The Stable Roommates Problem with Short Lists
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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, egaldsri, 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 NPhard even if d = 3. On the positive side we give a \(\frac {2d+3}{7}\)approximation algorithm for d ∈{3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called dsrti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of dsrti admits a stable matching is NPcomplete 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.
Keywords
Stable matching Bounded length preference lists Complexity Approximation algorithm1 Introduction
In the Stable Roommates problem with Incomplete lists (sri), a graph G = (A,E) and a set of preference lists \(\mathcal O\) are given, where the vertices A = {a_{1},…,a_{n}} correspond to agents, and \(\mathcal O=\{\prec _{1},\dots ,\prec _{n}\}\), where ≺_{i} is a linear order on the vertices adjacent to a_{i} in G (1 ≤ i ≤ n). We refer to ≺_{i} as a_{i}’s preference list. The agents that are adjacent to a_{i} in G are said to be acceptable to a_{i}. If a_{j} and a_{k} are two acceptable agents for a_{i} where a_{j}≺_{i}a_{k} then we say that a_{i}prefersa_{j} to a_{k}.
Let M be a matching in G. If a_{i}a_{j} ∈ M then we let M(a_{i}) denote a_{j}. An edge a_{i}a_{j} ∉ M blocks M, or forms a blocking edge of M, if a_{i} is unmatched or prefers a_{j} to M(a_{i}), and similarly a_{j} is unmatched or prefers a_{i} to M(a_{j}). A matching is called stable if no edge blocks it. Denote by sr the special case of sri in which G = K_{n}. Gale and Shapley [9] observed that an instance of sr need not admit a stable matching. Irving [15] gave a lineartime algorithm to find a stable matching or report that none exists, given an instance of sr. The straightforward modification of this algorithm to the sri case is described in [12]. We call an sri instance solvable if it admits a stable matching.
In practice agents may find it difficult to rank a large number of alternatives in strict order of preference. One natural assumption, therefore, is that preference lists are short, which corresponds to the graph being of bounded degree. Given an integer d ≥ 1, we define dsri to be the restriction of sri in which G is of bounded degree d. This special case of sri problem has potential applications in organising tournaments. As already pointed out in a paper of Kujansuu et al. [18], sri can model a pairing process similar to the Swiss system, which is used in largescale chess competitions. The assumption on short lists is reasonable, because according to the Swiss system, players can be matched only to other players with approximately the same score.
A second variant of sri, which can be motivated in a similar fashion, arises if we allow ties in the preference lists, i.e., ≺_{i} (1 ≤ i ≤ n) is now a strict weak ordering. That is, ≺_{i} is a strict partial order in which incomparability is transitive. We refer to this problem as the Stable Roommates problem with Ties and Incomplete lists (srti) [17]. As in the sri case, define dsrti to be the restriction of srti in which G is of bounded degree d. Denote by srt the special case of srti in which G = K_{n}. In the context of the motivating application of chess tournament construction as mentioned in the previous paragraph, dsrti is naturally obtained if a chess player has several potential partners of the same score and match history in the tournament.
In the srti context, ties correspond to indifference in the preference lists. In particular, if a_{i}a_{j} ∈ E and a_{i}a_{k} ∈ E where a_{j}⊀_{i}a_{k} and a_{k}⊀_{i}a_{j} then a_{i} is said to be indifferent betweena_{j} and a_{k}. Thus preference in the sri context corresponds to strict preference in the case of srti. Relative to the strict weak orders in \(\mathcal O\), we can define stability in srti instances in exactly the same way as for sri. This means, for example, that if a_{i}a_{j} ∈ M for some matching M, and a_{i} is indifferent between a_{j} and some agent a_{k}, then a_{i}a_{k} cannot block M. The term solvable can be defined in the srti context in an analogous fashion to sri. Using a highly technical reduction from a restriction of 3sat, Ronn [23] proved that the problem of deciding whether a given srt instance is solvable is NPcomplete. A simpler reduction was given by Irving and Manlove [17].
For solvable instances of sri there can be many stable matchings. Often it is beneficial to work with a stable matching that is fair to all agents in a precise sense [11, 16]. One such fairness concept can be defined as follows. Given two agents a_{i}, a_{j} in an instance \(\mathcal {I}\) of sri, where a_{i}a_{j} ∈ E, let rank(a_{i},a_{j}) denote the rank of a_{j} in a_{i}’s preference list (that is, 1 plus the number of agents that a_{i} prefers to a_{j}). Let A_{M} denote the set of agents who are matched in a given stable matching M. (Note that this set depends only on \(\mathcal I\) and is independent of M by [12, Theorem 4.5.2].) Define \(c(M)={\sum }_{a_{i}\in A_{M}} \text {rank}(a_{i},M(a_{i}))\) to be the cost of M. An egalitarian stable matching is a stable matching M that minimises c(M) over the set of stable matchings in \(\mathcal {I}\). Finding an egalitarian stable matching in sr was shown to be NPhard by Feder [7]. Feder [7, 8] also gave a 2approximation algorithm for this problem in the sri setting. He also showed that an egalitarian stable matching in sr can be approximated within a factor of α of the optimum if and only if Minimum Vertex Cover can be approximated within the same factor α. It was proved later that, assuming the Unique Games Conjecture, Minimum Vertex Cover cannot be approximated within 2 − ε for any ε > 0 [19].
Given an unsolvable instance \(\mathcal I\) of sri or srti, a natural approximation to a stable matching is a moststable matching [1]. Relative to a matching M in \(\mathcal I\), define bp(M) to be the set of blocking edges of M and let \(bp(\mathcal I)\) denote the minimum value of bp(M^{′}), taken over all matchings M^{′} in \(\mathcal I\). Then M is a moststable matching in \(\mathcal I\) if \(bp(M)=bp(\mathcal I)\). The problem of finding a moststable matching was shown to be NPhard and not approximable within n^{k−ε}, for any ε > 0, unless P = NP, where \(k=\frac {1}{2}\) if \(\mathcal I\) is an instance of sr and k = 1 if \(\mathcal I\) is an instance of srt [1].
To the best of our knowledge, there has not been any previous work published on either the problem of finding an egalitarian stable matching in a solvable instance of sri with boundedlength preference lists or the solvability of srti with boundedlength preference lists. This paper provides contributions in both of these directions, focusing on instances of dsri and dsrti for d ≥ 2, with the aim of drawing the line between polynomialtime solvability and NPhardness for the associated problems in terms of d.
Our Contribution
Summary of results for dsri and dsrti
Finding a stable matching  finding an egalitarian stable matching  

