Local Approximability of Max-Min and Min-Max Linear Programs

Abstract

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.

Linear-in-delta lower bounds in the LOCAL model

“…− they are logarithmic in ∆ [16][17][18], − they analyse the complexity as a function of n for a fixed ∆ [7,8,10,11,20,21,24], − they only hold in a model that is strictly weaker than LOCAL [14,19].…”

Linear-in- $$\varDelta $$ Δ lower bounds in the LOCAL model

“…-they are logarithmic in Δ [17][18][19], -they analyse the complexity as a function of n for a fixed Δ [7][8][9]11,21,22,25], -they only hold in a model that is strictly weaker than LOCAL [15,20].…”

Distributed assignment with limited communication for multi-robot multi-target tracking

“…In this section, we show how to solve the Bottleneck version of the SATA problem using a local algorithm. We adapt the local algorithm for solving max-min LPs given by Floréen et al [12] to solve the SATA problem in a distributed manner.…”

“…The proof is given in Appendix A. Floréen et al [12] presented a local algorithm to solve MPCP in Equation (5) in a distributed fashion. They presented both positive and negative results for MPCP.…”