Permutation Reconstruction from MinMax-Betweenness Constraints

TL;DR

This paper explores reconstructing permutations from specific betweenness constraints involving min and max elements, proposing algorithms and conditions for uniqueness in both directed and undirected cases.

Contribution

It introduces a new permutation reconstruction problem based on MinMax-betweenness constraints, providing polynomial algorithms and conditions for uniqueness.

Findings

Polynomial algorithm for a special case of the directed problem

Necessary and sufficient conditions for unique reconstruction

Analysis of constraints' stringency via a parameter k

Abstract

In this paper, we investigate the reconstruction of permutations on {1, 2, ..., n} from betweenness constraints involving the minimum and the maximum element located between t and t+1, for all t=1, 2, ..., n-1. We propose two variants of the problem (directed and undirected), and focus first on the directed version, for which we draw up general features and design a polynomial algorithm in a particular case. Then, we investigate necessary and sufficient conditions for the uniqueness of the reconstruction in both directed and undirected versions, using a parameter k whose variation controls the stringency of the betweenness constraints. We finally point out open problems.