dsri  in P for d = 2 (∗) NPhard even for d = 3 (∗) \(\frac {2d+3}{7}\)approximation for d ∈{3, 4, 5} (∗)  
dsrti  in P for d = 2 (∗) NPhard even for d = 3 (∗)  not welldefined (see Section 4) 
Related Work
Degreebounded graphs, moststable matchings and egalitarian stable matchings are widely studied concepts in the literature on matching under preferences [21]. As already mentioned, the problem of finding a moststable matching has been studied previously in the context of sri [1]. In addition to the results surveyed already, the authors of [1] gave an O(m^{k+1}) algorithm to find a matching M with bp(M)≤ k or report that no such matching exists, where m = E and k ≥ 1 is any integer. Moststable matchings have also been considered in the context of dsri [4]. The authors showed that, if d = 3, there is some constant c > 1 such that the problem of finding a moststable matching is not approximable within c unless P = NP. On the other hand, they proved that the problem is solvable in polynomial time for d ≤ 2. The authors also gave a (2d − 3)approximation algorithm for the problem for fixed d ≥ 3. This bound was improved to 2d − 4 if the given instance satisfies an additional condition (namely the absence of a structure called an elitist odd party). Moststable matchings have also been studied in the bipartite restriction of sri called the Stable Marriage problem with Incomplete lists (smi) [5, 14]. Since every instance of smi admits a stable matching M (and hence bp(M) = ∅), the focus in [5, 14] was on finding maximum cardinality matchings with the minimum number of blocking edges.
Regarding the problem of finding an egalitarian stable matching in an instance of sri, as already mentioned Feder [7, 8] showed that this problem is NPhard, though approximable within a factor of 2. A 2approximation algorithm for this problem was also given independently by Gusfield and Pitt [13], and by Teo and Sethuraman [26]. These approximation algorithms can also be extended to the more general setting where we are given a weight function on the edges, and we seek a stable matching of minimum weight. Feder’s 2approximation algorithm requires monotone, nonnegative and integral edge weights, whereas with the help of LP techniques [25, 26], the integrality constraint can be dropped, while the monotonicity constraint can be partially relaxed. Chen et al. [6] study the fixedparameter tractability of computing egalitarian stable matchings in the setting of srti.
2 The Egalitarian Stable Roommates Problem
In this section we consider the complexity and approximability of the problem of computing an egalitarian stable matching in instances of dsri. We begin by defining the following problems.
Problem 1
 Input:

A solvable instance \(\mathcal {I} = \langle G,\mathcal O\rangle \) of d sri , where G is a graph and \(\mathcal O\) is a set of preference lists, each of length at most d.
 Output:

An egalitarian stable matching M in\(\mathcal I\).
The decision version of egaldsri is defined as follows:
Problem 2
 Input:

\(\mathcal {I} = \langle G,\mathcal O,K^{\prime }\rangle \), where\(\langle G,\mathcal O\rangle \)is a solvable instance\(\mathcal I^{\prime }\)of dsriandK^{′}is an integer.
 Question:

Does\(\mathcal I^{\prime }\)admit a stable matching M with c(M) ≤ K^{′}?
In the following we give a reduction from the NPcomplete decision version of Minimum Vertex Cover in cubic graphs to egal 3sri dec, deriving the hardness of the latter problem.
Theorem 1
egal 3sri decisNPcomplete.
Proof
Clearly egal 3sri dec belongs to NP. To show NPhardness, we begin by defining the NPcomplete problem that we will reduce to egal 3sri dec. □
Problem 3
 Input:

