Markov bases and subbases for bounded contingency tables

TL;DR

This paper investigates the computation of Markov bases for bounded contingency tables, showing that basic moves of all 2x2 minors connect tables with positive bounds under the independence model, and discusses conditions involving structural zeros.

Contribution

It demonstrates that for bounded two-way tables with positive bounds, basic 2x2 moves suffice to connect tables under the independence model, simplifying computations.

Findings

Basic 2x2 moves connect all tables with positive bounds.

Lawrence lifting and Gr"obner basis computations are often infeasible.

Open problem on structural zeros and connectivity conditions.

Abstract

In this paper we study the computation of Markov bases for contingency tables whose cell entries have an upper bound. In general a Markov basis for unbounded contingency table under a certain model differs from a Markov basis for bounded tables. Rapallo, (2007) applied Lawrence lifting to compute a Markov basis for contingency tables whose cell entries are bounded. However, in the process, one has to compute the universal Gr\"obner basis of the ideal associated with the design matrix for a model which is, in general, larger than any reduced Gr\"obner basis. Thus, this is also infeasible in small- and medium-sized problems. In this paper we focus on bounded two-way contingency tables under independence model and show that if these bounds on cells are positive, i.e., they are not structural zeros, the set of basic moves of all $2 \times 2$ minors connects all tables with given margins. We end this paper with an open problem that if we know the given margins are positive, we want to find the necessary and sufficient condition on the set of structural zeros so that the set of basic moves of all $2 \times 2$ minors connects all incomplete contingency tables with given margins.