Edge-Disjoint Paths and Unsplittable Flow

Kolliopoulos, Stavros G.

doi:10.1201/9781420010749.ch57

“…We briefly review known results for general graphs. For a more comprehensive view, including results for special classes of EDP, see [7,10,17,11]. In discussing results for EDP we normally assume that the underlying graph is a simple graph and use n and m to refer to the number of nodes and edges (arcs) respectively.…”

Section: Introductionmentioning

confidence: 99%

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Chekuri¹,

Khanna²,

Shepherd³

2006

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We consider the maximization version of the edge-disjoint path problem (EDP). In undirected graphs and directed acyclic graphs, we obtain an O(√ n) upper bound on the approximation ratio where n is the number of nodes in the graph. We show this by establishing the upper bound on the integrality gap of the natural relaxation based on multicommodity flows. Our upper bound matches within a constant factor a lower bound of Ω(√ n) that is known for both undirected and directed acyclic graphs. The best previous upper bounds on the integrality gaps were O(min{n 2/3 , √ m}) for undirected graphs and O(min{ √ n log n, √ m}) for directed acyclic graphs; here m is the number of edges in the graph. These bounds are also the best known approximation ratios for these problems. Our bound also extends to the unsplittable flow problem (UFP) when the maximum demand is at most the minimum capacity.

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“…Moreover, under similar assumptions, there is no log O(1/c) n-approximation even if congestion c is allowed [3]. We refer the reader to [21,23] for pointers to work on algorithms for EDP.…”

Section: Related Work and Discussionmentioning

confidence: 97%

A Note on Multiflows and Treewidth

Chekuri

¹

,

Khanna

²

,

Shepherd

³

2007

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We consider multicommodity flow problems in capacitated graphs where the treewidth of the underlying graph is bounded by r. The parameter r is allowed to be a function of the input size. An instance of the problem consists of a capacitated graph and a collection of terminal pairs. Each terminal pair has a non-negative demand that is to be routed between the nodes in the pair. A class of optimization problems is obtained when the goal is to route a maximum number of the pairs in the graph subject to the capacity constraints on the edges. Depending on whether routings are fractional, integral or unsplittable, three different versions are obtained; these are commonly referred to respectively as maximum MCF, EDP (the demands are further constrained to be one) and UFP. We obtain the following results in such graphs.• An O(r log r log n) approximation for EDP and UFP.• The integrality gap of the multicommodity flow relaxation for EDP and UFP is O(min{r log n, √ n}).The integrality gap result above is essentially tight since there exist (planar) instances on which the gap is (min{r, √ n}). These results extend the rather limited number of graph classes that admit poly-logarithmic approximations for maximum EDP. Another related question is whether the cut-condition, a necessary condition for (frac-C. Chekuri ( ) Algorithmica (2009) 54: 400-412 401 tionally) routing all pairs, is approximately sufficient. We show the following result in this context.• The flow-cut gap for product multicommodity flow instances is O(log r). This was shown earlier by Rabinovich; we obtain a different proof.

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“…For instance, if we are given arc capacities and demands for each commodity and look for an unsplittable flow of minimum congestion, i.e., of minimum overload of arc capacities, Raghavan and Thompson [13,14] present an approximation algorithm based on their randomized rounding technique. We refer to the survey [10] by Kolliopoulos for an overview of results on general unsplittable flows.…”

Section: Related Literaturementioning

confidence: 99%

Single source unsplittable flows with arc-wise lower and upper bounds

Morell

¹

,

Skutella

²

2021

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In a digraph with a source and several destination nodes with associated demands, an unsplittable flow routes each demand along a single path from the common source to its destination. Given some flow x that is not necessarily unsplittable but satisfies all demands, it is a natural question to ask for an unsplittable flow y that does not deviate from x by too much, i.e., $$y_a\approx x_a$$ y a ≈ x a for all arcs a. Twenty years ago, in a landmark paper, Dinitz et al. (Combinatorica 19:17–41, 1999) proved that there exists an unsplittable flow y such that $$y_a\le x_a+d_{\max }$$ y a ≤ x a + d max for all arcs a, where $$d_{\max }$$ d max denotes the maximum demand value. Our first contribution is a considerably simpler one-page proof for this classical result, based upon an entirely new approach. Secondly, using a subtle variant of this approach, we obtain a new result: There is an unsplittable flow y such that $$y_a\ge x_a-d_{\max }$$ y a ≥ x a - d max for all arcs a. Finally, building upon an iterative rounding technique previously introduced by Kolliopoulos and Stein (SIAM J Comput 31:919–946, 2002) and Skutella (Math Program 91:493–514, 2002), we prove existence of an unsplittable flow that simultaneously satisfies the upper and lower bounds for the special case when demands are integer multiples of each other. For arbitrary demand values, we prove the weaker simultaneous bounds $$x_a/2-d_{\max }\le y_a\le 2x_a+d_{\max }$$ x a / 2 - d max ≤ y a ≤ 2 x a + d max for all arcs a.

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Edge-Disjoint Paths and Unsplittable Flow

Cited by 19 publications

References 76 publications

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A Note on Multiflows and Treewidth

Single source unsplittable flows with arc-wise lower and upper bounds

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