Results 1  10
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11
Approximation Algorithms for Disjoint Paths Problems
, 1996
"... The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for w ..."
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Cited by 140 (0 self)
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The construction of disjoint paths in a network is a basic issue in combinatorial optimization: given a network, and specified pairs of nodes in it, we are interested in finding disjoint paths between as many of these pairs as possible. This leads to a variety of classical NPcomplete problems for which very little is known from the point of view of approximation algorithms. It has recently been brought into focus in work on problems such as VLSI layout and routing in highspeed networks; in these settings, the current lack of understanding of the disjoint paths problem is often an obstacle to the design of practical heuristics.
Hardness of the undirected edgedisjoint paths problem
 Proc. of STOC
, 2005
"... In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c ..."
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Cited by 50 (8 self)
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In the EdgeDisjoint Paths problem with Congestion (EDPwC), we are given a graph with n nodes, a set of terminal pairs and an integer c. The objective is to route as many terminal pairs as possible, subject to the constraint that at most c demands can be routed through any edge in the graph. When c = 1, the problem is simply referred to as the EdgeDisjoint Paths (EDP) problem. In this paper, we study the hardness of EDPwC in undirected graphs. We obtain an improved hardness result for EDP, and also show the first polylogarithmic integrality gaps and hardness of approximation results for EDPwC. Specifically, we prove that EDP is (log 1 2 −ε n)hard to approximate for any constant ε> 0, unless NP ⊆ ZP T IME(n polylog n). We also show that for any congestion c = o(log log n / log log log n), there is no (log 1−ε c+1 n)approximation algorithm for EDPwC, unless NP ⊆ ZP T IME(npolylog n). For larger congestion, where c ≤ η log log n / log log log n for some constant η, we obtain superconstant inapproximability ratios. All of our hardness results can be converted into integrality gaps for the multicommodity flow relaxation. We also present a separate elementary direct proof of this integrality gap result. Finally, we note that similar results can be obtained for the AllorNothing Flow (ANF) problem, a relaxation of EDP, in which the flow unit routed between the sourcesink pairs does not have follow a single path, so the resulting flow is not necessarily integral. Using standard transformations, our results also extend to the nodedisjoint versions of these problems as well as to the directed setting. 1
Algorithmic Graph Minor Theory: Decomposition, Approximation, and Coloring
 In 46th Annual IEEE Symposium on Foundations of Computer Science
, 2005
"... At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topolog ..."
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Cited by 47 (12 self)
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At the core of the seminal Graph Minor Theory of Robertson and Seymour is a powerful structural theorem capturing the structure of graphs excluding a fixed minor. This result is used throughout graph theory and graph algorithms, but is existential. We develop a polynomialtime algorithm using topological graph theory to decompose a graph into the structure guaranteed by the theorem: a cliquesum of pieces almostembeddable into boundedgenus surfaces. This result has many applications. In particular, we show applications to developing many approximation algorithms, including a 2approximation to graph coloring, constantfactor approximations to treewidth and the largest grid minor, combinatorial polylogarithmicapproximation to halfintegral multicommodity flow, subexponential fixedparameter algorithms, and PTASs for many minimization and maximization problems, on graphs excluding a fixed minor. 1.
Linearity of Grid Minors in Treewidth with Applications through Bidimensionality
, 2005
"... We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cas ..."
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Cited by 21 (1 self)
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We prove that any Hminorfree graph, for a fixed graph H, of treewidth w has an \Omega (w) *\Omega ( w) grid graph as a minor. Thus grid minors suffice to certify that Hminorfree graphs havelarge treewidth, up to constant factors. This strong relationship was previously known for the special cases of planar graphs and boundedgenus graphs, and is known not to hold for generalgraphs. The approach of this paper can be viewed more generally as a framework for extending combinatorial results on planar graphs to hold on Hminorfree graphs for any fixed H. Ourresult has many combinatorial consequences on bidimensionality theory, parametertreewidth bounds, separator theorems, and bounded local treewidth; each of these combinatorial resultshas several algorithmic consequences including subexponential fixedparameter algorithms and approximation algorithms.
