages, 2 figure
Local approximability of max-min and min-max linear programs
In a max-min LP, the objective is to maximise Ï subject to Ax †1, Cx â„ Ï1, and x â„ 0. In a min-max LP, the objective is to minimise Ï subject to Ax †Ï1, Cx â„ 1, and x â„ 0. The matrices A and C are nonnegative and sparse: each row ai of A has at most ÎI positive elements, and each row ck of C has at most ÎK positive elements. We study the approximability of max-min LPs and min-max LPs in a distributed setting; in particular, we focus on local algorithms (constant-time distributed algorithms). We show that for any ÎI â„ 2, ÎK â„ 2, and Δ > 0 there exists a local algorithm that achieves the approximation ratio ÎI (1 â 1/ÎK) + Δ. We also show that this result is the best possible: no local algorithm can achieve the approximation ratio ÎI (1 â 1/ÎK) for any ÎI â„ 2 and ÎK â„ 2.Peer reviewe