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Convex hulls of superincreasing knapsacks and lexicographic orderings

Akshay Gupte
2016 Discrete Applied Mathematics  
We also establish a distributive property by proving that the convex hull of <- and >-type superincreasing knapsacks can be obtained by intersecting the convex hulls of <- and >-sets taken individually  ...  The elements of this superincreasing knapsack are exactly those vectors that are lexicographically smaller than the greedy solution to optimizing over this knapsack.  ...  In Section 4, we prove that the convex hull of intersection of two superincreasing knapsacks is given by the facets of the individual knapsack polytopes.  ... 
doi:10.1016/j.dam.2015.08.010 fatcat:lzpvcljrc5bjfgy7pn7fjzbwjm

On lexicographic approximations of integer programs [article]

Michael Eldredge, Akshay Gupte
2017 arXiv   pre-print
We use the lexicographic order to define a hierarchy of primal and dual bounds on the optimum of a bounded integer program.  ...  The latter result implies a stronger polyhedral representation for the integer feasible points and a new approach for deriving strong valid inequalities to the integer hull.  ...  Thus the superincreasing property is sufficient for a knapsack to be a lex-ordered set.  ... 
arXiv:1610.06470v3 fatcat:jwqazyg3vnh3bhbfwjaj7tzngu

Scanning integer points with lex-inequalities: a finite cutting plane algorithm for integer programming with linear objective

Michele Conforti, Marianna De Santis, Marco Di Summa, Francesco Rinaldi
2020 4OR  
To each integer point x in K we associate a family of inequalities (lex-inequalities) that define the convex hull of the integer points in K that are not lexicographically smaller than x.  ...  AbstractWe consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis.  ...  We thank both of them. We are also grateful to Akshay Gupte for his constructive comments and his pointers to the existing literature.  ... 
doi:10.1007/s10288-020-00459-6 fatcat:s57tueauundszdnm7qfarj74su

Scanning integer points with lex-inequalities: A finite cutting plane algorithm for integer programming with linear objective [article]

Michele Conforti, Marianna De Santis, Marco Di Summa, Francesco Rinaldi
2020 arXiv   pre-print
To each integer point x in K we associate a family of inequalities (lex-cuts) that defines the convex hull of the integer points in K that are not lexicographically smaller than x.  ...  We consider the integer points in a unimodular cone K ordered by a lexicographic rule defined by a lattice basis.  ...  We thank both of them. We are also grateful to Akshay Gupte for his constructive comments and his pointers to the existing literature.  ... 
arXiv:1811.02345v2 fatcat:hmfu2hilizcnvce4cip3ijv4zi

On Dantzig figures from graded lexicographic orders

Akshay Gupte, Svetlana Poznanović
2018 Discrete Mathematics  
We construct two families of Dantzig figures, which are (d,2d)-polytopes with an antipodal vertex pair, from convex hulls of initial subsets for the graded lexicographic (grlex) and graded reverse lexicographic  ...  (grevlex) orders on Z^d_≥ 0.  ...  Q) yields the convex hull of all the integral vectors that belong to a standard integral simplex and are lexicographically smaller (resp. greater) than a fixed integer vector.  ... 
doi:10.1016/j.disc.2018.02.016 fatcat:tix5g3kf5jbtfpvgjeyh5mtlcu

Exact Lexicographic Scheduling and Approximate Rescheduling [article]

Dimitrios Letsios, Miten Mistry, Ruth Misener
2020 arXiv   pre-print
Further, we revisit state-of-the-art exact lexicographic optimization methods and propose a lexicographic branch-and-bound algorithm whose performance is validated computationally.  ...  Our approach is substantiated analytically, with a price of robustness characterization parameterized by the degree of uncertainty, and numerically.  ...  Convex hulls of superincreasing knapsacks and lexicographic orderings. Discrete Applied Mathematics, 201 , 150-163. [29] Hanasusanto, G. A., Kuhn, D., & Wiesemann, W. (2015).  ... 
arXiv:1805.03437v3 fatcat:x6f632viijfhrpsireri7ugaim

Limits to parallel computation: P-completeness theory

1996 ChoiceReviews  
This book is an introduction to the rapidly growing theory of Pcompleteness -the branch of complexity theory that focuses on identifying the "hardest" problems in the class P of problems solvable in polynomial  ...  That is, algorithm designers have failed to find NC algorithms, feasible highly parallel solutions that take time polynomial in the logarithm of the problem size while using only a polynomial number of  ...  Acknowledgments The numbers in parentheses at the end of each entry in the bibliography are the page numbers on which that item is referenced.  ... 
doi:10.5860/choice.33-3959 fatcat:qjoueeu225gr7jvwdf6gkoxfbq

Subject Index to Volumes 1–75

2001 Information Processing Letters  
, 1468, 1536, 1997, 2762, 3092, 3219 ordering, 974, 1162 lexicographic path ordering, 2437 constraints, 2666 lexicographic relation, 506 semicommutations, 1468 lexicographical, 684, 970 metrics  ...  , 1528 topological order problem, 1873 lexicographically greatest string, 3844 least circular substrings, 576 lexicographically minimal maximal path, 1332 strings, 3988 lexicographically ordered  ...  subregion, 46 vertex, 1777 unimodality, 1676 , 1777 of convex polygons, 1676 Subject Index / Information Processing Letters 78 (2001 ) 5-336 unimonotone polygon, 2426 uninterpreted, 2879 parallel system  ... 
doi:10.1016/s0020-0190(01)00175-2 fatcat:5y67tfm6yfbblakrus5nnhs73e