论文信息 - Colourful Linear Programming and its Relatives

Colourful Linear Programming and its Relatives

We consider the following Colourful generalization of Linear Programming: given sets of points S1,..., Sk ⊂ Rd, referred to as colours, and a point b ∈ Rd, decide whether there is a colourfulT = {s1,..., sk} such that b ∈ convT, and if there is one, find it. Linear Programming is obtained by taking k = d + 1 and S1 =... = Sd+1. If k = d + 1 and b ∈ ∩i=1d+1 convSi then a solution always exists: we describe an efficient iterative approximation algorithm for this problem, that finds a colourful T whose convex hull contains a point e-close to b, and analyze its real arithmetic and Turing time complexities. In contrast, we show that Colourful Linear Programming is strongly NP-complete. We consider a class of linear algebraic relatives of Colourful Linear Programming, and give a computational complexity classification of the related decision and counting problems that arise. We also introduce and discuss the complexity of a hierarchy of w1, w2-Matroid-Basis-Nonbasis problems, and give an application of Colourful Linear Programming to the algorithmic problems of Tverberg's theorem in combinatorial geometry.

Imre Bárány | Shmuel Onn | I. Bárány | S. Onn

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