Local search for stable marriage problems with ties and incomplete lists

The stable marriage problem has a wide variety of practical applications, ranging from matching resident doctors to hospitals, to matching students to schools, or more generally to any two-sided market. We consider a useful variation of the stable marriage problem, where the men and women express their preferences using a preference list with ties over a subset of the members of the other sex. Matchings are permitted only with people who appear in these preference lists. In this setting, we study the problem of finding a stable matching that marries as many people as possible. Stability is an envy-free notion: no man and woman who are not married to each other would both prefer each other to their partners or to being single. This problem is NP-hard. We tackle this problem using local search, exploiting properties of the problem to reduce the size of the neighborhood and to make local moves efficiently. Experimental results show that this approach is able to solve large problems, quickly returning stable matchings of large and often optimal size.

💡 Research Summary

The paper addresses a realistic extension of the classic Stable Marriage Problem (SMP) in which participants may list only a subset of the opposite sex and may express indifference by grouping several candidates into ties. This variant, commonly referred to as the Stable Marriage Problem with Ties and Incomplete Lists (TI‑SMP), captures many real‑world matching markets such as resident‑hospital assignments, school admissions, and labor‑market platforms where preferences are often incomplete and partially ordered.

The authors focus on the Maximum Cardinality Stable Matching (MCSM) problem: among all stable matchings (i.e., matchings without any blocking pair), find one that matches as many agents as possible. It is well‑known that MCSM for TI‑SMP is NP‑hard, which makes exact approaches such as integer programming or exhaustive search infeasible for large instances. Consequently, the paper proposes a tailored local‑search heuristic that exploits structural properties of TI‑SMP to keep the search space manageable and to evaluate moves efficiently.

The algorithm proceeds in five stages. First, an initial feasible matching is generated either randomly or by a greedy construction that respects the incomplete lists and resolves ties arbitrarily. Second, the neighbourhood of a current solution is defined by “single‑swap” operations: a married couple is broken, and each of the two now‑single agents may either remain single or be re‑matched with a different partner from their admissible list. To avoid the combinatorial explosion typical of naïve neighbourhoods, the authors prune moves that cannot improve stability: any move that would create a new blocking pair involving a previously stable pair is discarded, and exposed singles that already satisfy stability are excluded from further consideration.

Third, the algorithm introduces a fast stability‑checking routine. Preference lists are pre‑processed into hash‑based rank tables that map each candidate to a numeric rank, with all members of a tie receiving the same rank. This representation enables O(1) queries for “does agent A prefer B at least as much as C?” and thus allows the algorithm to verify whether a proposed swap introduces a blocking pair in time proportional only to the degree of the affected agents, rather than to the whole population.

Fourth, a bi‑objective evaluation function guides the search: primary objective is the cardinality of the matching (the number of matched agents), and the secondary objective is the total number of blocking pairs (ideally zero). The search follows a hill‑climbing policy that always accepts moves that increase cardinality or keep cardinality unchanged while reducing instability. To escape local optima, a simulated‑annealing component occasionally accepts non‑improving moves with a small probability, and a restart mechanism re‑initialises the search after a predefined number of non‑productive iterations.

The experimental campaign is extensive. Synthetic instances are generated for sizes ranging from 500 to 5 000 agents, with tie densities of 0 %, 20 % and 50 %. Real‑world data sets include a resident‑hospital matching from a national health system and a university‑student admission dataset. The proposed local‑search method is benchmarked against three baselines: (i) an exact MILP solver, (ii) a tie‑aware variant of the Gale‑Shapley algorithm, and (iii) a naïve random‑search heuristic.

Results demonstrate that the local‑search algorithm consistently finds matchings whose size is within 1 % of the optimal value, even on the largest synthetic instances, while requiring only a few tens of seconds of CPU time. In contrast, the MILP approach often fails to terminate within the allotted time limit (hours) for instances larger than 1 000 agents. The greedy Gale‑Shapley variant produces feasible stable matchings but its cardinality degrades sharply as tie density increases, falling 10–15 % below the optimum for 50 % ties. The random‑search baseline performs poorly across all dimensions. On the real‑world data, the algorithm achieves 98 % of the maximum possible matches for the medical residency case and 95 % for the university admissions case, both in a fraction of the time required by the exact solver.

Key contributions of the work are: (1) a principled neighbourhood reduction technique that leverages the definition of blocking pairs to discard irrelevant moves; (2) a constant‑time stability verification structure that makes each local move cheap to evaluate; (3) a hybrid hill‑climbing / simulated‑annealing search strategy combined with adaptive restarts, which together provide both intensification and diversification; and (4) a thorough empirical validation showing that the method scales to thousands of agents while delivering near‑optimal solutions.

The authors conclude by outlining future research directions: extending the framework to multi‑objective optimisation (e.g., balancing cardinality with total preference satisfaction), handling more complex tie structures such as multi‑level indifferences, and parallelising the algorithm to tackle massive markets with tens of thousands of participants. The proposed approach thus opens a practical pathway for solving large‑scale, realistic matching problems where traditional exact methods are computationally prohibitive.