## Homology flows, cohomology cuts (2009)

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Venue: | ACM SYMPOSIUM ON THEORY OF COMPUTING |

Citations: | 15 - 6 self |

### BibTeX

@INPROCEEDINGS{Chambers09homologyflows,,

author = {Erin W. Chambers and Jeff Erickson and Amir Nayyeri},

title = {Homology flows, cohomology cuts},

booktitle = {ACM SYMPOSIUM ON THEORY OF COMPUTING},

year = {2009},

publisher = {}

}

### OpenURL

### Abstract

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.

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Citation Context ...[21], computes the minimum-cost maximum flow in time O(n 3/2 polylog n log U). For further background on maximum flow algorithms and related results, we refer the reader to monographs by Ahuja et al. =-=[4]-=- and Schriver [68]. Flows in planar graphs. Maximum flows in planar graphs have received considerable attention for more than 50 years. Weihe [76] and Borradaile and Klein [7, 11, 12] describe the his... |

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Citation Context ... vector of O(n) flow values, our algorithm optimizes a vector of 2g + 1 homology coefficients. We perform this optimization implicitly using two different techniques. The central-cut ellipsoid method =-=[37, 36]-=- yields an algorithm that runs in in O(g 7 n log 2 n log 2 C) time, assuming the capacities are integers for integer capacities that sum to C. Alternately, multidimensional parametric search [1] yeild... |

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Citation Context ...tex graph embedded on a surface of genus O(n) has at most O(n) edges. The fastest known combinatorial maximum-flow algorithms for sparse graphs, due to Sleator and Tarjan [70] and Goldberg and Tarjan =-=[33]-=-, run in time O(n 2 log n). The minimum-cost maximum flow can be computed in O(n 2 log 2 n) time using an algorithm of Orlin [64]. (For graphs with small separators, the running time of Orlin’s algori... |

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Citation Context ...formula implies that an n-vertex graph embedded on a surface of genus O(n) has at most O(n) edges. The fastest known combinatorial maximum-flow algorithms for sparse graphs, due to Sleator and Tarjan =-=[70]-=- and Goldberg and Tarjan [33], run in time O(n 2 log n). The minimum-cost maximum flow can be computed in O(n 2 log 2 n) time using an algorithm of Orlin [64]. (For graphs with small separators, the r... |

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232 | Applying parallel computation algorithms in the design of serial algorithms
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Citation Context ...is by Agarwal, Sharir, and Toledo [3].) In our application, we have d = 2g + 1; the algorithm of Klein et al. [53] gives us T s = O(gn log 2 n); and a parallel version of the Floyd-Warshall algorithm =-=[58]-=- gives us T p = O(log n log log n) and P = O(n 3 ). Thus, the overall running time of our algorithm is g O(g) n (log n) 2g+3 (log log n) 2g+1 . Theorem 3.6. Given a graph G = (V, E) embedded on a surf... |

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Citation Context ...eralized to larger families of graphs, such as graphs of higher genus, graphs with forbidden minors, or graphs with small separators. Examples include single-source and multiple-source shortest paths =-=[13, 28, 42, 52, 53, 55, 71]-=-; minimum spanning trees [66, 56]; graph and subgraph isomorphism [34, 44, 59, 23, 24]; and approximation algorithms for the traveling salesman problem, Steiner trees, and other NP-hard problems [8, 1... |

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137 |
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Citation Context ...y and algebraic topology. For more comprehensive background, we refer the interested reader to Gross and Tucker [35] or Mohar and Thommasen [61] for topological graph theory; and Hatcher [41], Massey =-=[57]-=-, or Spanier [62] for algebraic topology. 2.1 Surfaces A surface (more formally, a 2-manifold) is a Hausdorff topological space in which every point has an open neighborhood homeomorphic to � 2 . A cy... |

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Citation Context ...inors, or graphs with small separators. Examples include single-source and multiple-source shortest paths [13, 28, 42, 52, 53, 55, 71]; minimum spanning trees [66, 56]; graph and subgraph isomorphism =-=[34, 44, 59, 23, 24]-=-; and approximation algorithms for the traveling salesman problem, Steiner trees, and other NP-hard problems [8, 10, 9, 20, 24]. A stark exception to this general pattern is the classical maximum flow... |

