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The Intersection of Knapsack Polyhedra and Extensions [chapter]

Alexander Martin, Robert Weismantel
1998 Lecture Notes in Computer Science  
For the intersection of several knapsack polyhedra, incomparabilities between the column vectors of the associated matrix will be shown to transfer into inequalities of the associated polyhedron.  ...  The scheme is related to Chvatal-Gomory cutting planes and important special cases such as odd hole and clique inequalities for the stable set polyhedron or families of inequalities for the knapsack polyhedron  ...  We try to go one step further and investigate the intersection of two or more knapsack polyhedra.  ... 
doi:10.1007/3-540-69346-7_19 fatcat:hskhjhcxzvb65akofiqjglmbri

Page 3481 of Mathematical Reviews Vol. , Issue 91F [page]

1991 Mathematical Reviews  
The author discusses the connections with max flow min cut properties and provides a subadditive characterization of the facets of the underlying polyhedra.  ...  The problem of maximizing a concave integer function over a set of integer points of a polymatroid intersection is transformed to a problem of maximizing a linear function over an intersection of the same  ... 

Page 5955 of Mathematical Reviews Vol. , Issue 2000h [page]

2000 Mathematical Reviews  
Xue Liang Li (PRC-NWP-AM; Xian) 2000h:90066 90C27 90C10 90C57 Martin, Alexander (D-KOZU; Berlin); Weismantel, Robert (D-KOZU,; Berlin) The intersection of knapsack polyhedra and extensions.  ...  The authors address the question of when the facets of a single knapsack polyhedron define strong cutting planes of an integer program when several knapsack constraints intersect.  ... 

Page 10498 of Mathematical Reviews Vol. , Issue 2004m [page]

2004 Mathematical Reviews  
It is shown that the standard and primal separation problems are equivalent (in the Karp sense) for set packing, set covering, set partitioning and knapsack polyhedra, for general integer pro- grams, and  ...  Summary: “By solving a constrained matroid intersection prob- lem, we give a matroid generalization of the stable theorem of Gale and Shapley.  ... 

A Principled Approach to Mixed Integer/Linear Problem Formulation [chapter]

J. N. Hooker
2009 Operations Research and Cyber-Infrastructure  
We view mixed integer/linear problem formulation as a process of identifying disjunctive and knapsack constraints in a problem and converting them to mixed integer form.  ...  We show through a series of examples that following this process can yield mixed integer models that automatically incorporate some of the modeling devices that have been discovered over the years for  ...  The proof is a straightforward extension of Jeroslow's proof [4] .  ... 
doi:10.1007/978-0-387-88843-9_5 fatcat:jgwqjdtgurgwbpcsjkrf7r3icu

Page 2061 of Mathematical Reviews Vol. , Issue 84e [page]

1984 Mathematical Reviews  
It is shown that integral polymatroids have SID, polyhedra defined by totally unimodular matrices have MID, and branching polyhedra as well as intersections of two strongly-base-orderable matroids have  ...  Extensive computational tests have confirmed the efficiency of this approach for a wide variety of problems with as many as 5000 variables.” Baum, S.; Trotter, L.  ... 

On the existence of compact ϵ-approximated formulations for knapsack in the original space [article]

Yuri Faenza, Laura Sanità
2015 arXiv   pre-print
We show that there exists a family of Knapsack polytopes such that, for each polytope P from this family and each ϵ > 0, any ϵ-approximated formulation of P in the original space R^n requires a number  ...  This answers a question by Bienstock and McClosky (2012).  ...  Knapsack has been extensively studied in the combinatorial optimization and integer programming communities (see e.g. [9, 12] ).  ... 
arXiv:1503.04717v1 fatcat:v4c57ytzofaknohgxmf4hmf2rq

Randomized complexity of linear arrangements and polyhedra? [chapter]

