The cut property under demand uncertainty

Self Cite

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.

“…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%

“…An edge

e

K

. The two definitions are equivalent for the NL problem, as a problem with unbounded capacities becomes infeasible only when some source–destination pair is disconnected.…”

Section: Modelsmentioning

confidence: 99%

“…Let

T \subseteq δ (S)

be a minimal bridge set (possibly

δ (S)

itself). Since

d (S) / U \leq 1

, the CB inequality

\sum_{e \in T} x_{e} \geq 1

dominates the CB inequality for

\sum_{e \in normalδ false(S false)} x_{e} \geq 1

, which dominates the BE inequality

\sum_{e \in normalδ false(S false)} x_{e} \geq d (S) / U

. In general, the feasibility condition provided by the cuts is only necessary, as there may exist vectors

boldx

Section: Modelsmentioning

confidence: 99%

See 1 more Smart Citation

The capacity formulation of the capacitated edge activation problem

Mattia

2017

Self Cite

“…For papers dealing with the survivability of a network under some failure scenarios see , where several protection and restoration techniques are analyzed. Models for problems with uncertain demands are considered in . For a more complete overview and additional references on optimization models for telecommunications problems we address the reader to .…”

Section: Literature Reviewmentioning

confidence: 99%

An optimization approach for congestion control in network routing with quality of service requirements

Avella

Bernardi²,

Boccia

et al. 2019

Self Cite

In this paper we study a network design problem arising in the management of a carrier network. The aim is to route a traffic matrix, minimizing a measure of the network congestion while guaranteeing a prescribed quality of service. We formulate the problem, devise presolve procedures to reduce the size of the corresponding mixed-integer programming formulation and show that the proposed approach can efficiently solve some real-life problems, leading to an improvement with respect to the current practice in a real case study.

“…The result also applies to two‐layer networks . As for uncertain demands, in some properties about cuts and feasibility for deterministic network design are generalized to robust problems with splittable flows. In the polyhedron corresponding to the capacity formulation of the RNL problem has been studied, showing that the cutset inequalities are facets under static, volume, affine and dynamic routing, both for splittable and for unsplittable flows, whereas the three‐partition inequalities are facets for the dynamic/splittable case, but not in other settings.…”

Section: Introductionmentioning

confidence: 98%

A polyhedral analysis of the capacitated edge activation problem with uncertain demands

Mattia

2019

Self Cite

The capacitated edge activation problem consists of activating a minimum cost set of capacitated edges to ensure the routing of some traffic demands. If the demands are subject to uncertainty, we speak of the robust capacitated edge activation problem. We consider a capacity formulation of the problem and investigate, from a polyhedral perspective, the similarities and the differences between the robust capacitated edge activation and the robust network loading polyhedron, as well as between the polyhedra corresponding to different routing and flows policies. KEYWORDScapacitated edge activation problem, capacity formulation, demand uncertainty, facets, splittable and unsplittable flows, static and dynamic routing