Nearly-Linear Time Positive LP Solver with Faster Convergence Rate

Abstract

Positive linear programs (LP), also known as packing and covering linear programs, are an important class of problems that bridges computer science, operation research, and optimization. Efficient algorithms for solving such LPs have received significant attention in the past 20 years [2,3,4,6,7,9,11,15,16,18,19,21,24,25,26,29,30]. Unfortunately, all known nearly-linear time algorithms for producing (1+ε)-approximate solutions to positive LPs have a running time dependence that is at least proportional to ε −2 . This is also known as an O(1/ √ T ) convergence rate and is particularly poor in many applications.In this paper, we leverage insights from optimization theory to break this longstanding barrier. Our algorithms solve the packing LP in time O(N ε −1 ) and the covering LP in time O(N ε −1.5 ). At high level, they can be described as linear couplings of several first-order descent steps. This is the first application of our linear coupling technique (see [1]) to problems that are not amenable to blackbox applications known iterative algorithms in convex optimization. Our work also introduces a sequence of new techniques, including the stochastic and the non-symmetric execution of gradient truncation operations, which may be of independent interest.

Using Optimization to Break the Epsilon Barrier: A Faster and Simpler Width-Independent Algorithm for Solving Positive Linear Programs in Parallel

“…20 Suppose that there is an agent i controlling variable x i , and agent i is assumed to know (1) (upper bounds on) m and n, (2) 19 All known methods are implicitly 'smoothing' the LP objective by some parameter, and then performing the related updates. Therefore, none of our algorithms converge to the LP optimum.…”

“…This is the first nearly-linear-time approximate solver for positive LPs and also the first to run in parallel in nearly-linear-work and logarithmic depth. It was 2 Note that most width-dependent solvers are studied under the minmax form of positive LPs: min…”

Randomized MWU for Positive LPs

“…This was accomplished in [36] using a careful amortized data structure to lazily update the weights among other ideas. Some of these ideas were shown to be effective for implicit problems as well [6,7] t ← 0 // time goes from 0 to 1 [1] while t ≤ 1 and Q = ∅ [2] choose y ∈ R n ≥0 such that (*) v, Ay…”

“…We use randomization again in a different way to improve that step as well in Section 4. Figure 1 is incomplete, as we leave the implementation of lines [2] and [3] unspecified until Section 4. This part will also be randomized and the details are deferred primarily for ease of exposition.…”

“…The algorithm of Koufogiannakis and Young efficiently simulates a zero-sum game between two players solving the primal and dual problem respectively and draws inspiration from Grigoriadis and Khachiyan [15]. Recently, Allen-Zhu and Orecchia [2] gave a randomized coordinate descent algorithm that returns a (1 + )-relative approximation to (P) in O(N ln(N ) ln( )/ ) time and (C) in O N ln(n) ln( )/ 1.5 time.…”