Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing

Abstract

An elementary h-route flow, for an integer h ≥ 1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h ≤ 3: The size of a minimum h-route cut is at least f /h and at most O(log 4 k · f ) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h = 3 that has an approximation ratio of O(log 4 k). Previously, polylogarithmic approximation was known only for h-route cuts for h ≤ 2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem.

Approximation Algorithms and Hardness of the k -Route Cut Problem

“…When edge weights are arbitrary, we obtain a 2,Õ(log 2.5 r) -bi-criteria approximation in n O(k) time, and an O(log r), O(log 3 r) -bicriteria approximation in time polynomial in n and k. We also show an O(log 1.5 r)-approximation for the special case where k = 2, thus slightly improving the result of [BC10]. The previously known upper bounds and our results for EC-kRC are summarized in Table 1. Previous results Current paper k = 2 O(log 2 r) [BC10] O(log 1.5 r) k = 3 O(log 3 r) [KS11] arbitrary k, uniform -O(k log 1.5 r), 1 + δ, O 1 δ log 1.5 r for any constant 0 < δ < 1 arbitrary k, general -2, O(log 2.5 r log log r) in time n O(k) ; O(log r), O(log 3 r) in poly(n)-time Table 1: Upper bounds for EC-kRC. Running time is polynomial in n and k unless stated otherwise.…”

“…They note that it seems unlikely that their algorithm can be extended to handle higher values of k using similar techniques. Very recently, Kolman and Scheideler [KS11] obtained a O(log 3 r) approximation to EC-3RC (k = 3 case) from the linear program of [BC10] by using a multi-level ball growing rounding. To the best of our knowledge, no approximation algorithms with sub-polynomial (in n) guarantees are known for any variant of the problem, for any value k > 3, except in the single-source setting that we discuss later.…”

Approximation Algorithms and Hardness of the k-Route Cut Problem

“…They note that it seems unlikely that their algorithm can be extended to handle higher values of k using similar tech-niques. Very recently, Kolman and Scheideler [KS11] obtained a O(log 3 r) approximation to EC-3RC (k = 3 case) from the linear program of [BC10] by using a multi-level ball growing rounding. To the best of our knowledge, no approximation algorithms with subpolynomial (in n) guarantees are known for any variant of the problem, for any value k > 3, except in the single-source setting that we discuss later.…”

Improved Region-Growing and Combinatorial Algorithms for k-Route Cut Problems (Extended Abstract)

“…Multicut and multiway cut are APX-hard [12], with the former not admitting any constant-factor approximation assuming the uniquegames conjecture [9], and k-(s, t)-Cut is NP-hard; hence, we focus on approximation algorithms. Moreover, as highlighted in [10,5,23,11,22], k-route cut problems turn out to be much more challenging than their 1-route counterparts, especially for non-constant k, so (as in [11]) we consider bicriteria approximation guarantees. (This is further justified by our hardness result for k-(s, t)-Cut in Section 4.)…”

“…Our chief technical novelty is to prove a region-growing lemma (see Lemmas 3.1 and 3.3) applicable to settings with different metrics, that is inspired by, but more general, than the analogous lemma in [15], and much more sophisticated than the one used in [5,23,22]. This lemma, coupled with a subtle insight about the metrics derived from the LP solution, allows us to obtain the same kind of savings in our recursive region-growing algorithm that Even et al [15] obtain (via their region-growing lemma) in their divide-andconquer algorithms; this yields our improved approximation guarantees.…”