\(\mathcal {I} = \langle G, K \rangle \) ,where G is a cubic graph and K is an integer.
 Question:

Does G contain a vertex cover of size at most K?
Construction of the egal 3sri dec Instance
This finishes the construction of the egal 3sri dec instance \(\mathcal {I}\). In the remainder of the proof we show that G has a vertex cover C where C≤ K if and only if \(\mathcal {I}\) has a stable matching M where c(M) ≤ K^{′}.
Claim 2
IfGhas a vertex cover Csuch that C = k ≤ K, then there is a stable matchingMin\(\mathcal {I}\)such thatc(M) ≤ K^{′}.
Proof
Suppose that G has a vertex cover C such that C = k ≤ K. We construct a matching M in \(\mathcal {I}\) as follows. For each i (1 ≤ i ≤ n), if v_{i} ∈ C, add \({V_{i}^{c}}\) to M, otherwise add \({V_{i}^{u}}\) to M. For each j (1 ≤ j ≤ m), if v_{j,1} ∈ C, add \({E_{j}^{2}}\) to M, otherwise add \({E_{j}^{1}}\) to M. Finally add the pairs in M_{Z} to M.
We now argue that M is stable. Suppose that \({e_{j}^{1}} {e_{j}^{4}}\in M\) for some j (1 ≤ j ≤ m). Then \({E_{j}^{2}}\subseteq M\), so v_{j,1} ∈ C. Let v_{i} = v_{j,1}. Then by construction, \({V_{i}^{c}}\subseteq M\), and hence \({v_{i}^{r}}\) has his first choice for each r (1 ≤ r ≤ 4). Thus \({e_{j}^{1}}\) does not form a blocking edge of M with \(v({e_{j}^{1}})\). The argument is similar if \({e_{j}^{1}} {e_{j}^{2}}\in M\) for some j (1 ≤ j ≤ m). Then \({E_{j}^{1}}\subseteq M\), so v_{j,2} ∈ C. Let v_{i} = v_{j,2}. Then by construction, Vic ⊆ M, and hence \({v_{i}^{r}}\) has his first choice for each r (1 ≤ r ≤ 4). Thus \({e_{j}^{2}}\) does not form a blocking edge of M with \(v({e_{j}^{1}})\). Now suppose that \({v_{i}^{r}} w_{i}^{r+1}\in M\) for some i (1 ≤ i ≤ n) and r (1 ≤ r ≤ 3). Then \({V_{i}^{u}}\subseteq M\), so v_{i}∉C. Let \({e_{j}^{s}}=e({v_{i}^{r}})\). If s = 1 then v_{i} = v_{j,1}. Hence by construction of M, \({E_{j}^{1}}\subseteq M\). Then \({e_{j}^{1}}\) has his firstchoice partner, so \({v_{i}^{r}}\) does not block M with \(e({v_{i}^{r}})\). If s = 2 then v_{i} = v_{j,2}. As v_{j,2}∉C, it follows that v_{j,1} ∈ C as C is a vertex cover. Hence by construction of M, \({E_{j}^{2}}\subseteq M\). Then \({e_{j}^{2}}\) has its firstchoice partner, so \({v_{i}^{r}}\) does not block M with \(e({v_{i}^{r}})\). It is straightforward to verify that M cannot admit any other type of blocking edge, and thus M is stable in \(\mathcal {I}\).
Clearly every agent in \(\mathcal {I}\) is matched in M. We note that Theorem 4.5.2 of [12] implies that every stable matching in \(\mathcal {I}\) matches every agent in \(\mathcal {I}\) – we will use this fact in the next claim. We finally note that c(M) = 4k+12k+9(n−k)+2(n−k)+4(n−k)+7m+4n = 7m+19n + k ≤ K^{′}, considering the contributions from the agents matched in \({V_{i}^{c}}\), \({V_{i}^{u}}\) (1 ≤ i ≤ n), \({E_{j}^{1}}\), \({E_{j}^{2}}\) (1 ≤ j ≤ m) and M_{Z} respectively. □
Claim 3
If there is a stable matchingMin\(\mathcal {I}\)such thatc(M) ≤ K^{′}thenGhas a vertex coverCsuch that C = k ≤ K.
Proof
