Sequential and parallel algorithms for mixed packing and covering

Abstract

We describe sequential and parallel algorithms that approximately solve linear programs with no negative coefficients (a.k.a. mixed packing and covering problems).For BackgroundPacking and covering problems are problems that can be formulated as linear programs using only non-negative coefficients and non-negative variables. Special cases include pure packing problems, which are of the form Ñ Ü ¡ Ü Ü and pure covering problems, which are of the formLagrangian-relaxation algorithms are based on the following basic idea. Given an optimization problem specified as a collection of constraints, modify the problem by selecting some of the constraints and replacing them by a continuous "penalty" function that, given a partial solution Ü, measures how close Ü is to violating the removed constraints. Construct a solution iteratively in small steps, making each choice to maintain the remaining constraints while minimizing the increase in the penalty function.While Lagrangian-relaxation algorithms have the disadvantage of producing only approximately optimal (or approximately feasible) solutions, the algorithms have the following potential advantages in comparison to the simplex, interior point, and ellipsoid methods. They can be faster, easier to implement, and easier to parallelize. They can be particularly useful for problems that are sparse, or that have exponentially many variables or constraints (but still have some polynomial-size representation).Lagrangian relaxation was one of the first methods proposed for solving linear programs -as early as the 1950's, John von Neumann apparently proposed and analyzed an Ç´Ñ ¾ Ò ÐÓ ´ÑÒµ ¯¾µ-time Lagrangian-relaxation algorithm for solving two-person zero-sum matrix games (equivalent to pure packing or covering) [18]. The algorithm returned a solution with additive error¯assuming the matrix was scaled to lie between 0 and 1. In 1950, Brown and von Neumann also proposed a system of differential equations that converged to an optimal solution, with the suggestion that the equations could form the basis of an algorithm [3].Subsequent examples include a multicommodity flow algorithm by Ford and Fulkerson (1958), Dantzig-Wolfe decomposition (1960), Benders' decomposition (1962), and Held and Karp's lower bound for the traveling salesman problem (1971). In 1990, Shahrokhi and Matula proved polynomial-time convergence rates for a Lagrangianrelaxation algorithm for multicommodity flow. This caught the attention of the theoretical computer science research community, which has since produced a large body of research on the subject. Klein et al. [15] and Leighton et al. [19] (and many others) gave additional multicommodity flow results. Plotkin, Shmoys, and Tardos [21] and Grigoriadis and Khachiyan [10,11,8,9] adapted the techniques to the general class of packing/covering problems, including mixed packing and covering problems. These algorithms' running times depended linearly on the width -an unbounded function of the input instance. Relatively complicated techniques were de...

An Adaptive Memory Programming Framework for the Robust Capacitated Vehicle Routing Problem

“…The maximization of i∈S q i over a budget uncertainty set of the form (2) amounts to the solution of a fractional packing problem (Garg and Könemann 2007). Ignoring polylogarithmic factors and assuming that L ≤ n, a (1 + )-approximation to the fractional packing problem can be determined in time O( −2 Ln), see Young (2001).…”

Towards More Practical Linear Programming-based Techniques for Algorithmic Mechanism Design

“…Over the past 15 years, simple and fast methods have been developed for solving packing and covering linear programs [2,9,10,15,17,21,24] within an arbitrarily small error guarantee ε. These methods are based on the multiplicative weights update (MWU) method [1], in which a very simple update rule is repeatedly performed until a near-optimal solution is obtained.…”

An FPTAS for the fractional group Steiner tree problem

“…Garg and Khandekar in [11] pointed out that techniques in [23,26] cannot be applied to obtain a FPTAS for the min-hit problem on a collection of clutters since it cannot be formulated as a packing or covering problem. First, we want to present that nearly linear-time FPTAS for explicit fractional packing and covering linear programs by Koufogiannakis and Young in [20] is adoptable to an FPTAS for the FGST problem.…”