Extending Robinson Spaces: Complexity and Algorithmic Solutions for Non-Symmetric Dissimilarity Spaces

TL;DR

By generalizing Robinson spaces to non-symmetric dissimilarities, the paper introduces the notions of one-way-Robinson and two-way-Robinson for a set $X$ with dissimilarity $d$, and frames two algorithmic problems, Orientation and Assignment. It proves that Orientation is NP-hard and Assignment is NP-complete (even when the input tree is a path), but also provides an $O(n^3)$ recognition algorithm for two-way-Robinson spaces and several polynomial-time solvable tractable cases in symmetric settings (stars and paths). These contributions extend seriation techniques to directional data and identify concrete cases where efficient solutions exist, with implications for data representation and graph orientation problems. The work thus lays a theoretical and algorithmic groundwork for processing asymmetric dissimilarities via Robinson-like structure.

Abstract

In this work, we extend the concept of Robinson spaces to asymmetric dissimilarities, enhancing their applicability in representing and analyzing complex data. Within this generalized framework, we introduce two different problems that extend the classical seriation problem: an optimization problem and a decision problem. We establish that these problems are NP-hard and NP-complete, respectively. Despite this complexity results, we identify several non-trivial instances where these problems can be solved in polynomial time, providing valuable insights into their tractability.

Figures (2)

• Figure 1: The tree $T_{SAT}$ for a 3-cnf formula with $m$ clauses $c_1, c_2,\ldots, c_m$, all with three literals, on $n$ variables $v_1,v_2,\ldots, v_n$. \newlabelFIG_graph_SAT0
• Figure 1: The neighborhood of $x$ and $y$ \newlabelFIG_two_centrals_bis0

Theorems & Definitions (32)

• Proposition 2.1
• Proof 1
• Theorem 2.2
• Theorem 3.1
• Proof 2
• Lemma 3.2
• Proof 3
• Theorem 3.3
• Proof 4
• Lemma 4.1