Integer flows

Younger, D. H.

doi:10.1002/jgt.3190070307

Cited by 41 publications

(22 citation statements)

References 6 publications

(3 reference statements)

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“…There is another ways to derive the (2 + )-flow from the weak circular flow conjecture, as pointed out in [20]. If a graph has an orientation which is balanced modulo k where k is odd, then we can think of this as a flow where each flow value is (k − 1)/2 in the integers reduced modulo k. A result of Younger [19] then says that this flow can be replaced by an integer flow where each flow value is (k − 1)/2 or (k + 1)/2, possibly after reversing some edge directions. If we divide by k, then all flow values are 1 − 1/(2k) or 1 + 1/(2k).…”

Section: The (2 + )-Flow Conjecturementioning

confidence: 99%

Group flow, complex flow, unit vector flow, and the(2+ϵ)-flow conjecture

Thomassen¹

2014

Journal of Combinatorial Theory, Series B

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Section: The (2 + )-Flow Conjecturementioning

confidence: 99%

Group flow, complex flow, unit vector flow, and the(2+ϵ)-flow conjecture

Thomassen¹

2014

Journal of Combinatorial Theory, Series B

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“…For an undirected graph H, we say H has a nowhere-zero 6-flow if H has an orientation of its edges which has a nowhere-zero 6-flow. Seymour [41] proved that every 2-edgeconnected graph has a nowhere-zero 6-flow and such a flow can be found in polynomial time (see [44,40]). 4 Let H(U, F ) be a good density subgraph returned by D2ECS algorithm with more than 2k nodes, and let f be a nowhere-zero 6-flow on H. We can obtain a directed multigraph D(U, A) from f and H by placing f (e) copies of each edge e in the direction defined by the flow.…”

Section: Lemma 52 There Is An O(log N)-approximation For D2ecsmentioning

confidence: 99%

Survivable Network Design with Degree or Order Constraints

Lau¹,

Naor²,

Salavatipour³

et al. 2009

SIAM J. Comput.

101

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Abstract. We present algorithmic and hardness results for network design problems with degree or order constraints. The first problem we consider is the Survivable Network Design problem with degree constraints on vertices. The objective is to find a minimum cost subgraph which satisfies connectivity requirements between vertices and also degree upper bounds Bv on the vertices. This includes the well-studied Minimum Bounded Degree Spanning Tree problem as a special case. Our main result is a (2, 2Bv +3)-approximation algorithm for the edge-connectivity Survivable Network Design problem with degree constraints, where the cost of the returned solution is at most twice the cost of an optimum solution (satisfying the degree bounds) and the degree of each vertex v is at most 2Bv + 3. This implies the first constant factor (bicriteria) approximation algorithms for many degree constrained network design problems, including the Minimum Bounded Degree Steiner Forest problem. Our results also extend to directed graphs and provide the first constant factor (bicriteria) approximation algorithms for the Minimum Bounded Degree Arborescence problem and the Minimum Bounded Degree Strongly k-Edge-Connected Subgraph problem. In contrast, we show that the vertex-connectivity Survivable Network Design problem with degree constraints is hard to approximate, even when the cost of every edge is zero. A striking aspect of our algorithmic result is its simplicity. It is based on the iterative relaxation method, which is an extension of Jain's iterative rounding method. This provides an elegant and unifying algorithmic framework for a broad range of network design problems. We also study the problem of finding a minimum cost λ-edge-connected subgraph with at least k vertices, which we call the (k, λ)-subgraph problem. This generalizes some well-studied classical problems such as the k-MST and the minimum cost λ-edgeconnected subgraph problems. We give a polylogarithmic approximation for the (k, 2)-subgraph problem. However, by relating it to the Densest k-Subgraph problem, we provide evidence that the (k, λ)-subgraph problem might be hard to approximate for arbitrary λ.

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“…The circuit cover problem is related to problems involving graph embeddings [Arc, Hag, Lit, Tut], flow theory [Cel,Fan2,Jael,You], short circuit covers [Alo,Ber,Fanl,Gua,Jac,Jam2,Jam3,Tari,Zhal], the Chinese Postman Problem [Edm, Gua, Ita, Jac], perfect matchings [Ful,God2,p. 22] and decompositions of eulerian graphs [Fiel,Fle2,Sey3].…”

Section: Introductionmentioning

confidence: 99%

Graphs with the circuit cover property

Alspach¹,

Goddyn²,

Zhang³

1994

Trans. Amer. Math. Soc.

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Abstract. A circuit cover of an edge-weighted graph (G, p) is a multiset of circuits in G such that every edge e is contained in exactly p(e) circuits in the multiset. A nonnegative integer valued weight vector p is admissible if the total weight of any edge-cut is even, and no edge has more than half the total weight of any edge-cut containing it. A graph G has the circuit cover property if (G, p) has a circuit cover for every admissible weight vector p . We prove that a graph has the circuit cover property if and only if it contains no subgraph homeomorphic to Petersen's graph. In particular, every 2-edge-connected graph with no subgraph homeomorphic to Petersen's graph has a cycle double cover.

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Integer flows

Cited by 41 publications

References 6 publications

Group flow, complex flow, unit vector flow, and the(2+ϵ)-flow conjecture

Group flow, complex flow, unit vector flow, and the(2+ϵ)-flow conjecture

Survivable Network Design with Degree or Order Constraints

Graphs with the circuit cover property

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