A Note on Multiflows and Treewidth

Chekuri, Chandra; Khanna, Sanjeev; Shepherd, F. Bruce

doi:10.1007/s00453-007-9129-z

Cited by 15 publications

(17 citation statements)

References 35 publications

(63 reference statements)

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“…For MAX EDP, we have a constant factor integrality gap with congestion 2 for planar graphs. In [8] it is shown that the integrality gap of the LP for MAX EDP in graphs of treewidth at most k is O(k log k log n); note that this is with congestion 1. Existing integrality gap results, when interpreted in terms of treewidth k, show that the integrality gap is Ω(k) for congestion 1 and Ω(log O(1/c) k) for congestion c > 1.…”

Section: Introductionmentioning

confidence: 99%

“…It was asked in [8] whether the gap is O(k) with congestion 1. In particular, the question of whether the gap is O(1) for k = 2 (this is precisely the class of series parallel graphs) was open.…”

Section: Introductionmentioning

confidence: 99%

“…The work in [7] obtained a constant factor approximation for planar graphs by using a more refined decomposition that took advantage of the structure of planar graphs. For graphs of treewidth k, [8] used the well-linked decomposition framework to obtain an O(k log k log n)-approximation and integrality gap. Our work here shows that one can bypass the well-linked decomposition framework for bounded treewidth graphs, and more generally for k-sums over families of graphs.…”

Section: Introductionmentioning

confidence: 99%

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Maximum Edge-Disjoint Paths in k-Sums of Graphs

Chekuri

Naves

Shepherd

2013

Automata, Languages, and Programming

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We consider the approximability of the maximum edge-disjoint paths problem (MEDP) in undirected graphs, and in particular, the integrality gap of the natural multicommodity flow based relaxation for it. The integrality gap is known to be Ω( √ n) even for planar graphs [14] due to a simple topological obstruction and a major focus, following earlier work [17], has been understanding the gap if some constant congestion is allowed. In planar graphs the integrality gap is O(1) with congestion 2 [23, 7]. In general graphs, recent work has shown the gap to be polylog(n) [10, 11] with congestion 2. Moreover, the gap is log Ω(c) n in general graphs with congestion c for any constant c ≥ 1 [1]. It is natural to ask for which classes of graphs does a constant-factor constant-congestion property hold. It is easy to deduce that for given constant bounds on the approximation and congestion, the class of "nice" graphs is minor-closed. Is the converse true? Does every proper minor-closed family of graphs exhibit a constant factor, constant congestion bound relative to the LP relaxation? We conjecture that the answer is yes. One stumbling block has been that such bounds were not known for bounded treewidth graphs (or even treewidth 3). In this paper we give a polytime algorithm which takes a fractional routing solution in a graph of bounded treewidth and is able to integrally route a constant fraction of the LP solution's value. Note that we do not incur any edge congestion. Previously this was not known even for series parallel graphs which have treewidth 2. The algorithm is based on a more general argument that applies to k-sums of graphs in some graph family, as long as the graph family has a constant factor, constant congestion bound. We then use this to show that such bounds hold for the class of k-sums of bounded genus graphs.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Maximum Edge-Disjoint Paths in k-Sums of Graphs

Chekuri

Naves

Shepherd

2013

Automata, Languages, and Programming

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“…In this paper (Section 4.3), we show that the integer flow cut gap in this case is c O(k) . In this effort, we explicitly employ a second proof ingredient which is a simple rerouting lemma that was stated and used in [9] (see Section 4.2). Informally speaking the lemma says the following.…”