Marek Karpinski
1999 Lecture Notes in Computer Science  
We survey some of the recent results on the complexity of recognizing n{dimensional linear arrangements and convex polyhedra by randomized algebraic decision trees.  ...  In particular, we derive rst nontrivial, in fact quadratic, randomized lower bounds on the problems like Knapsack and Bounded Integer Programming.  ...  We shall deal here with the randomized complexity of linear arrangements, and convex polyhedra.  ... 
doi:10.1007/3-540-48321-7_1 fatcat:yatxihz3m5dl3c3254i44m2tfy

Page 4595 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
A consequence is the existence of a strongly polynomial nonuniform algorithm for the NP-complete knapsack problem.  ...  two re- sults: (i) the minimum number of simplices for a vertex-preserving triangulation of J; is 67, and (ii) the minimum number of sim- 52B_ Polytopes and polyhedra 94n:52029 plices for a vertex-preserving  ... 

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  
Here is a brief glimpse of the content of these papers. "Separation Algorithms for 0-1 Knapsack Polytopes" by Konstantinos Kaparis, and Adam N.  ...  Therefore, attention is focused on the former; extensive computational experiments are conducted to evaluate the efficiency and fairness achieved with it.  ... 
doi:10.1007/s10107-010-0369-3 fatcat:b7slvsqv4ncy7j45l3u4dcilz4

Linear and combinatorial sharing problems

Uwe Zimmermann
1986 Discrete Applied Mathematics  
., the minimization of separable objectives f(x)= max{~(xj) ]j-1 ..... n} on polyhedra from linear and combinatorial optimization.  ...  Finiteness and efficiency is derived for several applications, in particular on submodular flow and matching polyhedra.  ...  Submodular flows generalize network flows as well as polymatroid intersections.  ... 
doi:10.1016/0166-218x(86)90022-3 fatcat:elyksmaouje5hbmzhmkk6hzv6i

ZERO: Playing Mathematical Programming Games [article]

Gabriele Dragotto, Sriram Sankaranarayanan, Margarida Carvalho, Andrea Lodi
2021 arXiv   pre-print
We provide an overview of the software's key components and showcase a Knapsack Game, i.e., a game where each player solves a binary knapsack problem.  ...  We present ZERO, a modular and extensible C++ library interfacing Mathematical Programming and Game Theory.  ...  Acknowledgements This work has been supported by the Canada Excellence Research Chair in Real-time Decision-making (DS4DM).  ... 
arXiv:2111.07932v2 fatcat:hpm3p6vx5rgvbjhdf73kh53bz4

Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications

Chaitanya Swamy
2016 ACM Transactions on Algorithms  
open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs.  ...  Our techniques also yield an improvement for the knapsack median problem.  ...  I thank Chandra Chekuri for some useful discussions regarding the matroid-intersection median problem.  ... 
doi:10.1145/2963170 fatcat:guvwv2lubrfphemtq2l7l7vduu

Page 5224 of Mathematical Reviews Vol. , Issue 2000g [page]

2000 Mathematical Reviews  
Lenstra, Solving a linear Diophantine equation with lower and upper bounds on the variables (229-242); Alexander Martin and Robert Weismantel, The intersection of knapsack polyhedra and extensions (243  ...  They prove the continuous differentiability of L(s, S) and the existence of an interior minimiser and even though the minimisers might not satisfy the conditions assumed, the proof of the optimality of  ... 

The subdivision of large simplicial cones in Normaliz [article]

Winfried Bruns, Richard Sieg, Christof Söger
2016 arXiv   pre-print
In the homogeneous case, in which the polyhedron is a cone, the set of generators is the Hilbert basis of the intersection of the cone and the lattice, an affine monoid.  ...  Normaliz is an open-source software for the computation of lattice points in rational polyhedra, or, in a different language, the solutions of linear diophantine systems.  ...  Introduction Normaliz [3] is a software for the computation of lattice points in rational polyhedra.  ... 
arXiv:1605.07440v1 fatcat:o72ouxkhiffhrno3ojevbeybsu
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