Suppose that M is a stable matching in \(\mathcal {I}\) such that c(M) ≤ K^{′}. We construct a set of vertices C in G as follows. As M matches every agent in \(\mathcal {I}\), then for each i (1 ≤ i ≤ n), either \({V_{i}^{c}}\subseteq M\) or \({V_{i}^{u}}\subseteq M\). In the former case add v_{i} to C. Also, for each j (1 ≤ j ≤ m), as M matches every agent in \(\mathcal {I}\), either \({E_{j}^{1}}\subseteq M\) or \({E_{j}^{2}}\subseteq M\). Finally, it follows that M_{Z} ⊆ M.
We now argue that C is a vertex cover. Let j (1 ≤ j ≤ m) be given and suppose that v_{j,1}∉C and v_{j,2}∉C. Suppose firstly that \({E_{j}^{1}}\subseteq M\). Let v_{i} = v_{j,2}. Then \({V_{i}^{u}}\subseteq M\) by construction of C, so that \({e_{j}^{2}}\) blocks M with \(v({e_{j}^{2}})\), a contradiction. Now suppose that \({E_{j}^{2}}\subseteq M\). Let v_{i} = v_{j,1}. Then \({V_{i}^{u}}\subseteq M\) by construction of C, so that \({e_{j}^{1}}\) blocks M with \(v({e_{j}^{1}})\), a contradiction. Hence C is a vertex cover in G.
Moreover if k = C then given the composition of M, as noted in the previous claim, c(M) = 7m + 19n + k, and since c(M) ≤ K^{′} it follows that k ≤ K. □
Theorem 1 immediately implies the following result.
Corollary 4
egal 3sriisNPhard.
We remark that egal 2sri is trivially solvable in polynomial time: the components of the graph are paths and cycles in this case, and the cost of a stable matching selected in one component is not affected by the matching edges chosen in another component. Therefore we can deal with each path and cycle separately, minimising the cost of a stable matching in each. Paths and odd cycles admit exactly one stable matching (recall that (i) the instance is assumed to be solvable, and (ii) the set of matched agents is the same in all stable matchings [12, Theorem 4.5.2]), whilst even cycles admit at most two stable matchings (to find them, test each of the two perfect matchings for stability) – we can just pick the stable matching with lower cost in such a case. The following result is therefore immediate.
Proposition 5
egal 2sri admits a lineartime algorithm.
Corollary 4 naturally leads to the question of the approximabilty of egaldsri. As mentioned in the Introduction, Feder [7, 8] provided a 2approximation algorithm for the problem of finding an egalitarian stable matching in an instance of sri. As Theorems 6, 8 and 10 show, this bound can be improved for instances with boundedlength preference lists.
Theorem 6
egal 3sriis approximable within 9/7.
Proof
Let \(\mathcal I\) be an instance of 3sri and let M_{egal} denote an egalitarian stable matching in \(\mathcal I\). First we show that any stable matching in \(\mathcal I\) is a 4/3approximation to M_{egal}. We then focus on the worstcase scenario when this ratio 4/3 is in fact realised. Then we design a weight function on the edges of the graph and apply Teo and Sethuraman’s 2approximation algorithm [25, 26] to find an approximate solution M^{′} to a minimum weight stable matching M_{opt} for this weight function. This weight function helps M^{′} to avoid the worst case for the 4/3approximation for a significant amount of the matching edges. We will ultimately show that M^{′} is in fact a 9/7approximation to M_{egal}. □
Claim 7
In an instance ofegal 3sri, any stable matching approximatesc(M_{egal}) within a factor of 4/3.
Proof

