A Divide-and-Conquer Algorithm for Generating Markov Bases of Multi-way Tables

Summary

We describe a divide-and-conquer technique for generating a Markov basis that connects all tables of counts having a fixed set of marginal totals. This procedure is based on decomposing the independence graph induced by these marginals. We discuss the practical imports of using this method in conjunction with other algorithms for determining Markov bases.

A Decomposable and Reducible Graphs

A graph $\mathcal{G}$ is a pair (K, E), where K = {1, 2, …, k} is a finite set ofvertices and E ⊆ K × K is a set of edges linking the vertices. For any vertex set A ⊆ K, we define the edge set associated with it as

$$E(A) := \{(u, v) \in E|u, v \in A\}.$$

Let $\mathcal{G}(A) = (A, E(A))$ denote the subgraph of $\mathcal{G}$ induced by A. Two vertices u, v ∈ K are adjacent if (u, v) ∈ E. A set ofvertices of $\mathcal{G}$ is independent if no two of its elements are adjacent. An induced subgraph $\mathcal{G}(A)$ is complete if the vertices in A are pairwise adjacent in $\mathcal{G}$. We also say that A is complete in $\mathcal{G}$. A complete vertex set A in $\mathcal{G}$ that is maximal is a clique.

Let u, v ∈ K. A path (or chain) from u to v is a sequence u = v₀, …, v_n = v of distinct vertices such that (v_i−1, v_i) ∈ E for all i = 1, 2, …, n. The path is a cycle if the end points are allowed to be the same, u = v. If there is a path from u to v we say that u and v are connected. The sets A, B ⊂ K are disconnected if u and v are not connected for all u ∈ A, v ∈ B. The connected component of a vertex u ∈ K is the set of all vertices connected with u. A graph is connected if all the pairs of vertices are connected.

The set C ⊂ K is an uv-separator if all paths from u to v intersect C. The set C ⊂ K separates A from B if it is an uv-separator for every u ∈ A, v ∈ B. C is a separator of $\mathcal{G}$ if two vertices in the same connected component of $\mathcal{G}$ are in two distinct connected components of $\mathcal{G}\backslash{C}$ or, equivalently, if $\mathcal{G}\backslash{C}$ is disconnected. In addition, C is a minimal separator of $\mathcal{G}$ if C is a separator and no proper subset of C separates the graph. Unless otherwise stated, the separators we work with will be complete.

Decomposable graphs possess the special property that allows us to “decompose” them into components or subgraphs and work directly with these components. The idea is to decompose the graph $\mathcal{G}$ in two possibly overlapping subgraphs $\mathcal{G'}$ and $\mathcal{G''}$ so that no information of the graph is lost when transforming $\mathcal{G}$ into $\mathcal{G'}$ and $\mathcal{G''}$. Furthermore, by “correctly” decomposing $\mathcal{G'}$ and $\mathcal{G''}$, and so on, one ends up with a set of subgraphs of $\mathcal{G}$ which allow for no further decompositions. A set of subgraphs of $\mathcal{G}$ generated in this way is called a derived system of $\mathcal{G}$, while its elements are called atoms (Tarjan 1985). We define what we mean by “correct” decomposition.

Definition A.1. The partition (A₁, S, A₂) of K is said to form a decomposition of $\mathcal{G}$ if S is a minimal separator of A₁ and A₂.

In this case (A₁, S, A₂) decomposes $\mathcal{G}$ into the components $\mathcal{G}(A_1 \cup S)$ and $\mathcal{G}(S \cup A_2)$. The decomposition is proper if A₁ and A₂ are not empty.

Definition A.2. The graph $\mathcal{G}$ is decomposable if it is complete or if there exists a proper decomposition (A₁, S, A₂) into decomposable graphs $\mathcal{G}(A_1 \cup S)$ and $\mathcal{G}(S \cup A_2)$.

Graphs that are not decomposable, but can still be decomposed in sequences of atoms are described in Tarjan (1985) and Leimer (1993). In this case, the resulting atoms are not necessarily complete.

Definition A.3. A graph $\mathcal{G}$ is reducible if $\mathcal{G}$ admits a proper decomposition, otherwise $\mathcal{G}$ is a prime graph.

Given that every reducible graph $\mathcal{G}$ might have several derived systems (Tarjan 1985), we would like to be able to isolate one of them which could fully characterize the input graph $\mathcal{G}$.

Definition A.4. A subgraph $\mathcal{G}$ is a maximal prime (mp-) subgraph of $\mathcal{G}$, if $\mathcal{G}(A)$ is prime and $\mathcal{G}(B)$ is reducible for all B with A ⊂ B ⊆ K.

The set of mp-subgraphs of $\mathcal{G}$ is contained in every derived system of $\mathcal{G}$. Moreover, the set of mp-subgraphs of $\mathcal{G}$ is always a derived system of $\mathcal{G}$ (Leimer 1993), and consequently it is the unique minimal derived system. If $\mathcal{G}$ is decomposable, the mp-subgraphs of $\mathcal{G}$ are complete, hence the unique minimal derived system of a decomposable graph contains only its cliques (Leimer 1993).