k-Partition-based facets of the network design problem

Agarwal, Yash

doi:10.1002/net.20098

Cited by 39 publications

(24 citation statements)

References 26 publications

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“…Then, solution

boldz

obtained as

z_{t} = y_{t}

for

t \in E ∖ {e}

z_{e} = 1

is still feasible, but

{bolda}^{T} z = {bolda}^{T} y - a_{e} = b - a_{e} < b

. Hence, any feasible solution that satisfies

{bolda}^{T} x \geq b

with equality also satisfies facet

x_{e} \leq 1

with equality. Partitions of the node set provide facet‐defining inequalities for several network design problems . Here we give conditions for them to provide facets for the CEA problem.…”

Section: Polyhedral Resultsmentioning

confidence: 99%

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The capacity formulation of the capacitated edge activation problem

Mattia

2017

Networks

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Given a capacitated network, the Capacitated Edge Activation problem consists of choosing the edges to be activated to ensure the routing of a set of demands. We focus on capacity formulations of the problem, that is, formulations including only design variables, whereas variables corresponding to the routes are projected out. First, we investigate the combinatorial properties of the problem to derive a capacity formulation both for splittable flows (the routes are unrestricted) and for unsplittable ones (each demand must be routed on a single path). Then, we study the corresponding polyhedron, identifying valid and facet‐defining inequalities. Finally, we develop a branch‐and‐cut algorithm and present computational results.

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“…Then, solution

boldz

obtained as

z_{t} = y_{t}

for

t \in E ∖ {e}

z_{e} = 1

is still feasible, but

{bolda}^{T} z = {bolda}^{T} y - a_{e} = b - a_{e} < b

. Hence, any feasible solution that satisfies

{bolda}^{T} x \geq b

with equality also satisfies facet

x_{e} \leq 1

with equality. Partitions of the node set provide facet‐defining inequalities for several network design problems . Here we give conditions for them to provide facets for the CEA problem.…”

Section: Polyhedral Resultsmentioning

confidence: 99%

“…We note that the conditions for a facet of a

p

‐node subproblem to be a facet for the original problem are more restrictive for CEA than for NL, where only the connectivity of the subsets is required . We now consider inequalities deriving from 2‐ and 3‐partitions of the node set.…”

Section: Polyhedral Resultsmentioning

confidence: 99%

The capacity formulation of the capacitated edge activation problem

Mattia

2017

Networks

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“…Our solution approach is motivated by a key theorem proposed in Agarwal , according to which a facet inequality of the p ‐node subproblem resulting from a p ‐partition of the original problem translates into a facet of the original problem if certain mild conditions are satisfied. This theorem was originally proposed for the single facility NDP.…”

Section: The Capacity Formulation and A Key Theoremmentioning

confidence: 99%

“…They prove that partition inequalities and total capacity inequality completely define the polyhedron of the three node problem. A key theorem is presented for single‐facility NDP in according to which facet inequalities of p ‐partition problem translates to that of the original problem if certain mild conditions are satisfied. This theorem is used in to provide a complete polyhedral description of the 4‐node NDP and, therefore, a complete knowledge of 4‐partition‐based facets of larger NDPs.…”

Section: Introductionmentioning

confidence: 99%

Solving the two‐facility network design problem with 3‐partition facets

Hamid

Agarwal

2015

Networks

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The article studies the problem of designing telecommunication networks using transmission facilities of two different capacities. The point‐to‐point communication demands are met by installing a mix of facilities of both capacities on the edges to minimize total cost. We consider 3‐partitions of the original graph which results in smaller 3‐node subproblems. The extreme points of this subproblem polyhedron are characterized using a set of propositions. A new approach for computing the facets of the 3‐node subproblem is introduced based on polarity theory. The facets of the subproblem are then translated back to those of the original problem using a generalized version of a previously known theorem. The approach has been tested on several randomly generated and real life networks. The computational results show that the new family of facets significantly strengthen the linear programming formulation and reduce the integrality gap. Also, there is a substantial reduction in the size of the branch‐and‐bound (B&B) tree if these facets are used. Problems as large as 37 nodes and 57 edges have been solved to optimality within a few minutes of computer time. © 2015 Wiley Periodicals, Inc. NETWORKS, Vol. 66(1), 11–32 2015

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“…The special case of the network design problem with only integer capacities is called the network loading problem (NLP). This problem has been investigated intensely in the literature (Agarwal, 2006(Agarwal, , 2009Avella et al, 2007;Barahona, 1996;Magnanti et al, 1993). Different specifications of the NLP such as static (the same routing template for all considered traffic matrices) or dynamic (one routing per traffic matrix) routing, splittable or unsplittable flows can be considered.…”

Section: Introductionmentioning

confidence: 99%