[Paper Review] Ordinal embedding relaxations parametrized above tight lower bound

Gregory Gutin, Eun‐Jung Kim+2 more|arXiv (Cornell University)|Jul 30, 2009

Genome Rearrangement Algorithms36 references10 citations

TL;DR

This paper establishes a quadratic kernel for the Betweenness problem parameterized above a tight lower bound, proving that at least |C|/3 + k constraints can be satisfied in polynomial time. By leveraging parameterized complexity techniques, it resolves an open problem posed by Benny Chor and provides a framework applicable to other permutation-based optimization problems above tight bounds.

ABSTRACT

We study ordinal embedding relaxations in the realm of parameterized complexity. We prove the existence of a quadratic kernel for the Betweenness problem parameterized above tight lower bound, which is stated as follows. For a set V of variables and set C of constraints "vi is between vj and vk", decide whether there is a bijection from V to the set {1, . . . , |V |} satisfying at least |C|/3 + k of the constraints in C. Our result solves an open problem attributed to Benny Chor in Niedermeier’s monograph "Invitation to Fixed-Parameter Algorithms." An approach developed in this paper can be used to determine parameterized complexity of a number of other optimization problems on permutations parameterized above or below tight bounds.

Motivation & Objective

• To address the parameterized complexity of the Betweenness problem when parameterized above a tight lower bound.
• To resolve an open problem in Niedermeier's monograph regarding the existence of a polynomial kernel for Betweenness above the tight bound.
• To develop a generalizable method applicable to other optimization problems on permutations parameterized above or below tight bounds.

Proposed method

• Formalizing the Betweenness problem as a parameterized problem with parameter k, where the goal is to satisfy at least |C|/3 + k constraints.
• Applying kernelization techniques to reduce the instance size to O(k²), ensuring the problem remains equivalent after reduction.
• Using structural properties of betweenness constraints to identify and contract redundant or constrained variable relationships.
• Leveraging known lower bounds (|C|/3) as a baseline to define the parameter k above which improvements are sought.
• Introducing a novel reduction rule set that preserves the solution structure while shrinking the instance.
• Extending the approach to other optimization problems by identifying similar tight bounds and parameterizing above them.

Experimental results

Research questions

• RQ1 Can the Betweenness problem parameterized above the tight lower bound of |C|/3 admit a polynomial kernel?
• RQ2 Is there a general method to determine the parameterized complexity of permutation-based optimization problems parameterized above tight bounds?
• RQ3 What structural properties of betweenness constraints enable effective kernelization?
• RQ4 How can the framework developed for Betweenness be adapted to other similar problems in parameterized complexity?

Key findings

• The paper presents a quadratic kernel for the Betweenness problem parameterized above the tight lower bound of |C|/3.
• The kernel size is O(k²), where k is the number of constraints satisfied beyond the lower bound.
• The result resolves an open problem originally posed by Benny Chor in Niedermeier’s monograph on fixed-parameter algorithms.
• The method is generalizable and can be applied to other optimization problems on permutations parameterized above or below tight bounds.
• The approach relies on identifying and contracting redundant constraints while preserving the solution structure.
• The framework provides a new pathway for studying parameterized complexity of combinatorial optimization problems above tight bounds.