Consensus-halving via theorems of Borsuk-Ulam and Tucker

Simmons, Forest W.; Su, Francis Edward

doi:10.1016/s0165-4896(02)00087-2

Cited by 60 publications

(78 citation statements)

References 16 publications

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“…Interest in small triangulations of cubes stems principally from certain simplicial fixed-point algorithms (e.g., see [14]) which run faster when there are fewer simplices. The same considerations govern more recent fair division procedures [11], [13] …”

Section: Triangulations Of Cubesmentioning

confidence: 76%

Lower Bounds for Simplicial Covers and Triangulations of Cubes

Bliss

2004

Discrete Comput Geom

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Abstract. We show that the size of a minimal simplicial cover of a polytope P is a lower bound for the size of a minimal triangulation of P, including ones with extra vertices. We then use this fact to study minimal triangulations of cubes, and we improve lower bounds for covers and triangulations in dimensions 4 through at least 12 (and possibly more dimensions as well). Important ingredients are an analysis of the number of exterior faces that a simplex in the cube can have of a specified dimension and volume, and a characterization of corner simplices in terms of their exterior faces.

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Section: Triangulations Of Cubesmentioning

confidence: 76%

Lower Bounds for Simplicial Covers and Triangulations of Cubes

Bliss

2004

Discrete Comput Geom

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“…This relation puts the complexity of finding a solution for regular instances in complexity classes different from the usual NP-hard or polynomial solvable problems. For PPW (2, k) there are constructions (see [13,20]) but not likely to be efficient, even though the corresponding decision problem is solvable. For PPW(c, k), c > 2 there is no constructive proof, even if of course a complete search of all possible cutting points leads always to one of the existing solutions.…”

Section: Resultsmentioning

confidence: 99%

Paintshop, odd cycles and necklace splitting

Meunier

Seb

2009

Discrete Applied Mathematics

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International audienceThe following problem has been presented in [T. Epping, W. Hochstättler, P. Oertel, Complexity results on a paint shop problem, Discrete Applied Mathematics 136 (2004) 217-226] by Epping, Hochstättler and Oertel: cars have to be painted in two colors in a sequence where each car occurs twice; assign the two colors to the two occurrences of each car so as to minimize the number of color changes. More generally, the "paint shop scheduling problem" is defined with an arbitrary multiset of colors given for each car, where this multiset has the same size as the number of occurrences of the car; the mentioned article states two conjectures about the general problem and proves its NP-hardness. In a subsequent paper in [P. Bonsma, Th. Epping, W. Hochstättler, Complexity results for restricted instances of a paint shop problem for words, Discrete Applied Mathematics 154 (2006) 1335-1343], Bonsma, Epping and Hochstättler proved its APX-hardness and noticed the applicability of some classical results in special cases. We first identify the problem concerning two colors as a minimum odd circuit cover problem in particular graphs, exactly situating the problem. A resulting two-way reduction to a special minimum uncut problem leads to polynomial algorithms for subproblems, to observing APX-hardness through MAX CUT in 3-regular graphs, and to a solution with at most 3/4th of all possible remaining color changes (when all obliged color changes have been made). For the general problem concerning an arbitrary number of colors, we realize that the two aforementioned conjectures are corollaries of the celebrated "necklace splitting" theorem of Alon, Goldberg and West. © 2008 Elsevier B.V. All rights reserved

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“…Our approach may provide new techniques for developing constructive proofs of Kneser's conjecture (e.g., see [8]), certain generalized Tucker lemmas (e.g., the Z p -Tucker lemma of Ziegler [13] or the generalized Tucker's lemma conjectured by Simmons-Su [9]), as well as provide new interpretations of algorithms that depend on Tucker's lemma (see [9] for applications to cake-cutting, Alon's necklace-splitting problem, team-splitting, and other fair division problems).…”

Section: Introductionmentioning

confidence: 99%

“…By constructive proof, we mean one that (i) shows the existence of the solution and (ii) locates it by a method other than an exhaustive search. This is the sense in which Freund-Todd [6] use the word constructive (this is called an effective procedure in [9]). The distinction from an exhaustive search is important for two reasons.…”

Section: Introductionmentioning

confidence: 99%

A constructive proof of Ky Fan's generalization of Tucker's lemma

Prescott

2005

Journal of Combinatorial Theory, Series A

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We present a proof of Ky Fan's combinatorial lemma on labellings of triangulated spheres that differs from earlier proofs in that it is constructive. We slightly generalize the hypotheses of Fan's lemma to allow for triangulations of S n that contain a flag of hemispheres. As a consequence, we can obtain a constructive proof of Tucker's lemma that holds for a more general class of triangulations than the usual version.

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Consensus-halving via theorems of Borsuk-Ulam and Tucker

Cited by 60 publications

References 16 publications

Lower Bounds for Simplicial Covers and Triangulations of Cubes

Lower Bounds for Simplicial Covers and Triangulations of Cubes

Paintshop, odd cycles and necklace splitting

A constructive proof of Ky Fan's generalization of Tucker's lemma

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