Results 1  10
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17
Minimum Cuts and Shortest Homologous Cycles
 SYMPOSIUM ON COMPUTATIONAL GEOMETRY
, 2009
"... We describe the first algorithms to compute minimum cuts in surfaceembedded 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 ..."
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Cited by 33 (11 self)
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We describe the first algorithms to compute minimum cuts in surfaceembedded 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 minimumcut algorithm computes a minimumcost subgraph in every Z2homology class. We also prove that finding a minimumcost subgraph homologous to a single input cycle is NPhard.
Embeddings of topological graphs: Lossy invariants, linearization, and 2sums
"... We study the properties of embeddings, multicommodity flows, and sparse cuts in minorclosed families of graphs which are also closed under 2sums; this includes planar graphs, graphs of bounded treewidth, and constructions based on recursive edge replacement. In particular, we show the following. • ..."
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Cited by 14 (2 self)
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We study the properties of embeddings, multicommodity flows, and sparse cuts in minorclosed families of graphs which are also closed under 2sums; this includes planar graphs, graphs of bounded treewidth, and constructions based on recursive edge replacement. In particular, we show the following. • Every graph which excludes K4 as a minor (in particular, seriesparallel graphs) admits an embedding into L1 with distortion at most 2, confirming a conjecture of Gupta, Newman, Rabinovich, and Sinclair, and improving over their upper bound of 14. This shows that in every multicommodity flow instance on such a graph, one can route a maximum concurrent flow whose value is at least half the cut bound. Our upper bound is optimal, as it matches a recent lower bound of Lee and Raghavendra. • We move beyond K4minorfree graphs by showing that every W4minorfreegraph embeds into L1 with O(1) distortion, where W4 is the 4wheel. By a characterization of Seymour, these graphs are precisely subgraphs of 2sums of K4’s. • We prove that if G and H are two minorclosed families and G is closed under taking 2sums, then members of G embed nontrivially into noncontracting distributions over members of H if and only if G ⊆ H. This significantly generalizes a result of Gupta, et al. where G and H are the families of K4minorfree graphs and trees, respectively.
Randomly Removing g Handles at Once
, 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 ..."
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Cited by 11 (2 self)
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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 nonorientable graphs. Our result is obtained by showing that the minimumcut graph of [6] has low dilation, and then randomly cutting this graph out of the surface using the Peeling Lemma from [13].
On the Geometry of Graphs with a Forbidden Minor
"... We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the GuptaNewmanRabinovichSinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)approximate mul ..."
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Cited by 11 (3 self)
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We study the topological simplification of graphs via random embeddings, leading ultimately to a reduction of the GuptaNewmanRabinovichSinclair (GNRS) L1 embedding conjecture to a pair of manifestly simpler conjectures. The GNRS conjecture characterizes all graphs that have an O(1)approximate multicommodity maxflow/mincut theorem. In particular, its resolution would imply a constant factor approximation for the general Sparsest Cut problem in every family of graphs which forbids some minor. In the course of our study, we prove a number of results of independent interest. • Every metric on a graph of pathwidth k embeds into a distribution over trees with distortion depending only on k. This is equivalent to the statement that any family of graphs excluding a fixed tree embeds into a distribution over trees with O(1) distortion. For graphs of treewidth k, GNRS showed that this is impossible even for k = 2. In particular, our result implies that pathwidthk metrics embed into L1 with bounded distortion, which resolves an open question even for k = 3. • We prove a generic peeling lemma which uses random retractions to peel simple structures like handles and apices off of graphs. This allows a number of new topological reductions. For example, if X is any metric space in which the removal of O(1) points leaves a bounded genus metric, then X embeds into a distribution over planar graphs. • Using these techniques, we show that the GNRS embedding conjecture is equivalent to two simpler conjectures: (1) The wellknown planar embedding conjecture, and (2) a conjecture about embeddings of ksums of graphs.
Computing the shortest essential cycle
, 2008
"... An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of ..."
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Cited by 11 (4 self)
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An essential cycle on a surface is a simple cycle that cannot be continuously deformed to a point or a single boundary. We describe algorithms to compute the shortest essential cycle in an orientable combinatorial surface in O(n 2 log n) time, or in O(n log n) time when both the genus and number of boundaries are fixed. Our result corrects an error in a paper of Erickson and HarPeled.
