Combinatorial Structure of the Deterministic Seriation Method with Multiple Subset Solutions

TL;DR

This paper analyzes the combinatorial complexity of classical and iterative frequency seriation methods, highlighting the exponential growth of solutions and the need for new algorithms to efficiently find optimal subset solutions.

Contribution

It extends the understanding of the combinatorial structure of seriation to include multiple subset solutions, emphasizing the complexity and computational challenges involved.

Findings

Number of solutions exceeds n! for multiple subset seriation

Highlights the need for new algorithms and heuristics

Provides a theoretical foundation for the combinatorial structure

Abstract

Seriation methods order a set of descriptions given some criterion (e.g., unimodality or minimum distance between similarity scores). Seriation is thus inherently a problem of finding the optimal solution among a set of permutations of objects. In this short technical note, we review the combinatorial structure of the classical seriation problem, which seeks a single solution out of a set of objects. We then extend those results to the iterative frequency seriation approach introduced by Lipo (1997), which finds optimal subsets of objects which each satisfy the unimodality criterion within each subset. The number of possible solutions across multiple solution subsets is larger than $n!$, which underscores the need to find new algorithms and heuristics to assist in the deterministic frequency seriation problem.