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63
Minimal EdgeCoverings of Pairs of Sets
, 1995
"... A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matr ..."
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Cited by 70 (15 self)
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A new minmax theorem concerning bisupermodular functions on pairs of sets is proved. As a special case, we derive an extension of (A. Lubiw's extension of) E. Györi's theorem on intervals, W. Mader's theorem on splitting off edges in directed graphs, J. Edmonds' theorem on matroid partitions, and an earlier result of the first author on the minimum number of new directed edges whose addition makes a digraph kedgeconnected. As another consequence, we solve the corresponding nodeconnectivity augmentation problem in directed graphs.
Minimum Bounded Degree Spanning Trees
, 2006
"... We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. Thi ..."
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Cited by 45 (0 self)
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We consider the minimum cost spanning tree problem under the restriction that all degrees must be at most a given value k. We show that we can efficiently find a spanning tree of maximum degree at most k + 2 whose cost is at most the cost of the optimum spanning tree of maximum degree at most k. This is almost best possible. The approach uses a sequence of simple algebraic, polyhedral and combinatorial arguments. It illustrates many techniques and ideas in combinatorial optimization as it involves polyhedral characterizations, uncrossing, matroid intersection, and graph orientations (or packing of spanning trees). The result generalizes to the setting where every vertex has both upper and lower bounds and gives then a spanning tree which violates the bounds by at most two units and whose cost is at most the cost of the optimum tree. It also gives a better understanding of the subtour relaxation for both the symmetric and asymmetric traveling salesman problems. The generalization to ledgeconnected subgraphs is briefly discussed.
The Cycle Space of an Infinite Graph
 COMB., PROBAB. COMPUT
, 2004
"... Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph togethe ..."
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Cited by 40 (10 self)
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Finite graph homology may seem trivial, but for infinite graphs things become interesting. We present a new approach that builds the cycle space of a graph not on its finite cycles but on its topological circles, the homeomorphic images of the unit circle in the space formed by the graph together with its ends. Our approach
Deterministic distributed vertex coloring in polylogarithmic time
 In Proc. of the 29th ACM Symp. on Principles of Distributed Computing
, 2010
"... Consider an nvertex graph G = (V,E) of maximum degree ∆, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds. In the distributed vertex coloring problem ..."
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Cited by 29 (6 self)
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Consider an nvertex graph G = (V,E) of maximum degree ∆, and suppose that each vertex v ∈ V hosts a processor. The processors are allowed to communicate only with their neighbors in G. The communication is synchronous, i.e., it proceeds in discrete rounds. In the distributed vertex coloring problem the objective is to color G with ∆ + 1, or slightly more than ∆ + 1, colors using as few rounds of communication as possible. (The number of rounds of communication will be henceforth referred to as running time.) Efficient randomized algorithms for this problem are known for more than twenty years [1, 19]. Specifically, these algorithms produce a (∆+1)coloring within O(log n) time, with high probability. On the other hand, the best known deterministic algorithm that requires
Applications Of Submodular Functions
, 1993
"... Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization. ..."
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Cited by 26 (2 self)
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Submodular functions and related polyhedra play an increasing role in combinatorial optimization. The present surveytype paper is intended to provide a brief account of this theory along with several applications in graph theory and combinatorial optimization.
Locally finite graphs with ends: a topological approach
"... This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. Thi ..."
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Cited by 24 (7 self)
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This paper is intended as an introductory survey of a newly emerging field: a topological approach to the study of locally finite graphs that crucially incorporates their ends. Topological arcs and circles, which may pass through ends, assume the role played in finite graphs by paths and cycles. This approach has made it possible to extend to locally finite graphs many classical theorems of finite graph theory that do not extend verbatim. The shift of paradigm it proposes is thus as much an answer to old questions as a source of new ones; many concrete problems of both types are suggested in the paper. This paper attempts to provide an entry point to this field for readers that have not followed the literature that has emerged in the last 10 years or so. It takes them on a quick route through what appear to be the most important lasting results, introduces them to key proof techniques, identifies the most promising open
Sublogarithmic Distributed MIS Algorithm for Sparse Graphs using NashWilliams Decomposition
 In Journal of Distributed Computing Special Issue of selected papers from PODC
, 2008
"... We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on gr ..."
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Cited by 23 (8 self)
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We study the distributed maximal independent set (henceforth, MIS) problem on sparse graphs. Currently, there are known algorithms with a sublogarithmic running time for this problem on oriented trees and graphs of bounded degrees. We devise the first sublogarithmic algorithm for computing MIS on graphs of bounded arboricity. This is a large family of graphs that includes graphs of bounded degree, planar graphs, graphs of bounded genus, graphs of bounded treewidth, graphs that exclude a fixed minor, and many other graphs. We also devise efficient algorithms for coloring graphs from these families. These results are achieved by the following technique that may be of independent interest. Our algorithm starts with computing a certain graphtheoretic structure, called NashWilliams forestsdecomposition. Then this structure is used to compute the MIS or coloring. Our results demonstrate that this methodology is very powerful. Finally, we show nearlytight lower bounds on the running time of any distributed algorithm for computing a forestsdecomposition.
Characterizations of arboricity of graphs
 Ars Combinatorica
"... The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spannin ..."
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Cited by 17 (1 self)
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The aim of this paper is to give several characterizations for the following two classes of graphs: (i) graphs for which adding any l edges produces a graph which is decomposible into k spanning trees and (ii) graphs for which adding some l edges produces a graph which is decomposible into k spanning trees. Introduction and Theorems The concept of decomposing a graph into the minimum number of trees or forests dates back to NashWilliams and Tutte [6, 7, 11]. Since then, many authors have examined various tree decompositions of classes of graphs (for example [2, 8]). The aim of this paper is to give several characterizations for
The Locality of Distributed Symmetry Breaking
"... We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include 1) A randomized algorithm for computing a maximal matching (MM) in O(log ∆+(log log n) 4) rounds, where ∆ is the maximum degree. This improves a 25year old ..."
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Cited by 16 (2 self)
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We present new bounds on the locality of several classical symmetry breaking tasks in distributed networks. A sampling of the results include 1) A randomized algorithm for computing a maximal matching (MM) in O(log ∆+(log log n) 4) rounds, where ∆ is the maximum degree. This improves a 25year old randomized algorithm of Israeli and Itai that takes O(log n) rounds and is provably optimal for all log ∆ in the range [(log log n) 4, √ log n]. 2) A randomized maximal independent set (MIS) algorithm requiring O(log ∆ √ log n) rounds, for all ∆, and only 2 O(√log log n) rounds when ∆ = poly(log n). These improve on the 25year old O(log n)round randomized MIS algorithms of Luby and Alon, Babai, and Itai when log ∆ ≪ √ log n. 3) A randomized ( ∆ + 1)coloring algorithm requiring O(log ∆ + 2 O(√log log n)) rounds, improving on an algorithm √ of Schneider and Wattenhofer that takes O(log ∆+ log n) rounds. This result implies that an O(∆)coloring can be computed in 2 O(√log log n) rounds for all ∆, improving on Kothapalli et al.’s O ( √ log n)round algorithm. We also introduce a new technique for reducing symmetry breaking problems on low arboricity graphs to low degree graphs. Corollaries of this reduction include MM and MIS algorithms for low arboricity graphs (e.g., planar graphs and graphs that exclude any fixed minor) requiring O ( √ log n) and O(log 2/3 n) rounds w.h.p., respectively.