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31
Hadwiger’s conjecture for K6free graphs
 COMBINATORICA
, 1993
"... In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ..."
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In 1943, Hadwiger made the conjecture that every loopless graph not contractible to the complete graph on t + 1 vertices is tcolourable. When t ≤ 3 this is easy, and when t = 4, Wagner’s theorem of 1937 shows the conjecture to be equivalent to the fourcolour conjecture (the 4CC). However, when t ≥ 5 it has remained open. Here we show that when t = 5 it is also equivalent to the 4CC. More precisely, we show (without assuming the 4CC) that every minimal counterexample to Hadwiger’s conjecture when t = 5 is “apex”, that is, it consists of a planar graph with one additional vertex. Consequently, the 4CC implies Hadwiger’s conjecture when t = 5, because it implies that apex graphs are 5colourable.
Detecting a Network Failure
 Proc. 41st Annual IEEE Symposium on Foundations of Computer Science
, 2000
"... Abstract. Measuring the properties of a large, unstructured network can be difficult: One may not have full knowledge of the network topology, and detailed global measurements may be infeasible. A valuable approach to such problems is to take measurements from selected locations within the network a ..."
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Cited by 40 (1 self)
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Abstract. Measuring the properties of a large, unstructured network can be difficult: One may not have full knowledge of the network topology, and detailed global measurements may be infeasible. A valuable approach to such problems is to take measurements from selected locations within the network and then aggregate them to infer largescale properties. One sees this notion applied in settings that range from Internet topology discovery tools to remote software agents that estimate the download times of popular web pages. Some of the most basic questions about this type of approach, however, are largely unresolved at an analytical level. How reliable are the results? How much does the choice of measurement locations affect the aggregate information one infers about the network? We describe algorithms that yield provable guarantees for a particular problem of this type: detecting a network failure. Suppose we want to detect events of the following form in an nnode network: An adversary destroys up to k nodes or edges, after which two subsets of the nodes, each of size at least εn, are disconnected from one another. We call such an event an (ε,k)partition. One method for detecting such events would be to place “agents ” at a set D of nodes, and record a fault whenever two of them become separated from each other. To be a good detection set, D should become disconnected whenever there is an (ε,k)partition; in this way, it “witnesses ” all such events. We show that every graph has a detection set of size polynomial in k and ε −1,and independent of the size of the graph itself. Moreover, random sampling provides an effective way to construct such a set. Our analysis establishes a connection between graph separators and the notion of VCdimension, using techniques based on matchings and disjoint paths.
AN ALGORITHM FOR PACKING NONZERO APATHS IN GROUPLABELLED GRAPHS
 COMBINATORICA 28 (2) (2008) 145–161
, 2008
"... Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊆V. An Apath is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian ..."
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Cited by 19 (2 self)
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Let G=(V,E) be an oriented graph whose edges are labelled by the elements of a group Γ and let A⊆V. An Apath is a path whose ends are both in A. The weight of a path P in G is the sum of the group values on forward oriented arcs minus the sum of the backward oriented arcs in P. (If Γ is not abelian, we sum the labels in their order along the path.) We give an efficient algorithm for finding a maximum collection of vertexdisjoint Apaths each of nonzero weight. When A=V this problem is equivalent to the maximum matching problem.
A short proof of Mader's Spaths theorem
"... For an undirected graph G = (V; E) and a collection S of disjoint subsets of V , an Spath is a path connecting different sets in S. We give a short proof of Mader's minmax theorem for the maximum number of disjoint Spaths. Let G = (V; E) be an undirected graph and let S be a collection of d ..."
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Cited by 13 (1 self)
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For an undirected graph G = (V; E) and a collection S of disjoint subsets of V , an Spath is a path connecting different sets in S. We give a short proof of Mader's minmax theorem for the maximum number of disjoint Spaths. Let G = (V; E) be an undirected graph and let S be a collection of disjoint subsets of V . An Spath is a path connecting two different sets in S. Mader [4] gave the following minmax relation for the maximum number of (vertex)disjoint Spaths, where S := S S.
Packing nonreturning Apaths
, 2005
"... Chudnovsky et al. gave a minmax formula for the maximum number of nodedisjoint nonzero Apaths in grouplabeled graphs [1], which is a generalization of Mader’s theorem on nodedisjoint Apaths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is ..."
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Cited by 7 (4 self)
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Chudnovsky et al. gave a minmax formula for the maximum number of nodedisjoint nonzero Apaths in grouplabeled graphs [1], which is a generalization of Mader’s theorem on nodedisjoint Apaths [3]. Here we present a further generalization with a shorter proof. The main feature of Theorem 2.1 is that parity is “hidden” inside �ν, which is given by an oracle for nonbipartite matching.
Parameterized Tractability of Multiway Cut with Parity Constraints
"... Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether ..."
