Let C 1 , ..., C d+1 be d + 1 point sets in R d , each containing the origin in its convex hull. A subset C of d+1 i=1 C i is called a colorful choice (or rainbow) for C 1 , . . . , C d+1 , if it contains exactly one point from each set C i . The colorful Carathéodory theorem states that there always exists a colorful choice for C 1 , . . . , C d+1 that has the origin in its convex hull. This theorem is very general and can be used to prove several other existence theorems in high-dimensional discrete geometry, such as the centerpoint theorem or Tverberg's theorem. The colorful Carathéodory problem (ColorfulCarathéodory) is the computational problem of finding such a colorful choice. Despite several efforts in the past, the computational complexity of ColorfulCarathéodory in arbitrary dimension is still open.We show that ColorfulCarathéodory lies in the intersection of the complexity classes PPAD and PLS. This makes it one of the few geometric problems in PPAD and PLS that are not known to be solvable in polynomial time. Moreover, it implies that the problem of computing centerpoints, computing Tverberg partitions, and computing points with large simplicial depth is contained in PPAD ∩ PLS. This is the first nontrivial upper bound on the complexity of these problems.Finally, we show that our PPAD formulation leads to a polynomial-time algorithm for a special case of ColorfulCarathéodory in which we have only two color classes C 1 and C 2 in d dimensions, each with the origin in its convex hull, and we would like to find a set with half the points from each color class that contains the origin in its convex hull.
We show that finding minimally intersecting n paths from s to t in a directed graph or n perfect matchings in a bipartite graph can be done in polynomial time. This holds more generally for unimodular set systems.
Optimization over l × m × n integer 3-way tables with given line-sums is NP-hard already for fixed l = 3, but is polynomial time solvable with both l, m fixed. In the huge version of the problem, the variable dimension n is encoded in binary, with t layer types. It was recently shown that the huge problem can be solved in polynomial time for fixed t, and the complexity of the problem for variable t was raised as an open problem. Here we solve this problem and show that the huge table problem can be solved in polynomial time even when the number t of types is variable. The complexity of the problem over 4-way tables with variable t remains open. Our treatment goes through the more general class of huge n-fold integer programming problems. We show that huge integer programs over n-fold products of totally unimodular matrices can be solved in polynomial time even when the number t of brick types is variable.
Given d + 1 sets of points, or colours, S 1 , . . . ,The colourful Carathéodory theorem states that, if 0 is in the convex hull of each S i , then there exists a colourful simplex T containing 0 in its convex hull. Deza, Huang, Stephen, and Terlaky (Colourful simplicial depth, Discrete Comput. Geom., 35, 597-604 (2006)) conjectured that, when |S i | = d + 1 for all i ∈ {1, . . . , d + 1}, there are always at least d 2 + 1 colourful simplices containing 0 in their convex hulls. We prove this conjecture via a combinatorial approach.
ABSTRACT. The colourful simplicial depth conjecture states that any point in the convex hull of each of d +1 sets, or colours, of d +1 points in general position in R d is contained in at least d 2 +1 simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral systems. We present properties of octahedral systems generalizing earlier results on colourful point configurations and exhibit an octahedral system which cannot arise from a colourful point configuration. The number of octahedral systems is also given.1. INTRODUCTION 1.1. Preliminaries. An n-uniform hypergraph is said to be n-partite if its vertex set is the disjoint union of n sets V 1 , . . . ,V n and each edge intersects each V i at exactly one vertex. Such an hypergraph is an (n + 1)-tuple (V 1 , . . . ,V n , E ) where E is the set of edges. An octahedral system Ω is an n-partite hypergraph (V 1 , . . . ,V n , E ) with |V i | ≥ 2 for i = 1, . . . , n and satisfying the following parity condition: the number of edges ofi =1 S i of points such that |X ∩ S i | = 2 for i = 1, . . . , d + 1, there is an even number of colourful simplices generated by X and containing the origin 0. Therefore, the hypergraph Ω = (V 1 , . . . ,V d +1 , E ), with V i = S i for i = 1, . . . , d + 1 and where the edges in E correspond to the colourful simplices containing 0 forms an octahedral system. This property motivated Bárány to suggest octahedral systems as a combinatorial generalization of colourful point configurations, see [8].Let µ(d ) denote the minimum number of colourful simplices containing 0 over all colourful point configurations satisfying 0 ∈ d +1 i =1 conv(S i ) and Stephen, and Xie [8] . The lower bound was slightly improved in dimension 4 to µ(4) ≥ 14 via a computational approach presented in [9].An octahedral system arising from a colourful point configuration S 1 , . . . ,and |S i | = d + 1 for all i , is without isolated vertex; that is, each vertex belongs to at least one edge. Indeed, according to a strengthening of the colourful Carathéodory theorem [2], any point of such a colourful configuration is the vertex of at least one colourful simplex containing 0. Theorem 1.1, whose proof is given in Section 4, provides a lower bound for the number of edges of an octahedral system without isolated vertex.
