[Paper Review] Stable Marriage with Covering Constraints: A Complete Computational Trichotomy

Matthias Mnich, Ildikó Schlotter|Repository of the Academy's Library (Library of the Hungarian Academy of Sciences)|Feb 26, 2016

Game Theory and Voting Systems40 references4 citations

TL;DR

This paper presents a complete computational trichotomy for the Stable Marriage with Covering Constraints (SMC) problem, analyzing the impact of five key parameters—distinguished individuals, preference list lengths, and blocking pair limits—on tractability. It establishes that every parameter combination falls into one of three categories: polynomial-time solvable, NP-hard with fixed-parameter tractable solutions, or NP-hard and W[1]-hard, resolving an open question on parameterized intractability.

ABSTRACT

We consider Stable Marriage with Covering Constraints (SMC): in this variant of Stable Marriage, we distinguish a subset of women as well as a subset of men, and we seek a matching with fewest number of blocking pairs that matches all of the distinguished people. We investigate how a set of natural parameters, namely the maximum length of preference lists for men and women, the number of distinguished men and women, and the number of blocking pairs allowed determine the computational tractability of this problem. Our main result is a complete complexity trichotomy that, for each choice of the studied parameters, classifies SMC as polynomial-time solvable, NP-hard and fixed-parameter tractable, or NP-hard and W[1]-hard. We also classify all cases of one-sided constraints where only women may be distinguished.

Motivation & Objective

• To systematically analyze the computational complexity of stable marriage with covering constraints under various parameter combinations.
• To resolve the open question of fixed-parameter intractability posed by Hamada et al. regarding solution size as a parameter.
• To provide a comprehensive classification of tractable and intractable cases for SMC using five natural parameters: distinguished individuals, preference list lengths, and blocking pair limits.
• To extend prior work on HRLQ by introducing fixed-parameter tractability as a relaxed notion of tractability.
• To establish a decision diagram that fully determines the complexity of SMC for all combinations of the five studied parameters.

Proposed method

• The authors analyze the SMC problem under five natural parameters: |M*|, |W*|, ΔM, ΔW, and b (blocking pairs allowed), using a systematic parameterized complexity approach.
• They employ a reduction-based framework to prove NP-hardness and W[1]-hardness, particularly for cases with bounded preference list lengths and small numbers of distinguished individuals.
• The paper introduces a novel algorithmic framework that combines dynamic programming and branching techniques to solve tractable cases in polynomial time or fixed-parameter time.
• A decision diagram (Figure 10) is constructed to classify all combinations of parameters into one of three complexity classes: P, FPT, or W[1]-hard.
• The authors use symmetry and reflection principles (e.g., Theorem x R) to generalize results between men and women, reducing the number of cases to analyze.
• They validate their results through multiple theorems and lemmas, including proofs of correctness for key algorithmic steps and parameterized reductions.

Experimental results

Research questions

• RQ1 Is the SMC problem fixed-parameter tractable when parameterized by the number of blocking pairs and the number of distinguished individuals?
• RQ2 What is the computational complexity of SMC when preference lists are bounded in length and only one gender has distinguished individuals?
• RQ3 Does the problem remain NP-hard and W[1]-hard when both genders have bounded preference list lengths and a small number of distinguished individuals?
• RQ4 Can a complete classification of all parameter combinations into P, FPT, or W[1]-hard be achieved for SMC with covering constraints?
• RQ5 Does the problem remain intractable even under strong restrictions such as ΔM ≤ 2 and |W*| = 1?

Key findings

• The paper establishes a complete computational trichotomy: every combination of the five parameters falls into one of three classes—polynomial-time solvable, NP-hard with fixed-parameter tractable solution, or NP-hard and W[1]-hard.
• The problem is shown to be W[1]-hard when parameterized by b + |W*| or b + ΔM, even under restricted settings such as ΔW = 2 and |M*| = 1.
• For cases with ΔM ≤ 2 and |W*| = 0, the problem is polynomial-time solvable, as proven in Theorem 8.
• The problem remains NP-hard when ΔM = 3 and |W*| = 1, even with ΔW = 2, establishing a minimal intractable case.
• The authors resolve an open question by Hamada et al. by proving that the problem is W[1]-hard when parameterized by the optimal solution size (i.e., number of blocking pairs), ruling out fixed-parameter tractability under this parameter.
• The decision diagram in Figure 10 provides a complete and systematic classification of all parameter combinations, confirming that no further complexity classes exist beyond the three identified.