Nearly linear-time packing and covering LP solvers: Achieving width-independence and -convergence

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 ε , 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 ε - 2 . In this paper, we break this longstanding barrier by designing a packing solver that runs in time O ~ ( N ε - 1 ) and covering LP solver that runs in time O ~ ( N ε - 1.5 ) . Our packing solver can be extended to run in time O ~ ( N ε - 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.