Survivable Network Design with Degree or Order Constraints

Lau, Lap Chi; Naor, Joseph; Salavatipour, Mohammad R.; Singh, Mohit

doi:10.1137/070700620

Cited by 64 publications

(104 citation statements)

References 37 publications

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“…This method can be enhanced by adding a relaxation step, where one relaxes a constraint that can be ignored without losing too much in the feasibility. The iterative relaxation method has been very successful for approximating degree-constrained network design problems [24,25,39,43] and directed network design problems [4]. Recently, using an iterative randomized rounding approach, Byrka et al developed an improved approximation algorithm for the Steiner tree problem [8] which was further developed in [16].…”

Section: Introductionmentioning

confidence: 99%

New approaches to multi-objective optimization

et al. 2013

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A natural way to deal with multiple, partially conflicting objectives is turning all the objectives but one into budget constraints. Many classical optimization problems, such as maximum spanning tree and forest, shortest path, maximum weight (perfect) matching, maximum weight independent set (basis) in a matroid or in the intersection of two matroids, become NP-hard even with one budget constraint. Still, for most of these problems efficient deterministic and randomized approximation schemes are known. Not much is known however about the case of two or more budgets: filling this gap, at least partially, is the main goal of this paper. In more detail, we obtain the following main results:• Using iterative rounding for the first time in multi-objective optimization, we obtain multi-criteria PTASs (which slightly violate the budget constraints) for spanning tree, matroid basis, and bipartite matching with k = O(1) budget constraints.• We present a simple mechanism to transform multi-criteria approximation schemes into pure approximation schemes for problems whose feasible solutions define an independence system. This gives improved algorithms for several problems. In particular, this mechanism can be applied to the above bipartite matching algorithm, hence obtaining a pure PTAS.• We show that points in low-dimensional faces of any matroid polytope are almost integral, an interesting result on its own. This gives a deterministic approximation scheme for k-budgeted matroid independent set. • We present a deterministic approximation scheme for k-budgeted matching (in general graphs), where k = O(1). Interestingly, to show that our procedure works, we rely on a non-constructive result by Stromquist and Woodall, which is based on the Ham Sandwich Theorem.

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Section: Introductionmentioning

confidence: 99%

New approaches to multi-objective optimization

et al. 2013

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“…Parallel to the resilience criteria we use in our model, Grotschel et al (1995) argue that two-connected networks are able to provide a sufficient level of survivability because in most cases, the probability of two nodes failing at the same time is significantly small. This resilience criterion is widely accepted throughout the operations research literature; the reader can refer to Steiglitz et al (1969), Monma and Shallcross (1989), Fortz et al (2000) and Lau et al (2009) for relevant studies and detailed surveys on this topic. Similar to our approach, Chimani et al (2008) show that supported by a strong MIP formulation, the branch and cut procedure can efficiently solve two-node-connected Steiner network problems of realistic sizes.…”

Section: Introductionmentioning

confidence: 99%

Regenerator Location Problem and survivable extensions: A hub covering location perspective

Yıldız

Karaşan

2015

Transportation Research Part B: Methodological

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Survivable network design Branch and cut a b s t r a c tIn a telecommunications network the reach of an optical signal is the maximum distance it can traverse before its quality degrades. Regenerators are devices to extend the optical reach. The regenerator placement problem seeks to place the minimum number of regenerators in an optical network so as to facilitate the communication of a signal between any node pair. In this study, the Regenerator Location Problem is revisited from the hub location perspective directing our focus to applications arising in transportation settings. Two new dimensions involving the challenges of survivability are introduced to the problem. Under partial survivability, our designs hedge against failures in the regeneration equipment only, whereas under full survivability failures on any of the network nodes are accounted for by the utilization of extra regeneration equipment. All three variations of the problem are studied in a unifying framework involving the introduction of individual flow-based compact formulations as well as cut formulations and the implementation of branch and cut algorithms based on the cut formulations. Extensive computational experiments are conducted in order to evaluate the performance of the proposed solution methodologies and to gain insights from realistic instances.

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“…For problems WDBoundedTree and WDBoundedNetwork, we propose algorithms which achieve approximation ratios (1, 4 + 3θ + κ) and (2, 7 + 5θ + 2κ) respectively in O(L(|V | + |E|)) time, where L is the time for solving a linear programming. Our algorithms take the approach successfully applied to the bounded degree spanning tree problem by Singh and Lau [17] and to the bounded-degree survivable network design problem by Lau et al [12], which correspond to instances with uniform w 1 and E 2 = E 3 = ∅ in our problems. Their approach is based on the iterative rounding originally used for the generalized Steiner network problem by Jain [8].…”

Section: Weighted Degree Bounded Survivable Network Problem (Wdboundementioning

confidence: 99%

“…Lau et al [12] considered the survivable network problem, and proposed an algorithm that outputs a network of cost at most twice the optimal and the degree of v ∈ V is at most 2b(v)+3. This result was improved in Lau and Singh [13].…”

Section: Previous Workmentioning

confidence: 99%

“…al. [12] designed their algorithm for the case with w 1 (e, u) = w 1 (e, v) = 1, e = uv ∈ E 1 and E 2 = E 3 = ∅ by observing that P N (I) is (2, 4)-bounded with such instances. However, this does not hold in our problem even if w 1 (e, u) = w 1 (e, v) for all e = uv ∈ E 1 and E 2 = E 3 = ∅ as indicated by the following counterexample; Let G be the graph in Figure 2, f (U ) = 1 for all nonempty U ⊂ V , and A = V .…”

Section: Algorithm For Problem Wdboundednetworkmentioning

confidence: 99%

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Network Design with Weighted Degree Constraints

Fukunaga

Nagamochi

2009

WALCOM: Algorithms and Computation

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In an undirected graph G = (V, E) with a weight function w : E × V → Q + , the weighted degree d w (v; E) of a vertex v is defined as {w(e, v) | e ∈ E incident with v}. In this paper, we consider a network design problem with upper-bound on weighted degree of each vertex. Inputs of the problem are an undirected graph G = (V, E) with E = E 1∪ E 2∪ E 3 , weights w 1 :skew supermodular set function f : 2 V → N, and a degree-bound b : A → Q + . A solution consists of F ⊆ E, and weights w i :It is defined to be feasible if the cut-size of U in (V, F ) is at least f (U ) for U ⊂ V , w 2 (e, u) + w 2 (e, v) = µ(e) for e = uv ∈ F 2 , {w 3 (e, u), w 3 (e, v)} = {0, ν(e)} for e = uv ∈ F 3 , and d w1 (v; The goal of this problem is to find a feasible solution that minimizes its cost e∈F c(e).Relaxing the constraints on weighted degree, we propose a bi-criteria approximation algorithm based on the iterative rounding, which has been successfully applied to the degreebounded spanning tree problem. Our algorithm computes a (2, 9 + 5θ)-approximate solution, where θ = max{b(u)/b(v), b(v)/b(u) | uv ∈ E 2 } if E 2 = ∅ and θ = 0 if E 2 = ∅, where (α, β)-approximate solution has the cost at most α times the optimal and the weighted degree of v at most βb(v). We also give a (1, 5 + 3θ)-approximation algorithm to the case of f (U ) = 1 for U ⊂ V . Moreover, a problem minimizing the maximum weighted degree of vertices is also discussed.

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Survivable Network Design with Degree or Order Constraints

Cited by 64 publications

References 37 publications

New approaches to multi-objective optimization

New approaches to multi-objective optimization

Regenerator Location Problem and survivable extensions: A hub covering location perspective

Network Design with Weighted Degree Constraints

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