Edgedisjoint paths in planar graphs with constant congestion
 IN PROCEEDINGS OF THE 38TH ANNUAL ACM SYMPOSIUM ON THEORY OF COMPUTING, 2006
, 2009
"... We study the maximum edgedisjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) s1t1, s2t2,..., sktk, the goal is to maximize the number of demands that can be connected (routed) by edgedisjoint paths. The natural multicommodity flow relaxation has an Ω ( √ ..."
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Cited by 11 (2 self)
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We study the maximum edgedisjoint paths problem in undirected planar graphs: given a graph G and node pairs (demands) s1t1, s2t2,..., sktk, the goal is to maximize the number of demands that can be connected (routed) by edgedisjoint paths. The natural multicommodity flow relaxation has an Ω ( √ n) integrality gap, where n is the number of nodes in G. Motivated by this, we consider solutions with small constant congestion c>1, that is, solutions in which up to c paths are allowed to use an edge (alternatively, each edge has a capacity of c). In previous work we obtained an O(log n) approximation with congestion 2 via the flow relaxation. This was based on a method of decomposing into welllinked subproblems. In this paper we obtain an O(1) approximation with congestion 4. To obtain this improvement we develop an alternative decomposition that is specific to planar graphs. The decomposition produces instances that we call Okamura–Seymour (OS) instances. These have the property that all terminals lie on a single face. Another ingredient we develop is a constant factor approximation for the allornothing flow problem on OS instances via the flow relaxation.
Edge Disjoint Paths in Moderately Connected Graphs
"... Abstract. We study the Edge Disjoint Paths (EDP) problem in undirected graphs: Given a graph G with n nodes and a set T of pairs of terminals, connect as many terminal pairs as possible using paths that are mutually edge disjoint. This leads to a variety of classic NPcomplete problems, for which ap ..."
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Cited by 10 (0 self)
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Abstract. We study the Edge Disjoint Paths (EDP) problem in undirected graphs: Given a graph G with n nodes and a set T of pairs of terminals, connect as many terminal pairs as possible using paths that are mutually edge disjoint. This leads to a variety of classic NPcomplete problems, for which approximability is not well understood. We show a polylogarithmic approximation algorithm for the undirected EDP problem in general graphs with a moderate restriction on graph connectivity; we require the global minimum cut of G to be Ω(log 5 n). Previously, constant or polylogarithmic approximation algorithms were known for trees with parallel edges, expanders, grids and gridlike graphs, and most recently, evendegree planar graphs. These graphs either have special structure (e.g., they exclude minors) or there are large numbers of short disjoint paths. Our algorithm extends previous techniques in that it applies to graphs with high diameters and asymptotically large minors. 1
Euclidean Prizecollecting Steiner Forest
, 2009
"... In this paper, we consider Steiner forest and its generalizations, prizecollecting Steiner forest and kSteiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomialtime approxi ..."
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Cited by 4 (3 self)
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In this paper, we consider Steiner forest and its generalizations, prizecollecting Steiner forest and kSteiner forest, when the vertices of the input graph are points in the Euclidean plane and the lengths are Euclidean distances. First, we present a simpler analysis of the polynomialtime approximation scheme (PTAS) of Borradaile et al. [12] for the Euclidean Steiner forest problem. This is done by proving a new structural property and modifying the dynamic programming by adding a new piece of information to each dynamic programming state. Next we develop a PTAS for a wellmotivated case, i.e., the multiplicative case, of prizecollecting and budgeted Steiner forest. The ideas used in the algorithm may have applications in design of a broad class of bicriteria PTASs. At the end, we demonstrate why PTASs for these problems can be hard in the general Euclidean case (and thus for PTASs we cannot go beyond the multiplicative case).
The edge disjoint paths problem in Eulerian graphs and 4edgeconnected graphs
, 2010
"... We consider the following wellknown problem, which is called the edgedisjoint paths problem. Input: A graph G with n vertices and m edges, k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Output: Edgedisjoint paths P1, P2,..., Pk in G such that Pi joins si and ti for i = 1, 2,..., k. R ..."
