Results 1 - 10 of 18

Minimum Cuts and Shortest Homologous Cycles

by Erin Chambers, Jeff Erickson, Amir Nayyeri - SYMPOSIUM ON COMPUTATIONAL GEOMETRY , 2009
"... We describe the first algorithms to compute minimum cuts in surface-embedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)-cut in g O(g) n log n time. Except for the spec ..."
Abstract - Cited by 33 (11 self) - Add to MetaCart
We describe the first algorithms to compute minimum cuts in surface-embedded graphs in nearlinear time. Given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, our algorithm computes a minimum (s, t)-cut in g O(g) n log n time. Except for the special case of planar graphs, for which O(n log n)-time algorithms have been known for more than 20 years, the best previous time bounds for finding minimum cuts in embedded graphs follow from algorithms for general sparse graphs. A slight generalization of our minimum-cut algorithm computes a minimum-cost subgraph in every Z2-homology class. We also prove that finding a minimum-cost subgraph homologous to a single input cycle is NP-hard.
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Homology flows, cohomology cuts

by Erin W. Chambers, Jeff Erickson, Amir Nayyeri - ACM SYMPOSIUM ON THEORY OF COMPUTING , 2009
"... We describe the first algorithms to compute maximum flows in surface-embedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)-flow in O(g 7 n log 2 n log 2 C) time fo ..."
Abstract - Cited by 30 (10 self) - Add to MetaCart
We describe the first algorithms to compute maximum flows in surface-embedded graphs in nearlinear time. Specifically, given an undirected graph embedded on an orientable surface of genus g, with two specified vertices s and t, we can compute a maximum (s, t)-flow in O(g 7 n log 2 n log 2 C) time for integer capacities that sum to C, or in (g log n) O(g) n time for real capacities. Except for the special case of planar graphs, for which an O(n log n)-time algorithm has been known for 20 years, the best previous time bounds for maximum flows in surface-embedded graphs follow from algorithms for general sparse graphs. Our key insight is to optimize the relative homology class of the flow, rather than directly optimizing the flow itself. A dual formulation of our algorithm computes the minimum-cost cycle or circulation in a given (real or integer) homology class.

Approximation Schemes for Steiner Forest on Planar Graphs and Graphs of Bounded Treewidth

by Mohammadhossein Bateni, Mohammadtaghi Hajiaghayi, Dániel Marx , 2010
"... We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called priz ..."
Abstract - Cited by 28 (8 self) - Add to MetaCart
We give the first polynomial-time approximation scheme (PTAS) for the Steiner forest problem on planar graphs and, more generally, on graphs of bounded genus. As a first step, we show how to build a Steiner forest spanner for such graphs. The crux of the process is a clustering procedure called prize-collecting clustering that breaks down the input instance into separate subinstances which are easier to handle; moreover, the terminals in different subinstances are far from each other. Each subinstance has a relatively inexpensive Steiner tree connecting all its terminals, and the subinstances can be solved (almost) separately. Another building block is a PTAS for Steiner forest on graphs of bounded treewidth. Surprisingly, Steiner forest is NP-hard even on graphs of treewidth 3. Therefore, our PTAS for bounded treewidth graphs needs a nontrivial combination of approximation arguments and dynamic programming on the tree decomposition. We further show that Steiner forest can be solved in polynomial time for series-parallel graphs (graphs of treewidth at most two) by a novel combination of dynamic programming and minimum cut computations, completing our thorough complexity study of Steiner forest in the range of bounded treewidth graphs, planar graphs, and bounded genus graphs.
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Minimum Cuts and Shortest Non-Separating Cycles via Homology Covers

by Jeff Erickson, Amir Nayyeri - SYMPOSIUM ON DISCRETE ALGORITHMS , 2011
"... Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2-homology class in 2 O(g+b) n log n time; this problem is NP-hard even for undirected graphs. We also present two ap ..."
Abstract - Cited by 18 (5 self) - Add to MetaCart
Let G be a directed graph with weighted edges, embedded on a surface of genus g with b boundaries. We describe an algorithm to compute the shortest directed cycle in G in any given � 2-homology class in 2 O(g+b) n log n time; this problem is NP-hard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the shortest non-separating directed cycle in G in 2 O(g) n log n time, improving the recent algorithm of Cabello et al. [SOCG 2010] for all g = o(log n). The second is a combinatorial algorithm to compute minimum (s, t)-cuts in undirected surface graphs in 2 O(g) n log n time, improving an algorithm of Chambers et al. [SOCG 2009] for all positive g. Unlike earlier algorithms for surface graphs that construct and search finite portions of the universal cover, our algorithms use another canonical covering space, called the Z 2-homology cover.
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Randomly Removing g Handles at Once

by Glencora Borradaile, James R. Lee, Anastasios Sidiropoulos , 2009
"... It was shown in [11] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g − 1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2 O(g). By removing all g handles at once, we presen ..."
Abstract - Cited by 11 (2 self) - Add to MetaCart
It was shown in [11] that any orientable graph of genus g can be probabilistically embedded into a graph of genus g − 1 with constant distortion. Removing handles one by one gives an embedding into a distribution over planar graphs with distortion 2 O(g). By removing all g handles at once, we present a probabilistic embedding with distortion O(g 2) for both orientable and non-orientable graphs. Our result is obtained by showing that the minimum-cut graph of [6] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [13].
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Contraction decomposition in H-minor-free graphs and algorithmic applications

