A Quadratic Lower Bound for Stable Roommates Solvability

TL;DR

This paper addresses the decision problem of SR solvability, asking whether a stable matching exists for a given SR instance. It develops an embedding-based reduction from the two-party set disjointness problem in communication complexity to SR solvability, establishing a $\Omega(n^2)$ lower bound on adaptive Boolean queries to agents' preferences. Consequently, any algorithm (randomized or deterministic) must perform $\Omega(n^2)$ queries in expectation, implying quadratic time lower bounds for Turing machines and $\Omega(n^2/\log n)$ memory accesses for RAMs, up to a logarithmic factor. The results show that Irving's $O(n^2)$ algorithm is optimal up to a polylogarithmic factor and connect SR solvability to fundamental limits in communication complexity.

Abstract

In their seminal work on the Stable Marriage Problem (SM), Gale and Shapley introduced a generalization of SM referred to as the Stable Roommates Problem (SR). An instance of SR consists of a set of $2n$ agents, and each agent has preferences in the form of a ranked list of all other agents. The goal is to find a one-to-one matching between the agents that is stable in the sense that no pair of agents have a mutual incentive to deviate from the matching. Unlike the (bipartite) stable marriage problem, in SR, stable matchings need not exist. Irving devised an algorithm that finds a stable matching or reports that none exists in $O(n^2)$ time. In their influential 1989 text, Gusfield and Irving posed the question of whether $Ω(n^2)$ time is required for SR solvability -- the task of deciding if an SR instance admits a stable matching. In this paper we provide an affirmative answer to Gusfield and Irving's question. We show that any (randomized) algorithm that decides SR solvability requires $Ω(n^2)$ adaptive Boolean queries to the agents' preferences (in expectation). Our argument follows from a reduction from the communication complexity of the set disjointness function. The query lower bound implies quadratic time lower bounds for Turing machines, and memory access lower bounds for random access machines. Thus, we establish that Irving's algorithm is optimal (up to a logarithmic factor) in a very strong sense.

Figures (2)

• Figure 1: An illustration of the embedding of disjointness for $N = 3 \times 3$ and $n = 3$. This instance corresponds to $x_{1,1} = x_{1,2} = x_{3, 2} = 1$, while the remaining values of $x_{ij} = 0$, and $y_{2,2} = y_{3,3} = 1$ with the remaining values of $y_{ij} = 0$. Thus, $\mathop{\mathrm{disj}}\nolimits(x, y) = 1.$ The edges between agents are colored by agent preferences: the dark blue edges from the $a_i$ correspond to their most preferred partners (preferred above $c_i$). Similarly, the dark red edges from the $b_j$ correspond to their most preferred edges. The light blue edges from the $c_i$ and violet edges from the $d_j$ correspond to those agent's most preferred partners. Other possible partners are not depicted. Sub-figure (a) represents the remaining pairs in the preference table before the first rounds of proposals, while (b), (c), and (d) depict the remaining pairs after each round of proposals. The final figure depicts the unique stable matching for this instance.
• Figure 2: An illustration of the embedding of disjointness for $N = 3 \times 3$ and $n = 6$. This instance corresponds to $x_{1,1} = x_{2,2} = x_{3, 2} = 1$, while the remaining values of $x_{ij} = 0$, and $y_{2,2} = y_{3,3} = 1$ with the remaining values of $y_{ij} = 0$. Thus, $\mathop{\mathrm{disj}}\nolimits(x, y) = 1$ with $x_{2,2} = y_{2,2} = 1$. Sub-figure (a) represents the remaining pairs in the preference table before the first rounds of proposals, while (b), (c), and (d) depict the remaining pairs after each round of proposals. Note that in all of the figures, the pair $\left\{a_2, b_2\right\}$ is preferred by $a_2$ to $c_2$ and preferred by $b_2$ to $d_2$. Therefore, this edge is not removed after either the $C$ proposals nor the $D$ proposals in figures (b) and (c). Subsequently, when agents in $A$ propose, $a_2$ proposes to $b_2$, after which $b_2$ rejects $d_2$. At this point $d_2$'s preference list is empty, hence the instance does not admit a stable matching.

Theorems & Definitions (12)

• Theorem 1
• Corollary 1.1
• Remark 1.2
• Lemma 2.1: Irving Irving1985-stable, c.f. GI89
• Example 2.2
• Theorem 2: KS92Razborov92
• Theorem 3: c.f. Eden2018-lower
• Proposition 3.1
• proof
• Theorem 4