Ex-post Stability under Two-Sided Matching: Complexity and Characterization

TL;DR

The computational complexity of testing ex-post stability is NP-complete and the central result is that when either side has ties in the preferences/priorities, testing ex-post stability is NP-complete.

Abstract

A probabilistic approach to the stable matching problem has been identified as an important research area with several important open problems. When considering random matchings, ex-post stability is a fundamental stability concept. A prominent open problem is characterizing ex-post stability and establishing its computational complexity. We investigate the computational complexity of testing ex-post stability. Our central result is that when either side has ties in the preferences/priorities, testing ex-post stability is NP-complete. The result even holds if both sides have dichotomous preferences. On the positive side, we give an algorithm using an integer programming approach, that can determine a decomposition with a maximum probability of being weakly stable. We also consider stronger versions of ex-post stability (in particular robust ex-post stability and ex-post strong stability) and prove that they can be tested in polynomial time.

Figures (6)

• Figure 1: The gadget for a set $C_3=\{ a_3,a_5,a_8\}$ with the important edges in Theorem \ref{['ex-post-verif:strict-dich']}. The red, blue and green edges are the edges of $M_1^k,M_2^k$ and $M_3^k$ respectively, when the set $C_3$ is part of the exact 3-cover. The $p$ value on each edge is $\frac{1}{3}$ times the number of colors the edge has. The dotted black lines represent the edges with $p$ value 0.
• Figure 2: The gadget for a set $C_3=\{ a_3,a_5,a_8\}$ with the important edges in Theorem \ref{['ex-post-verif:strict-dich']}. The red, blue and green edges are the edges of $M_1^k,M_2^k$ and $M_3^k$ respectively, when the set $C_3$ is NOT part of the exact 3-cover. The $p$ value on each edge is $\frac{1}{3}$ times the number of colors the edge has. The dotted black lines represent the edges with $p$ value 0.
• Figure 3: The construction for Theorem \ref{['thm:deg3']}. The red, blue and green edges are the edges of $M_1,M_2$ and $M_3$ respectively, when the set $C_j$ is part of the exact 3-cover. The $p$ value on each edge is $\frac{1}{3}$ times the number of colors the edge has. The dotted lines represent the edges with $p$ value 0.
• Figure 4: The construction for Theorem \ref{['thm:deg3']}. The red, blue and green edges are the edges of $M_1,M_2$ and $M_3$ respectively, when the set $C_j$ is not part of the exact 3-cover. The $p$ value on each edge is $\frac{1}{3}$ times the number of colors the edge has. The dotted lines represent the edges with $p$ value 0.
• Figure 5: The gadget $G_j^l$ with its neighbors $_1x_j^{l+1},_3x_j^l,a_{j_l}$ and $y_j^l$. The red, blue and green edges correspond to the matchings $\{ M_1,M_2,M_3\}$, depending on which copy of $c_j^l$ is matched to the outside. The $p$ value on each edge is $\frac{1}{3}$ times the number of colors the edge has.

Theorems & Definitions (40)

• Definition 2.1: Stability for deterministic matchings
• Definition 2.2: Ex-post stability
• Definition 2.3: Fractional stability and violations of fractional stability
• Proposition 2.1: TeSe98a,AzKl19b
• proof
• Definition 2.4: Robust ex-post stability
• Definition 2.5: Strong stability
• Definition 2.6: Ex-post strong stability
• Definition 2.7: Fractional strong stability
• Theorem 3.1