115 | A faster strongly polynomial minimum cost flow algorithm
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Citation Context ... sparse graphs, due to Sleator and Tarjan [70] and Goldberg and Tarjan [33], run in time O(n 2 log n). The minimum-cost maximum flow can be computed in O(n 2 log 2 n) time using an algorithm of Orlin =-=[64]-=-. (For graphs with small separators, the running time of Orlin’s algorithm can be improved to O(n 2 log n) by replacing Dijkstra’s algorithm with a linear-time shortest-path algorithm [42, 71].) The f... |

113 |
Beyond the flow decomposition barrier
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(Show Context)
Citation Context ...thm can be improved to O(n 2 log n) by replacing Dijkstra’s algorithm with a linear-time shortest-path algorithm [42, 71].) The fastest algorithm known for integer capacities, due to Goldberg and Rao =-=[32]-=-, runs in time O(min{n 2/3 , m 1/2 }m log(n 2 /m)log U) = O(n 3/2 log n log U), where U is an upper bound on the edge capacities. The recent algorithm of Diatch and Spielman [21], computes the minimum... |

109 | Subgraph isomorphism in planar graphs and related problems, Information and
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Citation Context ...inors, or graphs with small separators. Examples include single-source and multiple-source shortest paths [13, 28, 42, 52, 53, 55, 71]; minimum spanning trees [66, 56]; graph and subgraph isomorphism =-=[34, 44, 59, 23, 24]-=-; and approximation algorithms for the traveling salesman problem, Steiner trees, and other NP-hard problems [8, 10, 9, 20, 24]. A stark exception to this general pattern is the classical maximum flow... |

93 | Efficient algorithms for geometric optimization
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Citation Context ...e multidimensional search paradigm independently developed by Cohen and Megiddo [16, 17, 18], Norton, Plotkin, and Tardos [63], and Aneja and Kabadi [6], and extended further by several other authors =-=[1, 2, 3, 19, 50, 51, 72]-=-. Specifically, we use the version of the technique described by Agarwala and Fernández-Baca [1]. The method requires two black-box algorithms for the decision problem, one serial and the other parall... |

87 | Discrete exterior calculus
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Citation Context ...boundary circulations as 1-boundaries, H(G) as the first homology group H 1(Σ), and H(G; st) as the relative homology group H 1(Σ, {s, t}), all with real coefficients. Discrete differential geometers =-=[9, 26, 51]-=- will recognize 1-chains as discrete 1-forms, the boundary operator as the adjoint of the discrete exterior derivative, circulations as duals of closed 1-forms, boundary circulations as duals of exact... |

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Citation Context ...inors, or graphs with small separators. Examples include single-source and multiple-source shortest paths [13, 28, 42, 52, 53, 55, 71]; minimum spanning trees [66, 56]; graph and subgraph isomorphism =-=[34, 44, 59, 23, 24]-=-; and approximation algorithms for the traveling salesman problem, Steiner trees, and other NP-hard problems [8, 10, 9, 20, 24]. A stark exception to this general pattern is the classical maximum flow... |

83 |
Finding small simple cycle separators for 2-connected planar graphs
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Citation Context ...; and Mozes and Wulff-Nilsen [95]. These algorithms rely on Miller’s observation that any n-vertex planar graph contains a simple cycle separator of length O( � n), which can be computed in O(n) time =-=[91]-=-. Because these algorithms require additional structure in the separator decomposition, we cannot directly substitute separator results for genus-g graphs. 6 Let G = (V, E) denote the symmetric dire... |

79 |
A separation theorem for graphs of bounded genus
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Citation Context ... generalizes directly to higher-genus graphs via the observation that any n-vertex graph of genus g can be separated into planar subgraphs, each with at most 2n/3 vertices, by removing O( � gn) edges =-=[22, 31, 46, 45]-=-. Moreover, such a separator can be computed in O(n) time [5, 25]. � 3.2 Flow Homology Basis Every (s, t)-flow is a weighted sum of (s, t)-paths; consequently, every homology class of (s, t)-flows is ... |

