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

In a max-min LP, the objective is to maximise omega subject to A xa parts per thousand currency sign1, C xa parts per thousand yen omega 1, and xa parts per thousand yen0. In a min-max LP, the objective is to minimise rho subject to A xa parts per thousand currency sign rho 1, C xa parts per thousand yen1, and xa parts per thousand yen0. The matrices A and C are nonnegative and sparse: each row a (i) of A has at most Delta (I) positive elements, and each row c (k) of C has at most Delta (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 Delta (I) a parts per thousand yen2, Delta (K) a parts per thousand yen2, and epsilon > 0 there exists a local algorithm that achieves 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) ) for any Delta (I) a parts per thousand yen2 and Delta (K) a parts per thousand yen2.