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Abstract. In this paper, we study a parity based generalization of the classical MULTIWAY CUT problem. Formally, we study the PARITY MULTIWAY CUT problem, where the input is a graph G, vertex subsets Te and To (T = Te ∪ To) called terminals, a positive integer k and the objective is to test whether there exists a ksized vertex subset S such that S intersects all odd paths from v ∈ To to T \ {v} and all even paths from v ∈ Te to T \ {v}. When Te = To, this is precisely the classical MULTIWAY CUT problem. If To = ∅ then this is the EVEN MULTIWAY CUT problem and if Te = ∅ then this is the ODD MULTIWAY CUT problem. We remark that even the problem of deciding whether there is a set of at most k vertices that intersects all odd paths between a pair of vertices s and t is NPcomplete. Our primary motivation for studying this problem is the recently initiated parameterized study of parity versions of graphs minors (Kawarabayashi, Reed and Wollan, FOCS 2011) and separation problems similar to MULTIWAY CUT. The area of design of parameterized algorithms for graph separation problems has seen a lot of recent activity, which includes algorithms
Algebraic Algorithms for Linear Matroid Parity Problems
"... We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the ma ..."
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Cited by 4 (1 self)
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We present fast and simple algebraic algorithms for the linear matroid parity problem and its applications. For the linear matroid parity problem, we obtain a simple randomized algorithm with running time O(mrω−1) where m and r are the number of columns and the number of rows and ω ≈ 2.376 is the matrix multiplication exponent. This improves the O(mrω)time algorithm by Gabow and Stallmann, and matches the running time of the algebraic algorithm for linear matroid intersection, answering a question of Harvey. We also present a very simple alternative algorithm with running time O(mr2) which does not need fast matrix multiplication. We further improve the algebraic algorithms for some specific graph problems of interest. For the Mader’s disjoint Spath problem, we present an O(nω)time randomized algorithm where n is the number of vertices. This improves the running time of the existing results considerably, and matches the running time of the algebraic algorithms for graph matching. For the graphic matroid parity problem, we give an O(n4)time randomized algorithm where n is the number of vertices, and an O(n3)time randomized algorithm for a special case useful in designing approximation algorithms. These algorithms are optimal in terms of n as the input size could be Ω(n4) and Ω(n3) respectively. The techniques are based on the algebraic algorithmic framework developed by Mucha and Sankowski, Harvey, and Sankowski. While linear matroid parity and Mader’s disjoint Spath are challenging generalizations for the design of combinatorial algorithms, our results show that both the algebraic algorithms for linear matroid intersection and graph matching can be extended nicely to more general settings. All algorithms are still faster than the existing algorithms even if fast matrix multiplication is not used. These provide simple algorithms that can be easily implemented in practice.
The PathPacking Structure of Graphs
, 2003
"... We prove EdmondsGallai type structure theorems for Mader's edge and vertexdisjoint paths including also capacitated variants, and state a conjecture generalizing Mader's minimax theorems on path packings and Cunningham and Geelen's pathmatching theorem. ..."
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Cited by 3 (0 self)
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We prove EdmondsGallai type structure theorems for Mader's edge and vertexdisjoint paths including also capacitated variants, and state a conjecture generalizing Mader's minimax theorems on path packings and Cunningham and Geelen's pathmatching theorem.
Packing nonreturning Apaths algorithmically
, 2005
"... In this paper we present an algorithmic approach to packing Apaths. It is regarded as a generalization of Edmonds’ matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the socalled 3way lemma, which either provid ..."
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Cited by 3 (1 self)
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In this paper we present an algorithmic approach to packing Apaths. It is regarded as a generalization of Edmonds’ matching algorithm, however there is the significant difference that here we do not build up any kind of alternating tree. Instead we use the socalled 3way lemma, which either provides augmentation, or a dual, or a subgraph which can be used for contraction. The method works in the general setting of packing nonreturning Apaths. It also implies an eardecomposition of criticals, as a generalization of the odd eardecomposition of factorcritical graph.
Halfintegrality of nodecapacitated multiflows and treeshaped facility locations on trees
, 2010
"... In this paper, we establish a novel duality relationship between nodecapacitated multiflows and treeshaped facility locations. We prove that the maximum value of a treedistanceweighted maximum nodecapacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in ..."
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Cited by 2 (2 self)
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In this paper, we establish a novel duality relationship between nodecapacitated multiflows and treeshaped facility locations. We prove that the maximum value of a treedistanceweighted maximum nodecapacitated multiflow problem is equal to the minimum value of the problem of locating subtrees in a tree, and the maximum is attained by a halfintegral multiflow. Utilizing this duality, we show that a halfintegral optimal multiflow and an optimal location can be found in strongly polynomial time. These extend previously known results in the maximum free multiflow problems. We also show that the set of treedistance weights is the only class having bounded fractionality in maximum nodecapacitated multiflow problems.