If uv ∈ M_{egal} is a (1,1)pair then u and v contribute 2 to c(M_{egal}) and also 2 to c(M) since they must be also be matched in M (and in every stable matching).

If uv ∈ M_{egal} is a (1,2)pair then u and v contribute 3 to c(M_{egal}) and at most 4 to c(M). Since, if uv ∉ M, then v must be matched to his 1st choice and u to his 2nd or 3rd, because one of u and v must be better off and the other must be worse off in M than in M_{egal}.

If uv ∈ M_{egal} is a (1,3)pair then u and v contribute 4 to c(M_{egal}) and at most 5 to c(M). Since, if uv ∉ M, then v must be matched to his 1st or 2nd choice and u to his 2nd or 3rd.

If uv ∈ M_{egal} is a (2,2)pair then u and v contribute 4 to c(M_{egal}) and at most 4 to c(M). Since, if uv ∉ M, then one must be matched to his 1st choice and the other to his 3rd.

If uv ∈ M_{egal} is a (2,3)pair then u and v contribute 5 to c(M_{egal}) and at most 5 to c(M). Since, if uv ∉ M, then v must be matched to his 1st or 2nd choice and u to his 3rd.

If uv ∈ M_{egal} is a (3,3)pair then u and v contribute 6 to c(M_{egal}) and also 6 to c(M) since they must be also be matched in M (and in every stable matching – this follows by [12, Lemma 4.3.9]).
Hence c(M)/c(M_{egal}) ≤ 4/3 and Claim 7 is proved.
As shown in Claim 7, the only case when the approximation ratio 4/3 is reached is where M_{egal} consists of (1,2)pairs exclusively, while the stable matching output by the approximation algorithm contains (1,3)pairs only. We will now present an algorithm that either delivers a stable solution M^{′} containing at least a significant amount of the (1,2)pairs in M_{egal} or a certificate that M_{egal} contains only a few (1,2)pairs and thus any stable solution is a good approximation.
To simplify our proof, we execute some basic preprocessing of the input graph. If there are any (1,1)pairs in G, then these can be fixed, because they occur in every stable matching and thus can only lower the approximation ratio. Similarly, if an arbitrary stable matching contains a (3,3)pair, then this edge appears in all stable matchings and thus we can fix it. Those (3,3)pairs that do not belong to the set of stable edges can be deleted from the graph. From this point on, we assume that no edge is ranked first or last by both of its end vertices in G and prove the approximation ratio for such graphs.
We designed w(uv) to fit the necessary Ushaped condition of Teo and Sethuraman’s 2approximation algorithm [25, 26]. This condition on the weight function is as follows. We are given a function f_{p} on the neighbouring edges of a vertex p. Function f_{p} is Ushaped if it is nonnegative and there is a neighbour q of p so that f_{p} is monotone decreasing on neighbours in order of p’s preference until q, and f_{p} is monotone increasing on neighbours in order of p’s preference after q. The approximation guarantee of Teo and Sethuraman’s algorithm holds for an edge weight function w(uv) if for every edge uv ∈ E, w(uv) can be written as w(uv) = f_{u}(uv) + f_{v}(uv), where f_{u} and f_{v} are Ushaped functions.
Our w(uv) function is clearly Ushaped, because at each vertex the sequence of edges in order of preference is either monotone increasing or it is (1,0,1). Since w itself is Ushaped, it is easy to decompose it into a sum of Ushaped f_{v} functions, for example by setting \(f_{v}(uv) = f_{u}(uv) = \frac {w(uv)}{2}\) for every edge uv.
Let M denote an arbitrary stable matching, let M^{(1,2)} be the set of (1,2)pairs in M, and let M_{opt} be a minimum weight stable matching with respect to the weight function w(uv). Since M_{opt} is by definition the stable matching with the largest number of (1,2)pairs, M opt(1,2)≥M egal(1,2). We also know that w(M) = M−M^{(1,2)} for every stable matching M.
 1)
2M egal(1,2)−M≤ 0
The derived lower bound for M^{′(1,2)} is negative or zero in this case. Yet we know that at most half of the edges in M_{egal} are (1,2)pairs, and c(e) ≥ 4 for the rest of the edges in M_{egal}. Let us denote M− 2M egal(1,2)≥ 0 by x. Thus, \(M_{\text {egal}}^{(1,2)}=\frac {Mx}{2}\).$$ c(M_{\text{egal}}) \geq \frac{Mx}{2} \cdot 3 + \frac{M+x}{2} \cdot 4 = 3.5 M +0.5x $$(2)We use our arguments in the proof of Claim 7 to derive that an arbitrary stable matching approximates c(M_{egal}) on the \(\frac {Mx}{2}\) (1,2)edges within a ratio of \(\frac {4}{3}\), while its cost on the remaining \(\frac {M+x}{2}\) edges is at most 5. These imply the following inequalities for an arbitrary stable matching M.$$ c(M) \leq \frac{Mx}{2} \cdot 3 \cdot \frac{4}{3} + \frac{M+x}{2} \cdot 5 = 4.5 M + 0.5x $$(3)  2)
2M egal(1,2)−M > 0
Let us denote 2M egal(1,2)−M by \(\hat {x}\). Notice that \(M_{\text {egal}}^{(1,2)} = \frac {\hat {x} + M}{2}\). We can now express the number of edges with cost 3, and at least 4 in M_{egal}.Let M^{′(1,2)} = z_{1}. Then exactly z_{1} edges in M^{′} have cost 3. It follows from (1) that \(z_{1}\geq \hat {x}\). Suppose that z_{2} ≤ z_{1} edges in M^{′(1,2)} correspond to edges in M egal(1,2). Recall that \(M_{\text {egal}}^{(1,2)}=\frac {\hat {x}+M}{2}\). The remaining \(\frac {M+\hat {x}}{2}z_{2}\) edges in M egal(1,2) have cost at most 4 in M^{′}. This leaves \(MM_{\text {egal}}^{(1,2)}(z_{1}z_{2})=\frac {M\hat {x}}{2}z_{1}+z_{2}\) edges in M_{egal} that are as yet unaccounted for; these have cost at most 5 in both M_{egal} and M^{′}. We thus obtain:$$\begin{array}{@{}rcl@{}} c(M_{\text{egal}}) & \geq & 3 \cdot \frac{\hat{x} + M}{2} + 4 \cdot \left( M  \frac{\hat{x} + M}{2}\right) \\ & = & 3.5M 0.5\hat{x} \end{array} $$(4)$$\begin{array}{@{}rcl@{}} c(M^{\prime}) & \leq & 3z_{1} + 4 \left( \frac{M + \hat{x}}{2}z_{2}\right) + 5 \left( \frac{M  \hat{x}}{2}z_{1}+z_{2}\right) \\ &=& 4.5M  0.5\hat{x} 2 z_{1} + z_{2} \\ &\leq& 4.5M 1.5\hat{x} \end{array} $$(5)
We derived that M^{′}, the 2approximate solution with respect to the weight function w(uv) delivers a \(\frac {9}{7}\)approximation in both cases. □
Using analogous techniques we can establish similar approximation bounds for egal 4sri and egal 5sri, as follows.
Theorem 8
egal 4sriis approximable within 11/7.
Proof
We start with a statement analogous to Claim 7. □
Claim 9
In an instance ofegal 4sri, any stable matching approximatesc(M_{egal}) within a factor of 5/3.
Proof
uv edges and the corresponding costs in egal 4sri
uv  Worst case cost at u  Worst case cost at v  Cost ratio 

