Abstract
We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egal d-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is \(\textsf {NP}\)-hard even if \(d=3\). On the positive side we give a \(\frac{2d+3}{7}\)-approximation algorithm for \(d\in \{3,4,5\}\) which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is \(\textsf {NP}\)-complete even if \(d=3\). We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the \(d=3\) case. However for \(d=2\) we show that the latter problem can be solved in polynomial time.
Á. Cseh—Supported by Icelandic Research Fund grant no. 152679-051, the Hungarian Academy of Sciences under its Momentum Programme (LP2016-3) and COST Action IC1205 on Computational Social Choice. Part of this work was carried out whilst visiting the University of Glasgow.
D.F. Manlove—Supported by EPSRC grant EP/K010042/1.
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Notes
- 1.
That is, \(\prec _i\) is a strict partial order in which incomparability is transitive.
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Cseh, Á., Irving, R.W., Manlove, D.F. (2016). The Stable Roommates Problem with Short Lists. In: Gairing, M., Savani, R. (eds) Algorithmic Game Theory. SAGT 2016. Lecture Notes in Computer Science(), vol 9928. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-53354-3_17
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