Abstract. In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0. In a min-max LP, the objective is to minimise ρ subject to Ax ≤ ρ1, Cx ≥ 1, and x ≥ 0. The matrices A and C are nonnegative and sparse: each row a i of A has at most ∆ I positive elements, and each row c k of C has at most ∆ K 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 ∆ I ≥ 2, ∆ K ≥ 2, and ε > 0 there exists a local algorithm that achieves 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 ) for any ∆ I ≥ 2 and ∆ K ≥ 2.

a b s t r a c tA local algorithm with local horizon r is a distributed algorithm that runs in r synchronous communication rounds; here r is a constant that does not depend on the size of the network. As a consequence, the output of a node in a local algorithm only depends on the input within r hops from the node.We give tight bounds on the local horizon for a class of local algorithms for combinatorial problems on unit-disk graphs (UDGs). Most of our bounds are due to a refined analysis of existing approaches, while others are obtained by suggesting new algorithms. The algorithms we consider are based on network decompositions guided by a rectangular tiling of the plane. The algorithms are applied to matching, independent set, graph colouring, vertex cover, and dominating set.We also study local algorithms on quasi-UDGs, which are a popular generalisation of UDGs, aimed at more realistic modelling of communication between the network nodes. Analysing the local algorithms on quasi-UDGs allows one to assume that the nodes know their coordinates only approximately, up to an additive error. Despite the localisation error, the quality of the solution to problems on quasi-UDGs remains the same as for the case of UDGs with perfect location awareness. We analyse the increase in the local horizon that comes along with moving from UDGs to quasi-UDGs.

Abstract. We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0 for nonnegative matrices A and C. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.