Beating Simplex for Fractional Packing and Covering Linear Programs

Abstract

We give an approximation algorithm for fractional packing and covering linear programs (linear programs with non-negative coefficients). Given a constraint matrix with n non-zeros, r rows, and c columns, the algorithm (with high probability) computes feasible primal and dual solutions whose costs are within a factor of 1 + ε of opt (the optimal cost) in time O((r + c) log(n)/ε 2 + n). 1 1 Introduction A packing problem is a linear program of the form max{a • x : M x ≤ b, x ∈ P }, where the entries of the constraint matrix M are non-negative and P is a convex polytope admitting some form of optimization oracle. A covering problem is of the form min{a •x : Mx ≥ b,x ∈ P }. This paper focuses on explicitly given packing and covering problems, that is, max{a • x : M x ≤ b, x ≥ 0} and min{a•x : Mx ≥ b,x ≥ 0}, where the polytope P is just the positive orthant. Explicitly given packing and covering are important special cases of linear programming, including, for example, fractional set cover, multicommodity flow problems with given paths, two-player zero-sum matrix games with non-negative payoffs, and variants of these problems. The paper gives a (1 + ε)-approximation algorithm-that is, an algorithm that returns feasible primal and dual solutions whose costs are within a given factor 1 + ε of opt. With high probability, it runs in time O((r + c) log(n)/ε 2 + n), where n-the input size-is the number of non-zero entries in the constraint matrix and r + c is the number of rows plus columns (i.e., constraints plus variables). For dense instances, r + c can be as small as O(√ n). For moderately dense instances-as long as r + c = o(n/ log n)-the 1/ε 2 factor multiplies a sub-linear term. Generally, the time is linear in the input size n as long as ε ≥ Ω((r + c) log(n)/n).

An FPTAS for the fractional group Steiner tree problem

“…Unfortunately, improvements of approximate min-cut computation cannot significantly improve the running time of Algorithm 1 by decreasing the dominating polynomial factors. In Section 2 we give an essential overview of the Lagrangean relaxation algorithm given by Koufogiannakis and Young [19,20] based on the idea of a two players zero-sum game [15] and non-uniform increments [12,13]. In the spirit of their algorithm, we present an adaptation that involves the calling of an oracle routine that computes a (1 + ε)-approximate cut at each iteration.…”

“…The linear program (2) exhibits the structure of a fractional covering linear program that is more carefully studied later in this paper. Since (2) has exponentially many constraints, state-of-the-art approaches in [19,20,17] cannot be directly applied. Our main task is to adopt their approach and show that it yields a better worst case running time algorithm compared to the previous algorithms, when the number of groups k is large enough.…”

“…We present the Lagrangian relaxation algorithm based on ideas described in the papers of Koufogiannakis and Young [19,20] that iteratively improves the solution of the Primal (10). In order to present the main idea of the algorithm, we recall how to obtain a Lagrangian relaxation of the Dual (11)…”

“…) is calculated at each iteration t ′ with respect to the capacities c e (t ′ ), e ∈ E, from (19) it is obvious that…”

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.…”