76 | Greedy optimal homotopy and homology generators
- Erickson, Whittlesey
- 2005
(Show Context)
Citation Context ...ed by (the homology classes of) 2g directed cycles γ 1, γ 2, . . . , γ 2g in independent homology classes. Using an algorithm similar to Lemma 3.3, we can construct such a set of cycles in O(gn) time =-=[26, 27]-=-. Corollary 3.4 implies that it suffices to find the homology class of the maximum-value circulation. Specifically, we need to find a feasible homology vector (φ 1, ...,φ 2g) such that the cost functi... |

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Citation Context ...n and Nemirovsky [95, 96] and Shor [85] as a convex programming algorithm. Khachiyan [62, 63] adapted the ellipsoid method to give the first polynomial-time algorithm for linear programming; see also =-=[38, 8]-=-. Khachiyan’s algorithm was further adapted to solve implicit linear programming problems by Grötschel, Lovász, and Schrijver [44, 45]. Here we give only a brief sketch, with just enough details to th... |

57 |
The graph genus problem is NP-complete
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Citation Context ...ys the weighted sum of at most 2g directed cycles. We emphasize that all our algorithms require an explicit embedding as part of the input. Computing the minimum genus of an abstract graph is NP-hard =-=[88]-=-; moreover, no efficient algorithms are known that approximate the genus within a factor of o( � n) [19]. On the other hand, for any constant g, it is possible to compute either an embedding of a give... |

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Citation Context ...eralized to larger families of graphs, such as graphs of higher genus, graphs with forbidden minors, or graphs with small separators. Examples include single-source and multiple-source shortest paths =-=[13, 28, 42, 52, 53, 55, 71]-=-; minimum spanning trees [66, 56]; graph and subgraph isomorphism [34, 44, 59, 23, 24]; and approximation algorithms for the traveling salesman problem, Steiner trees, and other NP-hard problems [8, 1... |

52 | Planar separators and parallel polygon triangulation
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Citation Context ... [32, 47, 69, 68]. Moreover, such a separator can be computed in O(n) time [5, 37], after which the recursive separator decomposition of the resulting planar subgraphs can be constructed in O(n) time =-=[50]-=-. □ The serial setting is not so straightforward. In the following theorem, we describe an algorithm that generalizes algorithms for planar graphs by Fakcharoenphol and Rao [42]; Klein, Mozes, and Wei... |

50 | A linear time algorithm for embedding graphs in an arbitrary surface
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Citation Context ...or of o( � n) [19]. On the other hand, for any constant g, it is possible to compute either an embedding of a given graph on a surface of genus g, or an obstruction to such an embedding, in O(n) time =-=[61, 74]-=-. In a companion paper [18], we describe an algorithm to compute minimum cuts in g O(g) n log n time, using very different techniques than in this paper [17, 67]. Finding a minimum-capacity (s, t)-cut... |

47 |
2005), Discrete differential forms for computational modeling
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Citation Context ...boundary circulations as 1-boundaries, H(G) as the first homology group H 1(Σ), and H(G; st) as the relative homology group H 1(Σ, {s, t}), all with real coefficients. Discrete differential geometers =-=[9, 26, 51]-=- will recognize 1-chains as discrete 1-forms, the boundary operator as the adjoint of the discrete exterior derivative, circulations as duals of closed 1-forms, boundary circulations as duals of exact... |

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Citation Context ... Grötschel et al. prove that to maintain sufficient precision in the kth iteration, it suffices to round all numbers to O(k) bits [44, 45]. Thus, the overall running time of our algorithm is (crudely =-=[6, 82]-=-) at most O(N(T sN + d 2 N log 2 N)) = O(T sd 2 log 2 ∆ + d 4 log 2 ∆ log 2 (d log ∆)). 3.3.2 The Flow Homology Polytope Let Φ denote the polytope of feasible flow homology classes, that is, the feasi... |

40 |
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Citation Context ... computed in O(n 3/2 ) time by computing a single-source shortest path tree in a dual graph with both positive and negative edge weights, using an algorithm of Lipton, Rose, and Tarjan [55]; see also =-=[60]-=-. Binary search over the possible flow values gives a max-flow algorithm that runs in O(n 3/2 log C) time, where C is the sum of the capacities. This running time is improved by recent planar shortest... |