(1,2)  4  1  5/3 
(1,3)  4  2  6/4 
(1,4)  4  3  7/5 
(2,2)  4  1  5/4 
(2,3)  4  2  6/5 
(2,4)  4  3  7/6 
(3,3)  4  2  6/6 
(3,4)  4  3  7/7 
 1)
2M egal(1,2)−M≤ 0
Let us denote M− 2M egal(1,2)≥ 0 by x. Thus, \(M_{\text {egal}}^{(1,2)}=\frac {Mx}{2}\). Furthermore, let y denote the number of edges with cost at least 5 in M_{egal}.$$\begin{array}{@{}rcl@{}} c(M_{\text{egal}}) & \geq & \frac{Mx}{2} \cdot 3 + \left( \frac{M+x}{2} y \right)\cdot 4 +5y\\ & = & 3.5 M +0.5x +y \end{array} $$$$c(M) \leq \frac{Mx}{2} \cdot 3 \cdot \frac{5}{3} + \left( \frac{M+x}{2} y \right) \cdot 6 + 7y= 5.5 M + 0.5x +y $$$$\begin{array}{llllllll} \frac{c(M)}{c(M_{\text{egal}})} &\leq \frac{5.5M +0.5x+y}{3.5M +0.5x+y} \leq \frac{11}{7}\ \end{array} $$  2)
2M egal(1,2)−M > 0
Let \(\hat {x}\) denote 2M egal(1,2)−M and y the number of edges with cost at least 5 in M_{egal}. Due to Inequality (1), we know that at least \(\hat {x}\) (1,2)pairs in M_{egal} correspond to edges of cost 3 in M^{′}. The remaining \(\frac {M\hat {x}}{2}\) (1,2)pairs in M_{egal} correspond to edges of cost at most 5 in M^{′}.$$c(M_{\text{egal}}) \geq \frac{\hat{x}+M}{2} \cdot 3 +4 \cdot(\frac{M\hat{x}}{2}y) +5y = 3.5 M 0.5 \hat{x} +y$$$$c(M^{\prime}) \leq 3\hat{x} + 5\cdot \frac{M\hat{x}}{2} +6\cdot (\frac{M\hat{x}}{2} y) + 7y = 5.5M2.5\hat{x} +y$$
Theorem 10
egal 5sriis approximable within 13/7.
Proof
Again we start with a statement analogous to Claim 7. □
Claim 11
In an instance ofegal 5sri, any stable matching approximatesc(M_{egal}) within a factor of 2.
Proof
uv edges and the corresponding costs in egal 5sri
uv  Worst case cost at u  Worst case cost at v  Cost ratio 

(1,2)  5  1  6/3 
(1,3)  5  2  7/4 
(1,4)  5  3  8/5 
(1,5)  5  4  9/6 
(2,2)  5  1  6/4 
(2,3)  5  2  7/5 
(2,4)  5  3  8/6 
(2,5)  5  4  9/7 
(3,3)  5  2  7/6 
(3,4)  5  3  8/7 
(3,5)  5  4  9/8 
(4,4)  5  3  8/8 
(4,5)  5  4  9/9 
 1)
2M egal(1,2)−M≤ 0
Let us denote M− 2M egal(1,2)≥ 0 by x. Thus, \(M_{\text {egal}}^{(1,2)}=\frac {Mx}{2}\). Furthermore, let y be the number of edges with cost 5 and z the number of edges with cost at least 6 in M_{egal}.$$\begin{array}{@{}rcl@{}} c(M_{\text{egal}}) & \geq & \frac{Mx}{2} \cdot 3 + \left( \frac{M+x}{2} y z \right)\cdot 4 +5y + 6z \\ & = & 3.5 M +0.5x +y + 2z \end{array} $$$$\begin{array}{@{}rcl@{}} c(M) & \leq & \frac{Mx}{2} \cdot 3 \cdot \frac{6}{3} + \left( \frac{M+x}{2} y z \right) \cdot 7 + 8y +9z \\ & = & 6.5 M + 0.5x +y + 2z \end{array} $$$$\begin{array}{llllllll} \frac{c(M)}{c(M_{\text{egal}})} &\leq \frac{6.5 M + 0.5x +y + 2z}{3.5 M +0.5x +y + 2z} \leq \frac{13}{7}\ \end{array} $$  2)
2M egal(1,2)−M > 0
Let \(\hat {x}\) denote 2M egal(1,2)−M, y the number of edges with cost 5 and z the number of edges with cost at least 6 in M_{egal}.$$c(M_{\text{egal}}) \geq \frac{\hat{x}+M}{2} \cdot 3 +4 \cdot(\frac{M\hat{x}}{2}yz) +5y +6z= 3.5 M 0.5 \hat{x} +y +2z$$$$c(M^{\prime}) \leq 3\hat{x} + 6\cdot \frac{M\hat{x}}{2} +7\cdot (\frac{M\hat{x}}{2} yz) + 8y +9z = 6.5M3.5\hat{x} +y +2z$$
Using a similar reasoning for each d ≥ 6, our approach gives a c_{d}approximation algorithm for egaldsri where c_{d} > 2. In these cases the 2approximation algorithm of Feder [7, 8] should be used instead.
3 Solvability and Moststable Matchings in dsrti
In this section we study the complexity and approximability of the problem of deciding whether an instance of dsrti admits a stable matching, and the problem of finding a moststable matching given an instance of dsrti.
We begin by defining two problems that we will be studying in this section from the point of view of complexity and approximability.
Problem 4
 Input:

\(\mathcal {I} = \langle G, \mathcal O\rangle \),whereGis a graph and\(\mathcal O\)is a set of preference lists, each of length at most d, possibly involving ties.
 Question:

Is \(\mathcal {I}\) solvable?
Problem 5
 Input:

An instance\(\mathcal I\)of dsrti.s
 Output:

A matchingMin\(\mathcal {I}\)such that\(bp(M)=bp(\mathcal I)\).
We will show that solvable 3srti is NPcomplete and min bp 3srti is hard to approximate. In both cases we will use a reduction from the following satisfiability problem:
Problem 6
 Input:

\(\mathcal {I} = B\) ,where B is a Boolean formula in CNF, in which each clause comprisesexactly 3 literals and each variable appears exactly twice in unnegated andexactly twice in negated form.
 Question:

Is there a truth assignment satisfying B?
(2,2)e3sat is NPcomplete, as shown by Berman et al. [2]. We begin with the hardness of solvable 3srti.
Theorem 12
solvable 3srtiisNPcomplete.
Proof
When constructing G, we will keep track of the order of the three literals in each clause of B and the order of the two unnegated and two negated occurrences of each variable in B. Each of these four occurrences of each variable is represented by an interconnecting edge.
A variable gadget for a variable x_{i} (1 ≤ i ≤ n) of B comprises the 4cycle 〈\({v_{i}^{1}}\), \({v_{i}^{2}}\), \({v_{i}^{3}}\), \({v_{i}^{4}}\)〉 with cyclic preferences. Each of these four vertices is incident to an interconnecting edge. These edges end at specific vertices of clause gadgets. The clause gadget for a clause c_{j} (1 ≤ j ≤ m) contains 20 vertices, three of which correspond to the literals in c_{j}; these vertices are also incident to an interconnecting edge.
Due to the properties of (2,2)e3sat, x_{i} occurs twice in unnegated form, say in clauses c_{j} and c_{k} of B. Its first appearance, as the r th literal of c_{j} (1 ≤ r ≤ 3), is represented by the interconnecting edge between vertex \({v_{i}^{1}}\) in the variable gadget corresponding to x_{i} and vertex \({a_{j}^{r}}\) in the clause gadget corresponding to c_{j}. Similarly the second occurrence of x_{i}, say as the s th literal of c_{k} (1 ≤ s ≤ 3) is represented by the interconnecting edge between \({v_{i}^{3}}\) and \({a_{k}^{s}}\). The same variable x_{i} also appears twice in negated form. Appropriate avertices in the gadgets representing those clauses are connected to \({v_{i}^{2}}\) and \({v_{i}^{4}}\). We remark that this construction involves a gadget similar to one presented by Biró et al. [4] in their proof of the NPhardness of min bp 3sri.
Now we prove that there is a truth assignment satisfying B if and only if there is a stable matching M in \(\mathcal I\). □
Claim 13
For any truth assignment satisfying B,a stable matchingMcan be constructedin\(\mathcal I\).
Proof
If a variable x_{i} (1 ≤ i ≤ n) is assigned to be true, \({M_{i}^{T}}\) is added to M, otherwise \({M_{i}^{F}}\) is added. Similarly, since at least one literal in c_{j} (1 ≤ j ≤ m) is true, let r (1 ≤ r ≤ 3) be the minimum integer such that the literal at position r of c_{j} is true; add \({M_{j}^{r}}\) to M. The intuition behind this choice is that if a literal is true, then the vertex representing it in the variable gadget is matched to its best choice. On the other hand, if some literals in a clause are true, then the vertex representing the appearance of one of them in that clause is matched to its lastchoice vertex.
We claim that no edge blocks M. Checking the edges in the clause and variable gadgets is easy. The five special matchings were designed in such a way that no edge within the gadgets blocks them. More explanation is needed regarding the interconnecting edges. Suppose one of them, \({a_{j}^{r}} {v_{i}^{s}}\), (r ∈ {1,2,3}, s ∈ {1,2,3,4}) blocks M. Since M is a perfect matching, \({a_{j}^{r}}\) needs to be matched to its last choice, a qvertex. Similarly, \({v_{i}^{s}}\) has to be matched to its worst partner. While the partner of \({a_{j}^{r}}\) indicates that the literal represented by \({v_{i}^{s}}\) (x_{i} or \(\bar {x}_{i}\)) is true in the clause, the partner of \({v_{i}^{s}}\) means that the literal is false. □