Section: Introductionmentioning

confidence: 99%

Flow-Cut Gaps for Integer and Fractional Multiflows

Chekuri

Shepherd

Weibel

2010

Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms

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Consider a routing problem instance consisting of a demand graph H = (V, E(H)) and a supply graph G = (V, E(G)). If the pair obeys the cut condition, then the flow-cut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C. It is wellknown that the flow-cut gap may be greater than 1 even in the case where G is the (series-parallel) graph K 2,3 . In this paper we are primarily interested in the "integer" flow-cut gap. What is the minimum value C such that there exists a feasible integer valued multiflow for H if each edge of G is given capacity C?We formulate a conjecture that states that the integer flow-cut gap is quantitatively related to the fractional flow-cut gap. In particular this strengthens the well-known conjecture that the flow-cut gap in planar and minor-free graphs is O(1) [14] to suggest that the integer flow-cut gap is O(1). We give several technical tools and results on non-trivial special classes of graphs to give evidence for the conjecture and further explore the "primal" method for understanding flow-cut gaps; this is in contrast to and orthogonal to the highly successful metric embeddings approach. Our results include the following:• Let G be obtained by series-parallel operations starting from an edge st, and consider orienting all edges in G in the direction from s to t. A demand is compliant if its endpoints are joined by a directed path in the resulting oriented graph. We show that if the cut condition holds for a compliant instance and G + H is Eulerian, then an integral routing of H exists. This result includes as a special case, routing on a ring but is not a special case of the Okamura-Seymour theorem. • Using the above result, we show that the integer flow-cut gap in series-parallel graphs is 5. We also give an explicit class of routing instances that shows via elementary calculations that the flow-cut gap in series-parallel graphs is at least 2 − o(1); this is motivated by and simplifies the proof by Lee and Raghavendra [20]. • The integer flow-cut gap in k-Outerplanar graphs is c O(k) for some fixed constant c.• A simple proof that the flow-cut gap is O(log k * ) where k * is the size of a node-cover in H; this was previously shown by Günlük via a more intricate proof [13].

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“…The best known approximation for MEDP is O( √ n) [11], and there is a matching approximation for MNDP [39]; here n is the number of nodes in the input graph. Various special classes of graphs have been studied; constant factor and poly-logarithmic factor approximations for MEDP are known for trees [26,15], graphs with bounded treewidth [13], grids and grid-like graphs [34,37,38], Eulerian planar graphs [35,31], graphs with good expansion [5,24,36,40], and graphs with large global minimum cut [46]. MEDP and MNDP with congestion have also been well-studied, especially given the integrality gap of the flow relaxation; it is known that randomized rounding techniques give an O(d 1/c ) approximation, where d is the maximum flow path length in the fractional solution and c is the congestion parameter [53,39,4,6]; this holds even for directed graphs and it leads to an O(n 1/c ) approximation.…”

Section: High-level Overview Of Algorithm and Technical Contributionmentioning

confidence: 99%

Poly-logarithmic Approximation for Maximum Node Disjoint Paths with Constant Congestion

Chekuri¹,

Ene²

2013

Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms

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We consider the Maximum Node Disjoint Paths (MNDP) problem in undirected graphs. The input consists of an undirected graph G = (V, E) and a collection {(s 1 , t 1 ), . . . , (s k , t k )} of k source-sink pairs. The goal is to select a maximum cardinality subset of pairs that can be routed/connected via node-disjoint paths. A relaxed version of MNDP allows up to c paths to use a node, where c is the congestion parameter. We give a polynomial time algorithm that routes Ω(OPT/poly log k) pairs with O(1) congestion, where OPT is the value of an optimum fractional solution to a natural multicommodity flow relaxation. Our result builds on the recent breakthrough of Chuzhoy [17] who gave the first polylogarithmic approximation with constant congestion for the Maximum Edge Disjoint Paths (MEDP) problem.

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A Note on Multiflows and Treewidth

Cited by 15 publications

References 35 publications

Maximum Edge-Disjoint Paths in k-Sums of Graphs

Maximum Edge-Disjoint Paths in k-Sums of Graphs

Flow-Cut Gaps for Integer and Fractional Multiflows

Poly-logarithmic Approximation for Maximum Node Disjoint Paths with Constant Congestion

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