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Cited by 4 (2 self)
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We consider the following wellknown problem, which is called the edgedisjoint paths problem. Input: A graph G with n vertices and m edges, k pairs of vertices (s1, t1), (s2, t2),..., (sk, tk) in G. Output: Edgedisjoint paths P1, P2,..., Pk in G such that Pi joins si and ti for i = 1, 2,..., k. Robertson and Seymour’s graph minor project gives rise to an O(m 3) algorithm for this problem for any fixed k, but their proof of the correctness needs the whole Graph Minor project, spanning 23 papers and at least 500 pages proof. We give a faster algorithm and a simpler proof of the correctness for the edgedisjoint paths problem for any fixed k. Our results can be summarized as follows: 1. If an input graph G is either 4edgeconnected or Eulerian, then our algorithm only needs to look for the following three simple reductions: (i) Excluding vertices of high degree. (ii) Excluding ≤ 3edgecuts. (iii) Excluding large clique minors. 2. When an input graph G is either 4edgeconnected or Eulerian, the number of terminals k is allowed to be nontrivially superconstant number, up to k = O((log log log n) 1 2 −ε) for any ε> 0. Thus our hidden constant in this case is dramatically smaller than RobertsonSeymour’s. In addition, if an input graph G is either 4edgeconnected planar or Eulerian planar, k is allowed to be O((log n) 1 2 −ε) for any ε> 0. The same thing holds for bounded genus graphs. Moreover, if an input graph is either 4edgeconnected Hminorfree or Eulerian Hminorfree for fixed graph H, k is allowed to be O((log log n) 1 2 −ε) for any ε> 0. 3. We also give our own algorithm for the edgedisjoint paths problem in general graphs. We basically follow RobertsonSeymour’s algorithm, but we cut half of the proof of the correctness for their algorithm. In addition, the time complexity of our algorithm is O(n²), which is faster than Robertson and Seymour’s.
A polylogarithimic approximation algorithm for edgedisjoint paths with congestion 2
 In Proc. of IEEE FOCS
, 2012
"... Abstract—In the EdgeDisjoint Paths with Congestion problem (EDPwC), we are given an undirected nvertex graph G, a collection M = {(s1, t1),..., (sk, tk)} of demand pairs and an integer c. The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge cong ..."
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Cited by 4 (0 self)
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Abstract—In the EdgeDisjoint Paths with Congestion problem (EDPwC), we are given an undirected nvertex graph G, a collection M = {(s1, t1),..., (sk, tk)} of demand pairs and an integer c. The goal is to connect the maximum possible number of the demand pairs by paths, so that the maximum edge congestion the number of paths sharing any edge is bounded by c. When the maximum allowed congestion is c = 1, this is the classical EdgeDisjoint Paths problem (EDP). The best current approximation algorithm for EDP achieves an O ( √ n)approximation, by rounding the standard multicommodity flow relaxation of the problem. This matches the Ω ( √ n) lower bound on the integrality gap of this relaxation. We show an O(poly log k)approximation algorithm for EDPwC with congestion c = 2, by rounding the same multicommodity flow relaxation. This gives the best possible congestion for a subpolynomial approximation of EDPwC via this relaxation. Our results are also close to optimal in terms of the number of pairs routed, since EDPwC is known to be hard to approximate to within a factor of ˜ ( Ω (log n) 1/(c+1) for any constant congestion c. Prior to our work, the best approximation factor for EDPwC with congestion 2 was Õ(n 3/7), and the best algorithm achieving a polylogarithmic approximation required congestion 14. Keywordsapproximation algorithms; network routing; edgedisjoint paths I.
Edge disjoint paths and multicut problems in graphs generalizing the trees
, 2005
"... We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg et al. [Primaldual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieve ..."
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Cited by 1 (0 self)
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We generalize all the results obtained for maximum integer multiflow and minimum multicut problems in trees by Garg et al. [Primaldual approximation algorithms for integral flow and multicut in trees. Algorithmica 18 (1997) 3–20] to graphs with a fixed cyclomatic number, while this cannot be achieved for other classical generalizations of the trees. Moreover, we prove that the minimum multicut problem with a fixed number of sourcesink pairs is polynomialtime solvable in planar and in bounded treewidth graphs. Eventually, we introduce the class of kedgeouterplanar graphs and show that the integrality gap of the maximum edgedisjoint paths problem is bounded in these graphs. We also provide stronger results for cacti (k = 1).