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New Facets of the Linear Ordering Polytope

G. Bolotashvili, M. Kovalev, E. Girlich
1999 SIAM Journal on Discrete Mathematics  
We introduce ten collections of inequalities representing facets of the linear ordering polytope.  ...  The rotation method for the linear ordering polytope generalizes facets induced by subgraphs called m-fences, M obius ladders and Z m -facets introduced by Reinelt 12], (m; k)fences introduced by Bolotashvily  ...  The trivial rotation mapping does not generate new facets for the most known facets of the polytope P n , because in many cases the mapping generates another member of the same facet family, i.e. the corresponding  ... 
doi:10.1137/s0895480196300145 fatcat:6lvjdfd2qnecxh6omvfcvt3i2a

Blending simple polytopes at faces

Fred B Holt
2004 Discrete Mathematics  
In pursuing the Hirsch conjectures, vertex-blends produced polytopes which have large diameter and few diametral paths. Here we deÿne and explore blends at higher-dimensional faces.  ...  Blending two simple polytopes together at vertices, at edges, or at other supplementary faces, produces another simple polytope.  ...  The truncating hyperplane provides a needed orientation for the blend, and it helps avoid admissible-transformation problems.  ... 
doi:10.1016/j.disc.2004.02.006 fatcat:56tiohr33jem7aryj3gygz65um

Polyhedral results for the edge capacity polytope

Stan P.M. van Hoesel, Arie M.C.A. Koster, Robert L.M.J. van de Leensel, Martin W.P. Savelsbergh
2002 Mathematical programming  
In this paper, we investigate the polytopes of the problem restricted to one arc/edge of the network (the undirected/directed edge capacity problem) for the non-bifurcated routing case.  ...  We give conditions under which the inequalities of the edge capacity polytopes define facets of the network loading polytope.  ...  We show that every non-trivial facet defining inequality for the directed edge capacity polytope defines a facet of the overall network loading polytope as well.  ... 
doi:10.1007/s101070200292 fatcat:no7drbxmavgcjciq2hcgsj6rb4

Generalized multiple depot traveling salesmen problem - polyhedral study and exact algorithm [article]

Kaarthik Sundar, Sivakumar Rathinam
2015 arXiv   pre-print
The GMDTSP is an NP-hard problem as it generalizes the MDTSP and has practical applications in design of ring networks, vehicle routing, flexible manufacturing scheduling and postal routing.  ...  The generalized multiple depot traveling salesmen problem (GMDTSP) is a variant of the multiple depot traveling salesmen problem (MDTSP), where each salesman starts at a distinct depot, the targets are  ...  We also introduce a general theorem that allows one to lift any facet of the MDTSP polytope into a facet of the GMDTSP polytope.  ... 
arXiv:1508.01813v1 fatcat:4gefvohirbaubovw3xrzsszqm4

A polyhedral approach to multicommodity survivable network design

Mechthild Stoer, Geir Dahl
1994 Numerische Mathematik  
We discuss basic properties (dimension and trivial facets) of these polytopes in Section 3. The remaining part of the paper discusses stronger formulations of the problem.  ...  We study properties of polytopes that are naturally associated with the model.  ...  The polytope ICOV(g, b) can be linearly transformed into a knapsack polytope with generalized upper bounds, so all results pertaining to that polytope apply also to the ICOV-polytope.  ... 
doi:10.1007/s002110050054 fatcat:k4ozr6bzengyfh2usidsabm6ty

Page 5191 of Mathematical Reviews Vol. , Issue 911 [page]

1991 Mathematical Reviews  
Our result generalizes a result of Deza stating that if G is a complete graph with more than two vertices, then an inequality defines a facet of P-(G) if and only if it defines a facet of the cut polytope  ...  We give a sufficient condition for an inequality defining a facet of Pc(G) to define a facet of the cut polytope of a graph containing G as an induced subgraph.  ... 

Geometric Combinatorics of Transportation Polytopes and the Behavior of the Simplex Method [article]

Edward D. Kim
2010 arXiv   pre-print
Transportation problems are, in many ways, the simplest kind of linear programs and thus have a rich combinatorial structure.  ...  with n facets is bounded above by n-d.  ...  In [82] , Currin studies the transportation problem with inadmissible routes.  ... 
arXiv:1006.2416v1 fatcat:xahucjplsraa3gbzbwkrr2u5ty

The m-Cost ATSP [chapter]

Christoph Helmberg
1999 Lecture Notes in Computer Science  
We show that, for m 3 machines, all facets of the one machine subproblem also de ne facets of the m-ATSP polytope.  ...  We present rst results of a polyhedral analysis of the m-ATSP in full generality.  ...  Although there is considerable literature on the TSP/ATSP 10, 13{15, 3, 11, 7] as well as on the m-ATSP for vehicle routing (see 5] and references therein) it seems that the m-ATSP problem in full generality  ... 
doi:10.1007/3-540-48777-8_19 fatcat:oseglnrjyjhcnb4rrmn2n7bm2i

