[Paper Review] On the approximability of the stable marriage problem with one-sided ties

Bauckholt, Felix, Pashkovich, Kanstantsin+1 more|arXiv (Cornell University)|May 14, 2018

Game Theory and Voting Systems12 references5 citations

TL;DR

This paper presents a refined analysis of Huang and Telikepalli's approximation algorithm for the stable marriage problem with one-sided ties, proving it achieves a 13/9-approximation factor—improving upon the prior best bound of 19/13. The analysis introduces novel concepts like 'popular women' and 'jumps' in a directed auxiliary graph to strengthen the charging scheme, and demonstrates tightness via a constructed example where the ratio reaches exactly 13/9.

ABSTRACT

The classical stable marriage problem asks for a matching between a set of men and a set of women with no blocking pairs, which are pairs formed by a man and a woman who would both prefer switching from their current status to be paired up together. When both men and women have strict preferences over the opposite group, all stable matchings have the same cardinality, and the famous Gale-Shapley algorithm can be used to find one. Differently, if we allow ties in the preference lists, finding a stable matching of maximum cardinality is an NP-hard problem, already when the ties are one-sided, that is, they appear only in the preferences of one group. For this reason, many researchers have focused on developing approximation algorithm for this problem. In this paper, we give a refined analysis of an approximation algorithm given by Huang and Telikepalli (IPCO14) for the stable marriage problem with one-sided ties, which shows an improved 13/9 -approximation factor for the problem. Interestingly, our analysis is tight.

Motivation & Objective

• To improve the approximation factor for the stable marriage problem with one-sided ties, where only one gender group may have tied preferences.
• To provide a tighter analysis of the combinatorial algorithm by Huang and Telikepalli (2014), which allows men to make up to two proposals and women to accept up to two.
• To develop a new charging scheme that better accounts for 5-augmenting paths in the symmetric difference between the algorithm's output and an optimal matching.
• To demonstrate that the 13/9-approximation bound is tight by constructing an instance where the algorithm's solution is exactly 13/9 times smaller than the optimal matching.

Proposed method

• Construct a directed auxiliary graph H by orienting edges from the proposal graph G′ and contracting edges of the algorithm’s matching M that lie on 5-augmenting paths.
• Introduce the concept of 'jumps'—specific structural features in H that allow more precise tracking of augmenting paths and their contributions to the approximation ratio.
• Define 'popular women' as those who are not rejected by all their preferred partners in the optimal matching, enabling better charge distribution in the analysis.
• Develop a refined charging scheme that maps each 5-augmenting path to components in the symmetric difference M ⊕ OPT, ensuring no component is overcharged.
• Use path-charges and unpopularity-charges to bound the total contribution of women in each component, proving that the sum of charges per component is at most 3ℓq for components with 2ℓq + 5 edges.
• Prove the overall approximation ratio by showing |OPT| / |M| ≤ 13/9 through bounding the total number of augmenting paths and their contributions using the refined charge sums.

Experimental results

Research questions

• RQ1 Can the approximation factor of Huang and Telikepalli’s algorithm for the one-sided ties stable marriage problem be improved beyond the previously known 19/13 bound?
• RQ2 What structural properties of the symmetric difference between the algorithm’s output and an optimal matching can be exploited to refine the analysis?
• RQ3 Is the 13/9-approximation factor tight, and can a concrete instance be constructed where this ratio is achieved?
• RQ4 How can the charging scheme be enhanced by introducing new concepts like 'jumps' and 'popular women' to reduce overestimation of path contributions?

Key findings

• The algorithm by Huang and Telikepalli (2014) is proven to achieve a 13/9-approximation factor for the stable marriage problem with one-sided ties, improving upon the prior best bound of 19/13.
• The analysis introduces two key concepts—'jumps' and 'popular women'—that enable a tighter and more accurate charging scheme for 5-augmenting paths in the symmetric difference M ⊕ OPT.
• The 13/9-approximation bound is shown to be tight, as a constructed instance achieves exactly this ratio, meaning no further improvement via the same algorithm is possible without structural changes.
• The refined charging scheme ensures that the sum of charges over all women in any (2ℓ+5)-augmenting path is at most 3ℓ, which is critical for proving the improved approximation ratio.
• The proof demonstrates that the number of 5-augmenting paths t is at most 3ℓΣ and the number of longer augmenting paths k is at most ℓΣ, where ℓΣ is the sum of ℓq over all components in M ⊕ OPT.
• The final ratio |OPT| / |M| ≤ 13/9 is derived by combining bounds on matching sizes and charge sums, confirming the algorithm’s performance guarantee.