A branch and cut approach to the cardinality constrained circuit problem

Bauer, Petra; Linderoth, Jeff; Savelsbergh, Martin

doi:10.1007/s101070100209

Cited by 26 publications

(32 citation statements)

References 31 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The constraints (5), called rounded capacity (RC) inequalities, prevent the existence of infeasible routes, and also have the side-effect of preventing subtours. Finally, (6) are the integrality conditions on the x-variables. Many classes of valid inequalities have been presented for the integer polytope associated with the formulation (2)-(6).…”

Section: The Two-index Formulationmentioning

confidence: 99%

“…The constraints (13)- (15) themselves, and the bounds 0 ≤ x k ij ≤ 1 and 0 ≤ y k i ≤ 1, are trivial examples of one-vehicle inequalities. Other inequalities for the capacitated circuit polytope can be found in Bauer et al [6] and Bixby et al [7].…”

Section: Proposition 4 If We Relax the Equationsmentioning

confidence: 99%

“…Interesting multi-commodity flow formulations arise if we take the two-index vehicle flow formulation (1)- (6), and add extra variables representing one commodity per customer. One such formulation, based on another unpublished paper by Gavish and Graves [18], was explored by Laporte and Nobert [28] and Baldacci et al [5].…”

Section: Multi-commodity Flow Formulationsmentioning

confidence: 99%

See 2 more Smart Citations

Projection results for vehicle routing

2005

View full text Add to dashboard Cite

Abstract.A variety of integer programming formulations have been proposed for Vehicle Routing Problems (VRPs), including the so-called two-and three-index formulations, the set partitioning formulation, and various formulations based on extra variables representing the flow of one or more commodities. Until now, there has not been a systematic study of how these formulations relate to each other. An exception is a paper of Luis Gouveia, which shows that a one-commodity flow formulation of Gavish and Graves yields, by projection, certain 'multistar' inequalities in the two-index space.We give a survey of formulations for the capacitated VRP, and then present various results of a similar flavour to those of Gouveia. In particular, we show that:-the three-index formulation, augmented by certain families of valid inequalities, gives the same lower bound as the two-index formulation, augmented by certain simpler families of valid inequalities, -the two-commodity flow formulation of Baldacci et al. gives the same lower bound and the same multistar inequalities as the one-commodity Gavish and Graves formulation, -a certain non-standard multi-commodity flow formulation, with one commodity per customer, implies by projection certain 'hypotour-like' inequalities in the two-index space, -the set partitioning formulation implies by projection both multistar and hypotour-like inequalities in the two-index space.We also briefly look at some other variants of the VRP, such as the VRP with time windows, and derive multistar-like inequalities for them. We also present polynomial-time separation algorithms for some of the new inequalities.

show abstract

Section: The Two-index Formulationmentioning

confidence: 99%

Section: Proposition 4 If We Relax the Equationsmentioning

confidence: 99%

Section: Multi-commodity Flow Formulationsmentioning

confidence: 99%

See 1 more Smart Citation

Projection results for vehicle routing

2005

View full text Add to dashboard Cite

show abstract

“…Finally, let c = (3, n). Beside d n + 1 Hamiltonian cycles, consider the 3-cycles (1, 3), (3,4), (4, 1) and {(1, 2), (2, j), (j, 1)}, j = 3, . .…”

Section: Introductionmentioning

confidence: 99%

On cardinality constrained cycle and path polytopes

Kaibel

Stephan

2008

Math. Program.

View full text Add to dashboard Cite

Given a directed graph D = (N, A) and a sequence of positive integers 1 ≤ c1 < c2 < · · · < cm ≤ |N |, we consider those path and cycle polytopes that are defined as the convex hulls of simple paths and cycles of D of cardinality cp for some p ∈ {1, . . . , m}, respectively. We present integer characterizations of these polytopes by facet defining linear inequalities for which the separation problem can be solved in polynomial time. These inequalities can simply be transformed into inequalities that characterize the integer points of the undirected counterparts of cardinality constrained path and cycle polytopes. Beyond we investigate some further inequalities, in particular inequalities that are specific to odd/even paths and cycles.

show abstract

“…In the early years of integer optimization, considerable research activity was focused on identifying part (or all) of the list of facets for specific combinatorial optimization problems by exploiting the special structure of conv(S) (Balas and Padberg 1972;Balas 1975;Bauer et al 2002;Hammer et al 1975;Nemhauser and Sigismondi 1992;Nemhauser and Trotter 1974;Nemhauser and Vance 1994;Padberg 1973Padberg , 1974Padberg , 1979aPochet and Wolsey 1991;Wolsey 1975Wolsey , 1976. This led to a wide variety of problem-dependent algorithms that are nevertheless based on the underlying principle embodied in Weyl's theorem.…”

Section: Advanced Proceduresmentioning

confidence: 99%

Identity Matrix

2013

Encyclopedia of Operations Research and Management Science

View full text Add to dashboard Cite

show abstract

A branch and cut approach to the cardinality constrained circuit problem

Cited by 26 publications

References 31 publications

Projection results for vehicle routing

Projection results for vehicle routing

On cardinality constrained cycle and path polytopes

Identity Matrix

Contact Info

Product

Resources

About