Markov bases for two-way subtable sum problems

TL;DR

This paper investigates the conditions under which square-free degree-two moves form a Markov basis for two-way contingency tables with fixed row sums, column sums, and an additional subtable sum constraint.

Contribution

It provides a necessary and sufficient condition on the subtable for these moves to constitute a Markov basis, extending previous known results.

Findings

Characterizes when degree-two moves form a Markov basis with subtable constraints

Identifies conditions on the subtable for Markov basis formation

Extends understanding of Markov bases in contingency table analysis

Abstract

It has been well-known that for two-way contingency tables with fixed row sums and column sums the set of square-free moves of degree two forms a Markov basis. However when we impose an additional constraint that the sum of a subtable is also fixed, then these moves do not necessarily form a Markov basis. Thus, in this paper, we show a necessary and sufficient condition on a subtable so that the set of square-free moves of degree two forms a Markov basis.