by Erik D. Demaine, Mohammadtaghi Hajiaghayi, Ken-ichi Kawarabayashi - the 43rd ACM Symposium on Theory of Computing (STOC’11 , 2011
"... We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph of treewidth linear in k. This result is a natural finale to research in contraction decomposition, genera ..."
Abstract - Cited by 9 (3 self) - Add to MetaCart
We prove that any graph excluding a fixed minor can have its edges partitioned into a desired number k of color classes such that contracting the edges in any one color class results in a graph of treewidth linear in k. This result is a natural finale to research in contraction decomposition, generalizing previous such decompositions for planar and bounded-genus graphs, and solving the main open problem in this area (posed at SODA 2007). Our decomposition can be computed in polynomial time, resulting in a general framework for approximation algorithms, particularly PTASs (with k ≈ 1/ε), and fixed-parameter algorithms, for problems closed under contractions in graphs excluding a fixed minor. For example, our approximation framework gives the first PTAS for TSP in weighted H-minor-free graphs, solving a decade-old open problem of Grohe; and gives another fixed-parameter algorithm for k-cut in
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Shortest non-trivial cycles in directed surface graphs

by Jeff Erickson - In Proc. 27th Ann. Symp. Comput. Geom , 2011
"... Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest non-separating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest non-contractible cy ..."
Abstract - Cited by 8 (2 self) - Add to MetaCart
Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest non-separating cycle in G in O(g 2 n log n) time, exactly matching the fastest algorithm known for undirected graphs. We also describe an algorithm to compute the shortest non-contractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.
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Global Minimum Cuts in Surface Embedded Graphs

by Jeff Erickson, Kyle Fox, Amir Nayyeri
"... We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm kno ..."
Abstract - Cited by 8 (5 self) - Add to MetaCart
We give a deterministic algorithm to find the minimum cut in a surface-embedded graph in near-linear time. Given an undirected graph embedded on an orientable surface of genus g, our algorithm computes the minimum cut in g O(g) n log log n time, matching the running time of the fastest algorithm known for planar graphs, due to Ł ˛acki and Sankowski, for any constant g. Indeed, our algorithm calls Ł ˛acki and Sankowski’s recent O(n log log n) time planar algorithm as a subroutine. Previously, the best time bounds known for this problem followed from two algorithms for general sparse graphs: a randomized algorithm of Karger that runs in O(n log³ n) time and succeeds with high probability, and a deterministic algorithm of Nagamochi and Ibaraki that runs in O(n² log n) time. We can also achieve a deterministic g O(g) n² log log n time bound by repeatedly applying the best known algorithm for minimum (s, t)-cuts in surface graphs. The bulk of our work focuses on the case where the dual of the minimum cut splits the underlying surface into multiple components with positive genus.

Shortest Non-trivial Cycles in Directed and Undirected Surface Graphs

by Kyle Fox
"... Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest ..."
Abstract - Cited by 5 (3 self) - Add to MetaCart
Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of non-trivial cycles in G, using different techniques depending on whether or not G is an undirected graph. If G is undirected, then we give an algorithm to compute a shortest non-separating cycle in G in 2O(g) n log log n time. Similar algorithms are given to compute a shortest non-contractible or non-null-homologous cycle in 2O(g+b) n log log n time. Our algorithms for undirected G combine an algorithm of Kutz with known techniques for efficiently enumerating homotopy classes of curves that may be shortest non-trivial cycles. Our main technical contributions in this work arise from assuming G is a directed graph with possibly asymmetric edge weights. For this case, we give an algorithm to compute a shortest non-contractible cycle in G in O((g 3 + g b)n log n) time. In order to achieve this time bound, we use a restriction of the infinite cyclic cover that may be useful in other contexts. We also describe an algorithm to compute a shortest non-null-homologous cycle in G in O((g 2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest non-separating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b. 1
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Subexponential-Time Parameterized Algorithm for Steiner Tree on Planar Graphs ∗

by Marcin Pilipczuk, Michał Pilipczuk, Piotr Sankowski
"... The well-known bidimensionality theory provides a method for designing fast, subexponentialtime parameterized algorithms for a vast number of NP-hard problems on sparse graph classes such as planar graphs, bounded genus graphs, or, more generally, graphs with a fixed excluded minor. However, in orde ..."
Abstract - Cited by 3 (0 self) - Add to MetaCart
The well-known bidimensionality theory provides a method for designing fast, subexponentialtime parameterized algorithms for a vast number of NP-hard problems on sparse graph classes such as planar graphs, bounded genus graphs, or, more generally, graphs with a fixed excluded minor. However, in order to apply the bidimensionality framework the considered problem needs to fulfill a special density property. Some well-known problems do not have this property, unfortunately, with probably the most prominent and important example being the Steiner Tree problem. Hence the question whether a subexponential-time parameterized algorithm for Steiner Tree on planar graphs exists has remained open. In this paper, we answer this question positively and develop an algorithm running in O(2 O((k log k)2/3) n) time and polynomial space, where k is the size of the Steiner tree and n is the number of vertices of the graph. Our algorithm does not rely on tools from bidimensionality theory or graph minors theory, apart from Baker’s classical approach. Instead, we introduce new tools and concepts to the study of the parameterized complexity of problems on sparse graphs.