Sequential and Parallel Algorithms for Mixed Packing and Covering

Mixed packing and covering problems are problems that can be formulated as linear programs using only non-negative coefficients. Examples include multicommodity network flow, the Held-Karp lower bound on TSP, fractional relaxations of set cover, bin-packing, knapsack, scheduling problems, minimum-weight triangulation, etc. This paper gives approximation algorithms for the general class of problems. The sequential algorithm is a simple greedy algorithm that can be implemented to find an epsilon-approximate solution in O(epsilon^-2 log m) linear-time iterations. The parallel algorithm does comparable work but finishes in polylogarithmic time. The results generalize previous work on pure packing and covering (the special case when the constraints are all "less-than" or all "greater-than") by Michael Luby and Noam Nisan (1993) and Naveen Garg and Jochen Konemann (1998)

Nearly Linear-Work Algorithms for Mixed Packing/Covering and Facility-Location Linear Programs

We describe the first nearly linear-time approximation algorithms for explicitly given mixed packing/covering linear programs, and for (non-metric) fractional facility location. We also describe the first parallel algorithms requiring only near-linear total work and finishing in polylog time. The algorithms compute $(1+\epsilon)$-approximate solutions in time (and work) $O^*(N/\epsilon^2)$, where $N$ is the number of non-zeros in the constraint matrix. For facility location, $N$ is the number of eligible client/facility pairs

Oracle-Based Primal-Dual Algorithms for Packing and Covering Semidefinite Programs

Packing and covering semidefinite programs (SDPs) appear in natural relaxations of many combinatorial optimization problems as well as a number of other applications. Recently, several techniques were proposed, that utilize the particular structure of this class of problems, to obtain more efficient algorithms than those offered by general SDP solvers. For certain applications, such as those described in this paper, it maybe required to deal with SDP\u27s with exponentially or infinitely many constraints, which are accessible only via an oracle. In this paper, we give an efficient primal-dual algorithm to solve the problem in this case, which is an extension of a logarithmic-potential based algorithm of Grigoriadis, Khachiyan, Porkolab and Villavicencio (SIAM Journal of Optimization 41 (2001)) for packing/covering linear programs

Using Optimization to Obtain a Width-Independent, Parallel, Simpler, and Faster Positive SDP Solver

We study the design of polylogarithmic depth algorithms for approximately solving packing and covering semidefinite programs (or positive SDPs for short). This is a natural SDP generalization of the well-studied positive LP problem. Although positive LPs can be solved in polylogarithmic depth while using only $\tilde{O}(\log^{2} n/\varepsilon^2)$ parallelizable iterations, the best known positive SDP solvers due to Jain and Yao require $O(\log^{14} n /\varepsilon^{13})$ parallelizable iterations. Several alternative solvers have been proposed to reduce the exponents in the number of iterations. However, the correctness of the convergence analyses in these works has been called into question, as they both rely on algebraic monotonicity properties that do not generalize to matrix algebra. In this paper, we propose a very simple algorithm based on the optimization framework proposed for LP solvers. Our algorithm only needs $\tilde{O}(\log^2 n / \varepsilon^2)$ iterations, matching that of the best LP solver. To surmount the obstacles encountered by previous approaches, our analysis requires a new matrix inequality that extends Lieb-Thirring's inequality, and a sign-consistent, randomized variant of the gradient truncation technique proposed in