FlowCut Gaps for Integer and Fractional Multiflows
, 2009
"... Consider a routing problem instance consisting of a demand graph H = (V, E(H)) and a supply graph G = (V, E(G)). If the pair obeys the cut condition, then the flowcut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C ..."
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Cited by 9 (1 self)
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Consider a routing problem instance consisting of a demand graph H = (V, E(H)) and a supply graph G = (V, E(G)). If the pair obeys the cut condition, then the flowcut gap for this instance is the minimum value C such that there exists a feasible multiflow for H if each edge of G is given capacity C. It is wellknown that the flowcut gap may be greater than 1 even in the case where G is the (seriesparallel) graph K2,3. In this paper we are primarily interested in the “integer ” flowcut gap. What is the minimum value C such that there exists a feasible integer valued multiflow for H if each edge of G is given capacity C? We formulate a conjecture that states that the integer flowcut gap is quantitatively related to the fractional flowcut gap. In particular this strengthens the wellknown conjecture that the flowcut gap in planar and minorfree graphs is O(1) [12] to suggest that the integer flowcut gap is O(1). We give several technical tools and results on nontrivial special classes of graphs to give evidence for the conjecture and further explore the “primal ” method for understanding flowcut gaps; this is in contrast to and orthogonal to the highly successful metric embeddings approach. Our results include the following: • Let G be obtained by seriesparallel operations starting from an edge st, and consider orienting all edges in G in the direction from s to t. A demand is compliant if its endpoints are joined by a directed
Shortest nontrivial cycles in directed surface graphs
 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 nonseparating 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 noncontractible cy ..."
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Cited by 8 (2 self)
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Let G be a directed graph embedded on a surface of genus g. We describe an algorithm to compute the shortest nonseparating 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 noncontractible cycle in G in g O(g) n log n time, matching the fastest algorithm for undirected graphs of constant genus.
A Note on Multiflows and Treewidth
, 2007
"... 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 ..."
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Cited by 6 (1 self)
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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 nonnegative 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 Ω(min{r, √ n}). These results extend the rather limited number of graph classes that admit polylogarithmic approximations for maximum EDP. Another related question is whether the cutcondition, a necessary condition for (fractionally) routing all pairs, is approximately sufficient. We show the following result in this context. • The flowcut gap for product multicommodity flow instances is O(log r). This was shown earlier by Rabinovich; we obtain a different proof.
Shortest Nontrivial Cycles in Directed and Undirected Surface Graphs
"... Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial 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 ..."
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Cited by 5 (3 self)
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Let G be a graph embedded on a surface of genus g with b boundary cycles. We describe algorithms to compute multiple types of nontrivial 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 nonseparating cycle in G in 2O(g) n log log n time. Similar algorithms are given to compute a shortest noncontractible or nonnullhomologous 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 nontrivial 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 noncontractible 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 nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, extending a known algorithm of Erickson to compute a shortest nonseparating cycle. In both the undirected and directed cases, our algorithms improve the best time bounds known for many values of g and b. 1
Preserving Terminal Distances using Minors ⋆
"... Abstract. We introduce the following notion of compressing an undirected graph G with (nonnegative) edgelengths and terminal vertices R ⊆ V (G). A distancepreserving minor is a minor G ′ (of G) with possibly different edgelengths, such that R ⊆ V (G ′ ) and the shortestpath distance between ever ..."
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Cited by 4 (3 self)
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Abstract. We introduce the following notion of compressing an undirected graph G with (nonnegative) edgelengths and terminal vertices R ⊆ V (G). A distancepreserving minor is a minor G ′ (of G) with possibly different edgelengths, such that R ⊆ V (G ′ ) and the shortestpath distance between every pair of terminals is exactly the same in G and in G ′. We ask: what is the smallest f ∗ (k) such that every graph G with k = R  terminals admits a distancepreserving minor G ′ with at most f ∗ (k) vertices? Simple analysis shows that f ∗ (k) ≤ O(k 4). Our main result proves that f ∗ (k) ≥ Ω(k 2), significantly improving over the trivial f ∗ (k) ≥ k. Our lower bound holds even for planar graphs G, in contrast to graphs G of constant treewidth, for which we prove that O(k) vertices suffice. 1