Nearly linear-time packing and covering LP solvers

Abstract

Packing and covering linear programs (PC-LP s) form an important class of linear programs (LPs) across computer science, operations research, and optimization. Luby and Nisan (in: STOC, ACM Press, New York, 1993) constructed an iterative algorithm for approximately solving PC-LP s in nearly linear time, where the time complexity scales nearly linearly in N, the number of nonzero entries of the matrix, and polynomially in $$\varepsilon $$ , the (multiplicative) approximation error. Unfortunately, existing nearly linear-time algorithms (Plotkin et al. in Math Oper Res 20(2):257–301, 1995; Bartal et al., in: Proceedings 38th annual symposium on foundations of computer science, IEEE Computer Society, 1997; Young, in: 42nd annual IEEE symposium on foundations of computer science (FOCS’01), IEEE Computer Society, 2001; Koufogiannakis and Young in Algorithmica 70:494–506, 2013; Young in Nearly linear-time approximation schemes for mixed packing/covering and facility-location linear programs, 2014. arXiv:1407.3015 ; Allen-Zhu and Orecchia, in: SODA, 2015) for solving PC-LP s require time at least proportional to $$\varepsilon ^{-2}$$ . In this paper, we break this longstanding barrier by designing a packing solver that runs in time $$\widetilde{O}(N \varepsilon ^{-1})$$ and covering LP solver that runs in time $$\widetilde{O}(N \varepsilon ^{-1.5})$$ . Our packing solver can be extended to run in time $$\widetilde{O}(N \varepsilon ^{-1})$$ for a class of well-behaved covering programs. In a follow-up work, Wang et al. (in: ICALP, 2016) showed that all covering LPs can be converted into well-behaved ones by a reduction that blows up the problem size only logarithmically.