On Tractability of Ulams Metric in Highier Dimensions and Dually Related Hierarchies

TL;DR

This paper explores the computational complexity and properties of Ulam's metric extended to higher dimensions, analyzing hierarchies of permutation pairs with applications in VLSI design.

Contribution

It introduces a higher-dimensional version of Ulam's metric and studies related hierarchies, revealing their computational hardness and approximation challenges.

Findings

Higher-dimensional Ulam's metric is computationally hard.

The hierarchies exhibit complex approximability properties.

Applications include topological relations in VLSI design.

Abstract

The Ulam's metric is the minimal number of moves consisting in removal of one element from a permutation and its subsequent reinsertion in different place, to go between two given permutations. Thet elements that are not moved create longest common subsequence of permutations. Aldous and Diaconis, in their paper, pointed that Ulam's metric had been introduced in the context of questions concerning sorting and tossing cards. In this paper we define and study Ulam's metric in highier dimensions: for dimension one the considered object is a pair of permutations, for dimension k it is a pair of k-tuples of permutations. Over encodings by k-tuples of permutations we define two dually related hierarchies. Our very first motivation come from Murata at al. paper, in which pairs of permutations were used as representation of topological relation between rectangles packed into minimal area with application to VLSI physical design. Our results concern hardness, approximability, and parametrized complexity inside the hierarchies.