<p> Given a point p and d + 1 sets (i.e., colours) of points in dimension d, the Colourful Feasibility Problem is to decide whether there are d + 1 points of different colours containing p in their convex hull; and if yes, find such a point set. The monochrome version of this problem, expressing p as a linear combination of d + 1 points in a set S, can be solved using traditional linear optimization algorithms. The Colourful Feasibility Problem was presented by
Bárány and Onn in 1997, and it is still not known if a polynomial-time algorithm exists. The case where we have d colours in dimension d and no restriction on the size of the sets has been shown to be strongly NP-complete through a reduction of 3-SAT. We define the core of a configuration to be the intersection of the convex hulls of each colour. We start from the important subcase that we call Colourful Core Feasibility Problem where we have d + 1 points of each colour, and p in the core. By Bárány's 1982 Colourful Caratheodory Theorem, a solution is guaranteed to exist, and the problem is to exhibit one. This problem is described by Bárány and Onn as "an outstanding problem on the border line between tractable and intractable problems". Besides applications to combinatorics, The Colourful Feasibility Problem models a situation where we want to select a set of points that is both diverse and representative.</p> <p> While we have not found out whether the Colourful Core Feasibility Problem can be solved in polynomial time, our contributions are on both the theoretical and practical performance of algorithms to solve the Colourful Feasibility Problem. The algorithms proposed by Bárány and Onn are essentially geometric, and the complexity guarantees depend crucially on having p inside the core. We consider modifications of these algorithms which update multiple colours at each stage, as well a greedy heuristic where we choose the adjacent simplex of maximum volume in each iteration and a random sampling approach. Our test suite includes unstructured random problems, ill-conditioned problems, problems with a restricted number of solutions and infeasible problems. We conclude that the most robust and nearly fastest algorithm for the Colourful Core Feasibility Problem is the multi-update variant which yields substantial gains over the original ones. Alternative approaches based on nondefinite quadratic optimization problem and positive semidefinite relaxation, and a combinatorial algorithm not depending on having p in the core are also introduced. Finally,
we give the first upper bound for the minimal number of colourful simplices containing a core point and the first improvement of the lower bound since Bárány's result in 1982.</p> / Thesis / Master of Science (MSc)
| Identifer | oai:union.ndltd.org:mcmaster.ca/oai:macsphere.mcmaster.ca:11375/21379 |
| Date | 07 1900 |
| Creators | Huang, Sui |
| Contributors | Deza, Antoine, Terlaky, Tamás, Computing and Software |
| Source Sets | McMaster University |
| Language | en_US |
| Detected Language | English |
| Type | Thesis |
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