The colorful Carathéodory theorem, proved by Bárány in 1982, states that given d + 1 sets of points S 1 , . . . , S d+1 in R d , with each S i containing 0 in its convex hull, there exists a set T ⊆ d+1 i=1 S i containing 0 in its convex hull and such that |T ∩ S i | ≤ 1 for all i ∈ {1, . . . , d + 1}. An intriguing question -still open -is whether such a set T , whose existence is ensured, can be found in polynomial time. In 1997, Bárány and Onn defined colorful linear programming as algorithmic questions related to the colorful Carathéodory theorem. The question we just mentioned comes under colorful linear programming.The traditional applications of colorful linear programming lie in discrete geometry. In this paper, we study its relations with other areas, such as game theory, operations research, and combinatorics. Regarding game theory, we prove that computing a Nash equilibrium in a bimatrix game is a colorful linear programming problem. We also formulate an optimization problem for colorful linear programming and show that as for usual linear programming, deciding and optimizing are computationally equivalent. We discuss then a colorful version of Dantzig's diet problem. We also propose a variant of the Bárány algorithm, which is an algorithm computing a set T whose existence is ensured by the colorful Carathéodory theorem. Our algorithm makes a clear connection with the simplex algorithm and we discuss its computational efficiency. Related complexity and combinatorial results are also provided.
ABSTRACT. The colourful simplicial depth conjecture states that any point in the convex hull of each of d +1 sets, or colours, of d +1 points in general position in R d is contained in at least d 2 +1 simplices with one vertex from each set. We verify the conjecture in dimension 4 and strengthen the known lower bounds in higher dimensions. These results are obtained using a combinatorial generalization of colourful point configurations called octahedral systems. We present properties of octahedral systems generalizing earlier results on colourful point configurations and exhibit an octahedral system which cannot arise from a colourful point configuration. The number of octahedral systems is also given.1. INTRODUCTION 1.1. Preliminaries. An n-uniform hypergraph is said to be n-partite if its vertex set is the disjoint union of n sets V 1 , . . . ,V n and each edge intersects each V i at exactly one vertex. Such an hypergraph is an (n + 1)-tuple (V 1 , . . . ,V n , E ) where E is the set of edges. An octahedral system Ω is an n-partite hypergraph (V 1 , . . . ,V n , E ) with |V i | ≥ 2 for i = 1, . . . , n and satisfying the following parity condition: the number of edges of Ω induced by X ⊆is a collection of d + 1 sets, or colours, S 1 , . . . , S d +1 . A colourful simplex is defined as the convex hull of a subset S ofS i of points such that |X ∩ S i | = 2 for i = 1, . . . , d + 1, there is an even number of colourful simplices generated by X and containing the origin 0. Therefore, the hypergraph Ω = (V 1 , . . . ,V d +1 , E ), with V i = S i for i = 1, . . . , d + 1 and where the edges in E correspond to the colourful simplices containing 0 forms an octahedral system. This property motivated Bárány to suggest octahedral systems as a combinatorial generalization of colourful point configurations, see [8].Let µ(d ) denote the minimum number of colourful simplices containing 0 over all colourful point configurations satisfying 0 ∈ Subsequently, Bárány and Matoušek [3] verified the conjecture for d = 3 and provided a lower bound of . The lower bound was slightly improved in dimension 4 to µ(4) ≥ 14 via a computational approach presented in [9].An octahedral system arising from a colourful point configuration S 1 , . . . ,conv(S i ) and |S i | = d + 1 for all i , is without isolated vertex; that is, each vertex belongs to at least one edge. Indeed, according to a strengthening of the colourful Carathéodory theorem [2], any point of such a colourful configuration is the vertex of at least one colourful simplex containing 0. Theorem 1.1, whose proof is given in Section 4, provides a lower bound for the number of edges of an octahedral system without isolated vertex. Bárány's sufficient condition for the existence of a colourful simplex containing 0 has been recently generalized in [1,11,14]. The related algorithmic question of finding a colourful simplex containing 0 is presented and studied in [4,7]. For a recent breakthrough for a monocolour version [10, 13]. Definitions. Let E [X ] denote the set of edges induce...
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