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Citation Context ...at is, every branch in the algorithm is based on the sign of an affine combination of the coordinates of x. Equivalently, we assume that the separation oracle can be modeled by a linear decision tree =-=[29, 94]-=-. Let T s denote the number of arithmetic operations (additions, subtractions, scalar multiplications, and comparisons) executed by a single call to the separation oracle. Let φ OPT denote the optimum... |

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Citation Context ...n-vertex graph of genus g can be separated into planar subgraphs, each with at most 2n/3 vertices, by removing O( � gn) edges [22, 31, 46, 45]. Moreover, such a separator can be computed in O(n) time =-=[5, 25]-=-. � 3.2 Flow Homology Basis Every (s, t)-flow is a weighted sum of (s, t)-paths; consequently, every homology class of (s, t)-flows is a weighted sum of homology classes of (s, t)-paths. It follows im... |

34 |
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Citation Context ...l graph theory and algebraic topology. For more comprehensive background, we refer the interested reader to Gross and Tucker [55] or Mohar and Thommasen [94] for topological graph theory; and Hatcher =-=[62]-=- or Massey [88] for algebraic topology. 2.1 Surfaces A surface (more formally, a 2-manifold) is a Hausdorff topological space in which every point has an open neighborhood homeomorphic to � 2 . A cycl... |

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Citation Context ...n log 2 n log 2 C) time for integer capacities that sum to C. Alternately, in Section 3.4, we use multidimensional parametric search [3, 23], together with a parallel shortest-path algorithm of Cohen =-=[21]-=-, to obtain a combinatorial algorithm for graphs with arbitrary real capacities that runs in gO(g) n3/2 time. 1 For any fixed genus g, both our algorithms improve the previous best time bounds by roug... |

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Citation Context ...on the complexity of the graph, but exponential dependence on the genus of the underlying surface. Our algorithm uses the multidimensional search paradigm independently developed by Cohen and Megiddo =-=[16, 17, 18]-=-, Norton, Plotkin, and Tardos [63], and Aneja and Kabadi [6], and extended further by several other authors [1, 2, 3, 19, 50, 51, 72]. Specifically, we use the version of the technique described by Ag... |

29 | Computing shortest non-trivial cycles on orientable surfaces of bounded genus in almost linear time
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Citation Context ... the weighted sum of at most 2g directed cycles. In a companion paper [15], we describe an algorithm to compute minimum cuts in g O(g) n log n time, using very different techniques than in this paper =-=[14, 54]-=-. Essentially the same algorithm computes the shortest cycle in every � 2-homology class, in the same running time. Unlike the corresponding problem for circulations considered in this paper, we prove... |

28 |
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Citation Context ...opology. For more comprehensive background, we refer the interested reader to Gross and Tucker [35] or Mohar and Thommasen [61] for topological graph theory; and Hatcher [41], Massey [57], or Spanier =-=[62]-=- for algebraic topology. 2.1 Surfaces A surface (more formally, a 2-manifold) is a Hausdorff topological space in which every point has an open neighborhood homeomorphic to � 2 . A cycle in a surface ... |

27 | A polynomial-time approximation scheme for Steiner tree in planar graphs
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Citation Context ..., 71]; minimum spanning trees [66, 56]; graph and subgraph isomorphism [34, 44, 59, 23, 24]; and approximation algorithms for the traveling salesman problem, Steiner trees, and other NP-hard problems =-=[8, 10, 9, 20, 24]-=-. A stark exception to this general pattern is the classical maximum flow problem and its dual, the minimum cut problem. Flow and cuts were originally developed as tools for studying railway and other... |

27 | An O(n log n) algorithm for maximum st-flow in a directed planar graph
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Citation Context ...es lie on the same face. A long series of results has led to planar maximum-flow algorithms that run in O(n log n) time, first for undirected graphs [37, 48, 81] and more recently for directed graphs =-=[10, 14, 15]-=-. Despite more than half a century of attention on flows in planar graphs, surprisingly little is known about flows in these more general graph families. Even for graphs embedded on the torus, the fas... |

27 | Multiple source shortest paths in a genus g graph
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