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Maximum Stable Matching with Matroids and Partial Orders

Gergely Csáji, Tamás Király, Yu Yokoi

TL;DR

This work studies maximum stable matching under two generalized frameworks: SMTI and matroid kernels. It introduces a simple $1.5$-approximation for the matroid kernel problem when preferences are interval orders, achieved through a three-way replication and a Gale–Shapley–style algorithm, with a projection back to the original ground set. In the bipartite case, the approach also tightens the LP-relaxation gap to $1.5$, showing the integrality gap is at most $1.5$ for interval orders. The paper further establishes that for general partial orders, beating a trivial $2$-approximation is UGC-hard, and the $1.5$-approximation does not extend beyond interval orders, highlighting interval orders as a natural, tractable generalization for stable matching with matroid constraints.

Abstract

The Stable Marriage problem (SM), solved by the famous deferred acceptance algorithm of Gale and Shapley (GS), has many natural generalizations. If we allow ties in preferences, then the problem of finding a maximum stable matching becomes NP-hard, and the best known approximation ratio is 1.5 (McDermid 2009, Paluch 2011, Z. Király 2012), achievable by running GS on a cleverly constructed modified instance. Another elegant generalization of SM is the matroid kernel problem introduced by Fleiner (2001), which is solvable in polynomial time using an abstract matroidal version of GS. Our main result is a simple 1.5-approximation algorithm for the matroid kernel problem when preferences are given as interval orders -- a broad subclass of partial orders that covers many applications beyond preferences with ties. In addition, for the bipartite matching case, we show that the output of our algorithm also 1.5-approximates the LP-optimum of the relaxation of the corresponding Integer Program, which shows that the integrality gap is at most 1.5 for the interval order case. To contrast this with hardness results, we show that if arbitrary partial orders are allowed in the preferences, then even in the bipartite matching case, the problem becomes hard to approximate within a factor better than 2 assuming the Unique Games Conjecture, and the integrality gap becomes 2.

Maximum Stable Matching with Matroids and Partial Orders

TL;DR

This work studies maximum stable matching under two generalized frameworks: SMTI and matroid kernels. It introduces a simple -approximation for the matroid kernel problem when preferences are interval orders, achieved through a three-way replication and a Gale–Shapley–style algorithm, with a projection back to the original ground set. In the bipartite case, the approach also tightens the LP-relaxation gap to , showing the integrality gap is at most for interval orders. The paper further establishes that for general partial orders, beating a trivial -approximation is UGC-hard, and the -approximation does not extend beyond interval orders, highlighting interval orders as a natural, tractable generalization for stable matching with matroid constraints.

Abstract

The Stable Marriage problem (SM), solved by the famous deferred acceptance algorithm of Gale and Shapley (GS), has many natural generalizations. If we allow ties in preferences, then the problem of finding a maximum stable matching becomes NP-hard, and the best known approximation ratio is 1.5 (McDermid 2009, Paluch 2011, Z. Király 2012), achievable by running GS on a cleverly constructed modified instance. Another elegant generalization of SM is the matroid kernel problem introduced by Fleiner (2001), which is solvable in polynomial time using an abstract matroidal version of GS. Our main result is a simple 1.5-approximation algorithm for the matroid kernel problem when preferences are given as interval orders -- a broad subclass of partial orders that covers many applications beyond preferences with ties. In addition, for the bipartite matching case, we show that the output of our algorithm also 1.5-approximates the LP-optimum of the relaxation of the corresponding Integer Program, which shows that the integrality gap is at most 1.5 for the interval order case. To contrast this with hardness results, we show that if arbitrary partial orders are allowed in the preferences, then even in the bipartite matching case, the problem becomes hard to approximate within a factor better than 2 assuming the Unique Games Conjecture, and the integrality gap becomes 2.
Paper Structure (11 sections, 11 theorems, 4 equations, 3 figures, 1 table)

This paper contains 11 sections, 11 theorems, 4 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Basic definitions
  3. Our contribution
  4. Related work
  5. Existence of perfect matching of exchange edges in matroids
  6. Matroid kernel algorithm with interval orders
  7. Description of the algorithm
  8. Proof of $1.5$-approximation
  9. Inapproximability with General Partial Orders
  10. Structural observations
  11. UGC-hardness of $(2-\delta)$-approximation

Key Result

Theorem 1

Given two partially ordered matroids $M_1=(S,\mathcal{I}_1,\succ_1 )$ and $M_2=(S,\mathcal{I}_2,\succ_2 )$ with $\succ_1$ and $\succ_2$ being interval orders, one can find an $(M_1, M_2)$-kernel $A$ such that $|A|$ is at least $\frac{2}{3}$ of the size of a maximum $(M_1, M_2)$-kernel.

Figures (3)

  • Figure 1: The graphic matroid defined by $K_4$. When $e_1\succ e_2 \succ e_3\succ e_4\succ e_5\succ e_6$, the thick edges form the optimal base with respect to $\succ$.
  • Figure 2: Illustration of Example \ref{['ex:weird']}. The green stable matching is $M_2$ and the blue one is $M_3$.
  • Figure 3: The reduction for Theorem \ref{['thm:maxsmp']}. (a) An instance of independent set with two disjoint independent sets $V_1$ (blue) and $V_2$ (red). (b) The corresponding instance of max-smpo. Parallel edges $h_{i(\ell)}$ and $h'_{i(\ell)}$ with $\ell=1$ are drawn on the left and those with $\ell=2$ are drawn on the right. Blue and red edges represent the edges corresponding to the two independent sets $V_1,V_2$, respectively, which are chosen in the constructed matching.

Theorems & Definitions (29)

  • Theorem 1
  • Proposition 2
  • Theorem 3
  • Proposition 4: Strong circuit axiom
  • Theorem 5
  • proof
  • Claim 6
  • proof
  • Claim 7
  • proof
  • ...and 19 more