The k edge-disjoint 3-hop-constrained paths polytope

F. Bendali, I. Diarrassouba, A.R. Mahjoub, J. Mailfert
2010 Discrete Optimization  
In this paper we consider this problem from a polyhedral point of view. We give an integer programming formulation for the problem and discuss the associated polytope.  ...  This generalizes the results of Huygens et al. (2004) [1] for k = 2 and L = 2, 3 and those of Dahl et al. (2006) [2] for L = 2 and k ≥ 2. This also proves a conjecture in [1] .  ...  Acknowledgements The authors would like to thank the anonymous referees for their comments that permitted to considerably improve the presentation of the paper.  ... 
doi:10.1016/j.disopt.2010.05.001 fatcat:44dbxsqhsbcghimhauypucbip4

Multiple Depot Ring Star Problem: A polyhedral study and exact algorithm [article]

Kaarthik Sundar, Sivakumar Rathinam
2014 arXiv   pre-print
) assign each non-visited customer to a visited customer or a depot, and (iii) minimize the sum of the routing costs, i.e., the cost of the cycles and the assignment costs.  ...  The Multiple Depot Ring-Star Problem (MDRSP) is an important combinatorial optimization problem that arises in the context of optical fiber network design, and in applications pertaining to collecting  ...  problem (MDTSP) polytope [6] .  ... 
arXiv:1407.5080v2 fatcat:asvjbja3ifanpixivr4dfhcacq

Some integer programs arising in the design of main frame computers

Carlos Edwards Ferreira, Martin Gr�tschel, Alexander Martin, Robert Weismantel, Stefan Kiefl, Ludwig Krispenz
1993 Mathematical Methods of Operations Research  
We introduce several relaxations of the general model, which are also N P-hard, but seem to be more easily accessible.  ...  The mathematical relations between the relaxations and the exact formulation of the problem are discussed as well.  ...  Relaxations for the General Model The Multiple Knapsack Problem In this subsection we present a first relaxation of the general model.  ... 
doi:10.1007/bf01416008 fatcat:iqpxezgkrfhsnbg2cfm7rxluva

Multiple depot ring star problem: a polyhedral study and an exact algorithm

Kaarthik Sundar, Sivakumar Rathinam
2016 Journal of Global Optimization  
) assign each non-visited customer to a visited customer or a depot, and (iii) minimize the sum of the routing costs, i.e., the cost of the cycles and the assignment costs.  ...  The Multiple Depot Ring-Star Problem (MDRSP) is an important combinatorial optimization problem that arises in the context of optical fiber network design, and in applications pertaining to collecting  ...  Lemma 2 is that any valid inequality αx + βy ≤ γ that is facet-inducing to the MDTSP polytope Q and satisfying the conditions (1)-(4) of the lemma is valid and facet-inducing to the MDRSP polytope P .  ... 
doi:10.1007/s10898-016-0431-7 fatcat:ryfvtjq5cfhr7fblx6kqfyxoje

Traces of the XII Aussois Workshop on Combinatorial Optimization

Michael Jünger, Thomas M. Liebling, Denis Naddef, William R. Pulleyblank, Gerhard Reinelt, Giovanni Rinaldi, Laurence A. Wolsey
2010 Mathematical programming  
"On the Dominant of the s-t-cut Polytope" by Martin Skutella, and Alexia Weber gives a complete description of the dominant of the s-t cut polytope, by characterizing its vertices, facets, and adjacency  ...  Among other consequences, it permits us to determine whether an integer programming problem can be transformed to a b-matching problem on a bidirected graph, in which case it can be solved using Edmonds  ... 
doi:10.1007/s10107-010-0369-3 fatcat:b7slvsqv4ncy7j45l3u4dcilz4

Fourier transforms of polytopes, solid angle sums, and discrete volume [article]

Ricardo Diaz, Quang-Nhat Le, Sinai Robins
2018 arXiv   pre-print
Given a real closed polytope P, we first describe the Fourier transform of its indicator function by using iterations of Stokes' theorem.  ...  The combinatorics of the face poset of P plays a central role in the description of the Fourier transform of P.  ...  Introduction It is a classical problem in the geometry of numbers [30] to find ways of counting the number of lattice points in a general convex body.  ... 
arXiv:1602.08593v2 fatcat:neps7xerjrfhzmrr5bzauxoaem

A study of the quadratic semi-assignment polytope

Hiroo Saito, Tetsuya Fujie, Tomomi Matsui, Shiro Matuura
2009 Discrete Optimization  
We introduce an isomorphic projection and transform the polytope to a tractable full-dimensional polytope.  ...  We study a polytope which arises from a mixed integer programming formulation of the quadratic semi-assignment problem.  ...  Acknowledgments The authors thank anonymous referees for constructive suggestions for the revision of the paper.  ... 
doi:10.1016/j.disopt.2008.08.003 fatcat:tdkddlfcbvgjjafecyt6tygtem
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