Near-Linear Time Approximation Schemes for some Implicit Fractional Packing Problems

Abstract

We consider several implicit fractional packing problems and obtain faster implementations of approximation schemes based on multiplicative-weight updates. This leads to new algorithms with near-linear running times for some fundamental problems in combinatorial optimization. We highlight two concrete applications. The first is to find the maximum fractional packing of spanning trees in a capacitated graph; we obtain a (1 − )-approximation inÕ m/ 2 time, where m is the number of edges in the graph. Second, we consider the LP relaxation of the weighted unsplittable flow problem on a path and obtain a (1− )-approximation inÕ n/ 2 time, where n is the number of demands.

Randomized MWU for Positive LPs

“…We note that the randomized MWU algorithm is useful not just for mixed packing and covering but also pure packing and covering. In some recent work [6,7] we had developed faster algorithms for implicit packing problems by adapting the MWU framework via a combination of data structures. Our randomized variant of MWU was initially motivated by some examples of implicit problems in combinatorial and geometric settings that were not amenable to the data structure ideas in [6].…”

“…In some recent work [6,7] we had developed faster algorithms for implicit packing problems by adapting the MWU framework via a combination of data structures. Our randomized variant of MWU was initially motivated by some examples of implicit problems in combinatorial and geometric settings that were not amenable to the data structure ideas in [6]. In Section 5 we sketch a few applications of explicit and implicit problems and highlight how the randomized variant allows for handling new implicit problems as well as generalizing some prior results in [6] to the mixed packing and covering setting.…”

“…Our randomized variant of MWU was initially motivated by some examples of implicit problems in combinatorial and geometric settings that were not amenable to the data structure ideas in [6]. In Section 5 we sketch a few applications of explicit and implicit problems and highlight how the randomized variant allows for handling new implicit problems as well as generalizing some prior results in [6] to the mixed packing and covering setting. Our focus in this paper is on the highlevel randomized algorithm and its analysis for explicit problems.…”

“…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…”

“…Our work here is not only motivated by explicit problems of the form (NMPC), but other (implicit) mixed packing and covering problems contingent on the implementation of simple subroutines/oracles. In this regard, one advantage of the MWU-based iterative methods is the relative simplicity of both the algorithm and the analysis, which allows for subroutines to be approximated by heuristics and accelerated by data structures for a better total running time (see, for example, [27,1,36,6,7]). Our focus in this paper is on augmenting the MWU framework to obtain faster and/or simpler algorithms for several classes of problems.…”