Removing Degeneracies in LP-type Problems May Need to Increase Dimension

LP-type problems are an abstract model of linear programming. For some applications it is useful to have LP-type problems satisfying a special property: nondegeneracy. For any LP-type problem there is a nondegenerate refinement, however the complexity measured by the combinatorial dimension may need to increase. We construct an example of an LP-type problem of dimension 4, for which any nondegenerate refinement has dimension at least 6. This is the first known example where the dimension increases by more than 1. We show how this relates to the problem of covering of a certain poset by Boolean algebras. Introduction In optimization, one of the most common problems is linear programming. For its solving, the simplex algorithm is mostly used. Practically it performs very well, but it is known that for the common variants, some artificially crafted inputs can be made such that the running time is exponential. However, some randomized versions have been developed, which can be proved to be subexponential (see [Sharir and Welzl, 1992; Matoušek et al., 1996; or Kalai, 1992]). The analysis takes an abstract look at the linear programming. It introduces a new wider class of problems, the LP-type problems, and the simplex algorithm is presented in this more general setting. Later, more algorithms connected to LP-type problems have been proposed; e.g., finding the optimum solution satisfying all but k constraints, see [Matoušek, 1995]. The analysis of this last algorithm depends on a special property of the input problem, called nondegeneracy. If the problem is degenerate, we need to get rid of the degeneracy. More specifically, given an LP-type problem L, a nondegenerate LP-type problem L is said to refine L under certain circumstances, and then the optimum solution for the problem L (or whatever the result of the algorithm is) can be easily converted to the correct result for the problem L. However, the running time of the algorithm depends on another parameter of the input, called the combinatorial dimension of the problem (denoted by dimL for an input L). Unluckily, for given input L, the dimension of the refined variant L sometimes has to increase. In particular, an LP-type problem L had been presented in [Matoušek, 1995], such that dimL = 2 and dimL ≥ 3 for any nondegenerate refinement L of L. However, it has not been known, whether there are problems for which the dimension has to increase more than by one; i.e. dimL ≥ dimL + i for i ≥ 2. In this paper we partially answer this question affirmatively. We present a construction of a problem L for which dimL = 4 and dimL ≥ 6 for any nondegenerate refinement L of L. The combinatorial proof is based on the case analysis. We conjecture that the construction can be extended to get any dimL−dimL, but the proof presented here can not be generalized. As the problem is stated in an unfamiliar terminology of LP-type problems, we reformulate it in terms of decomposition of posets. We refer to Škovroň [2002] for more detailed introduction of LP-type problems and related concepts including the motivation. The next section contains the formal definitions of LP-type problems and other notions informally sketched in this introduction. Then the sections with actual construction of the problem L follow. WDS'06 Proceedings of Contributed Papers, Part I, 196–207, 2006. ISBN 80-86732-84-3 © MATFYZPRESS