Weakly-Popular and Super-Popular Matchings with Ties and Their Connection to Stable Matchings

TL;DR

This work investigates new popularity notions for two-sided matchings with ties, introducing weak-popularity, $\gamma$-popularity, and super-popularity to overcome nonexistence issues in traditional popular matchings. It develops an edge-duplication based algorithm that yields a polynomial-time $\,{\$3/4$}-approximation for max-$\gamma$-PM, which also achieves a ${\$4/5$}-approximation relative to the maximum $\gamma$-min stable matching, leveraging a connection to strict-stable instances via Gale–Shapley. The authors prove NP-hardness for maximum size weakly popular matchings (even with one-sided ties) and show the $3/4$-approximation bound is tight under the Strong-UGC or SSEH, while also establishing NP-hardness for the existence of super-popular matchings. The results illuminate a robust, efficient alternative to stability in the presence of ties and open directions on the tractability of stronger popularities (notably super-popularity) under restricted tie configurations.

Abstract

In this paper, we study a slightly different definition of popularity in bipartite graphs $G=(U,W,E)$ with two-sided preferences, when ties are present in the preference lists. This is motivated by the observation that if an agent $u$ is indifferent between his original partner $w$ in matching $M$ and his new partner $w'\ne w$ in matching $N$, then he may probably still prefer to stay with his original partner, as change requires effort, so he votes for $M$ in this case, instead of being indifferent. We show that this alternative definition of popularity, which we call weak-popularity allows us to guarantee the existence of such a matching and also to find a weakly-popular matching in polynomial-time that has size at least $\frac{3}{4}$ the size of the maximum weakly popular matching. We also show that this matching is at least $\frac{4}{5}$ times the size of the maximum (weakly) stable matching, so may provide a more desirable solution than the current best (and tight under certain assumptions) $\frac{2}{3}$-approximation for such a stable matching. We also show that unfortunately, finding a maximum size weakly popular matching is NP-hard, even with one-sided ties and that assuming some complexity theoretic assumptions, the $\frac{3}{4}$-approximation bound is tight. Then, we study a more general model than weak-popularity, where for each edge, we can specify independently for both endpoints the size of improvement the endpoint needs to vote in favor of a new matching $N$. We show that even in this more general model, a so-called $γ$-popular matching always exists and that the same positive results still hold. Finally, we define an other, stronger variant of popularity, called super-popularity, where even a weak improvement is enough to vote in favor of a new matching. We show that for this case, even the existence problem is NP-hard.

Figures (1)

• Figure 1: The created preferences in Theorem \ref{['thm:wpopinappr']}. Here, $i\in [n],j\in [\frac{n}{2}]$

Theorems & Definitions (25)

• Theorem 1
• proof
• Theorem 2
• proof
• Claim 1
• Claim 2
• Theorem 3
• proof
• Claim 3
• Claim 4