Results 1  10
of
11
Minimum Cuts and Shortest NonSeparating Cycles via Homology Covers
 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 � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two ap ..."
Abstract

Cited by 18 (5 self)
 Add to MetaCart
(Show Context)
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 � 2homology class in 2 O(g+b) n log n time; this problem is NPhard even for undirected graphs. We also present two applications of our algorithm. The first is an algorithm to compute the shortest nonseparating 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 2homology cover.
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 ..."
Abstract

Cited by 11 (4 self)
 Add to MetaCart
(Show Context)
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 ..."
Abstract

Cited by 9 (1 self)
 Add to MetaCart
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 ..."
Abstract

Cited by 8 (2 self)
 Add to MetaCart
(Show Context)
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.
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 ..."
Abstract

Cited by 5 (3 self)
 Add to MetaCart
(Show Context)
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
Optimal stochastic planarization
 In 2010 IEEE 51st Annual Symposium on Foundations of Computer Science
, 2010
"... It has been shown by Indyk and Sidiropoulos [IS07] that any graph of genus g> 0 can be stochastically embedded into a distribution over planar graphs with distortion 2O(g). This bound was later improved to O(g2) by Borradaile, Lee and Sidiropoulos [BLS09]. We give an embedding with distortion O(l ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
It has been shown by Indyk and Sidiropoulos [IS07] that any graph of genus g> 0 can be stochastically embedded into a distribution over planar graphs with distortion 2O(g). This bound was later improved to O(g2) by Borradaile, Lee and Sidiropoulos [BLS09]. We give an embedding with distortion O(log g), which is asymptotically optimal. Apart from the improved distortion, another advantage of our embedding is that it can be computed in polynomial time. In contrast, the algorithm of [BLS09] requires solving an NPhard problem. Our result implies in particular a reduction for a large class of geometric optimization problems from instances on genusg graphs, to corresponding ones on planar graphs, with a O(log g) loss factor in the approximation guarantee. ar X iv
Faster shortest noncontractible cycles in directed surface graphs
 CoRR
"... Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
Let G be a directed graph embedded on a surface of genus g with b boundary cycles. We describe an algorithm to compute the shortest noncontractible cycle in G in O((g 3 + g b)n log n) time. Our algorithm improves the previous best known time bound of (g + b) O(g+b) n log n for all positive g and b. We also describe an algorithm to compute the shortest nonnullhomologous cycle in G in O((g 2 + g b)n log n) time, generalizing a known algorithm to compute the shortest nonseparating cycle.
FAST ALGORITHMS FOR SURFACE EMBEDDED GRAPHS VIA HOMOLOGY
, 2013
"... We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the ..."
Abstract
 Add to MetaCart
We describe several results on combinatorial optimization problems for graphs where the input comes with an embedding on an orientable surface of small genus. While the specific techniques used differ between problems, all the algorithms we describe share one common feature in that they rely on the algebraic topology construct of homology. We describe algorithms to compute global minimum cuts and count minimum s, tcuts. We describe new algorithms to compute short cycles that are topologically nontrivial. Finally, we describe ongoing work in designing a new algorithm for computing maximum s, tflows in surface embedded graphs. We begin by describing an algorithm to compute global minimum cuts in edge weighted genus g graphs in gO(g)n log log n time. When the genus is a constant, our algorithm’s running time matches the best time bound known for planar graphs due to La̧cki and Sankowski. In our algorithm, we reduce to the problem of finding a minimum weight separating subgraph in the dual
FlowCut Gaps for Integer and Fractional Multiflows
, 2010
"... 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. ..."
Abstract
 Add to MetaCart
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) [14] 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
FlowCut Gaps for Integer and Fractional Multiflows
, 2010
"... 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. ..."
Abstract
 Add to MetaCart
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) [14] 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