The Stable Roommates Problem with Short Lists

Cseh, Ágnes; Irving, Robert W.; Manlove, David F.

doi:10.1007/978-3-662-53354-3_17

Cited by 8 publications

(14 citation statements)

References 22 publications

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“…Finding an egalitarian stable matching for a given instance of SRI whose preference lists are short was found to be NP-hard more recently by Cseh et al [9]. They showed that the problem remains NP-hard for instances with preference lists at most length 3.…”

Section: Np-hardness Results and Approximabilitymentioning

confidence: 99%

“…Proof. The proof in [9] is a reduction from vertex cover in the cubic graphs. We construct an instance I of SRI with preference lists at most length 3 as described in the proof.…”

Section: The First Proof Of Theoremmentioning

confidence: 99%

“…Theorem 5 (Theorem 1 from [9]). Egalitarian SRI is NP-hard even when the preference lists are of at most length 3.…”

Section: Egalitarian Sri With Short Preference Listsmentioning

confidence: 99%

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Profile-based optimal stable matchings in the Roommates problem

Simola,

Manlove

2021

Preprint

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The stable roommates problem can admit multiple different stable matchings. We have different criteria for deciding which one is optimal, but computing those is often NP-hard.We show that the problem of finding generous or rank-maximal stable matchings in an instance of the roommates problem with incomplete lists is NP-hard even when the preference lists are at most length 3. We show that just maximising the number of first choices or minimising the number of last choices is NP-hard with the short preference lists.We show that the number of R th choices, where R is the minimum-regret of a given instance of SRI, is 2-approximable among all the stable matchings. Additionally, we show that the problem of finding a stable matching that maximises the number of first choices does not admit a constant time approximation algorithm and is W[1]-hard with respect to the number of first choices.We implement integer programming and constraint programming formulations for the optimality criteria of SRI. We find that constraint programming outperforms integer programming and an earlier answer set programming approach by Erdam et. al. [11] for most optimality criteria. Integer programming outperforms constraint programming and answer set programming on the almost stable roommates problem.

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Section: Np-hardness Results and Approximabilitymentioning

confidence: 99%

“…Proof. The proof in [9] is a reduction from vertex cover in the cubic graphs. We construct an instance I of SRI with preference lists at most length 3 as described in the proof.…”

Section: The First Proof Of Theoremmentioning

confidence: 99%

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Profile-based optimal stable matchings in the Roommates problem

Simola,

Manlove

2021

Preprint

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“…The stable marriage problem has been extensively studied over the years [15,17,26,32] with several interesting results in the original model of [15] as well as several of its variants [8,18,20,22,27]. While the Gale-Shapley algorithm can find a stable matching in polynomial time, several of these variants are computationally more challenging [27]; we refer the interested reader to the survey of Iwama and Miyazaki [21] for a more detailed exposition.…”

Section: Further Related Workmentioning

confidence: 99%

“…Consider the mixed integer linear program (OPT-Stab) from Section 2.1 for finding an optimal stable fractional matching for I. Relaxing the integrality constraint (8) to (m, w) ∈ [0, 1]…”

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confidence: 99%

Stable fractional matchings

Caragiannis

Filos-Ratsikas

Kanellopoulos

et al. 2021

Artificial Intelligence

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We study a generalization of the classical stable matching problem that allows for cardinal preferences (as opposed to ordinal) and fractional matchings (as opposed to integral). After observing that, in this cardinal setting, stable fractional matchings can have much higher social welfare than stable integral ones, our goal is to understand the computational complexity of finding an optimal (i.e., welfare-maximizing) or nearlyoptimal stable fractional matching. We present simple approximation algorithms for this problem with weak welfare guarantees and, rather unexpectedly, we furthermore show that achieving better approximations is hard. This computational hardness persists even for approximate stability. To the best of our knowledge, these are the first computational complexity results for stable fractional matchings. En route to these results, we provide a number of structural observations.

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