Claim 14
For any stable matching M in \(\mathcal I\) , there is a truth assignment satisfying B.
Proof
In the next three paragraphs we show that the restriction of M to any variable or clause gadget is one of the above listed special matchings, and no interconnecting edge is in M.
First of all, if a vertex u is the only first choice of another vertex, then u certainly needs to be matched in M. This property is fulfilled for all vertices of all clause gadgets except for \({y_{j}^{3}}\) and \({z_{j}^{3}}\) for each c_{j} (1 ≤ j ≤ m). Let us first study clause gadget c_{j}. If \({y_{j}^{4}}\) is matched to \({y_{j}^{2}}\), then \({y_{j}^{2}}{y_{j}^{3}}\) blocks M. Thus, \({y_{j}^{3}} {y_{j}^{4}}\), and similarly, \({z_{j}^{3}}{z_{j}^{4}}\) are part of M for all clause gadgets.
Our proof for clause gadgets from this point involves considering matchings covering all twelve remaining vertices. We differentiate two possible cases, depending on the partner of \({p_{j}^{3}}\). In the first case, \({p_{j}^{3}}\)\({b_{j}^{3}}\) ∈ M. Therefore, \({p_{j}^{2}}\)\({p_{j}^{1}}\) ∈ M too, because \({p_{j}^{2}}\) has to be matched. For similar reasons, \(\{ {b_{j}^{1}}{a_{j}^{1}}, {b_{j}^{2}}{a_{j}^{2}}, {q_{j}^{1}}{q_{j}^{2}}, {q_{j}^{3}}{a_{j}^{3}}\}\) ⊆ M. This gives us matching \({M_{j}^{3}}\). In the second case, if \({p_{j}^{3}}\) is matched to \({p_{j}^{2}}\), then \(\{ {b_{j}^{3}}{a_{j}^{3}}, {q_{j}^{3}}{q_{j}^{2}} \} \subseteq M\). There are two possible matchings on the remaining six vertices: \(\{{p_{j}^{1}}{b_{j}^{1}}, {a_{j}^{1}}{q_{j}^{1}}, {b_{j}^{2}}{a_{j}^{2}}\}\) and \(\{{p_{j}^{1}}{b_{j}^{2}}, {q_{j}^{1}} {a_{j}^{2}},{b_{j}^{1}}{a_{j}^{1}}\}\). These two matchings together with the lower part of the gadget form \({M_{j}^{1}}\) and \({M_{j}^{2}}\).
Since all avertices have a partner within their clause gadgets, no interconnecting edge can be a part of M. For the variable gadgets, it is straightforward to see that \({M_{i}^{T}}\) and \({M_{i}^{F}}\) are the only matchings covering all vertices of the 4cycles.
The truth assignment to B is then defined in the following way. Each variable whose gadget has the edges of \({M_{i}^{T}}\) in M is assigned to be true, while all other variables with \({M_{i}^{F}}\) on their gadgets are false.
All that remains is to show that this is indeed a truth assignment. Suppose that there is an unsatisfied clause c_{j} in B. Since all three of c_{j}’s literals are false, every vertex \({v_{i}^{r}}\) (1 ≤ i ≤ n) such that \({v_{i}^{r}} {a_{j}^{s}}\) is an interconnecting edge prefers \({a_{j}^{s}}\) to its partner in M (1 ≤ s ≤ 3). Hence a blocking edge can only be avoided if \({a_{j}^{1}}{b_{j}^{1}}\), \({a_{j}^{2}}{b_{j}^{2}}\) and \({a_{j}^{3}}{b_{j}^{3}}\) are all in M, which never occurs in any stable matching as shown above.
This finishes the proof of Theorem 12. □
Our construction shows that the complexity result holds even if the preference lists are either strictly ordered or consist of a single tie of length two. Moreover, Theorem 12 also immediately implies the following result.
Corollary 15
min bp 3srtiisNPhard.
The following result strengthens Corollary 15.
Theorem 16
min bp 3srtiis not approximable within n^{1−ε}, for anyε > 0, unless P = NP, where n is the number of agents.
Proof
The core idea of the proof is to gather several copies of the 3srti instance created in the proof of Theorem 12, together with a small unsolvable 3srti instance. By doing so, we create a min bp 3srti instance \(\mathcal {I}\) in which \(bp(\mathcal {I})\) is large if the Boolean formula B (originally given as an instance of (2,2)e3sat) is not satisfiable, and \(bp(\mathcal {I}) = 1\) otherwise. Therefore, finding a good approximation for \(\mathcal I\) will imply a polynomialtime algorithm to decide the satisfiability of B. Our proof is similar to that of an analogous inapproximabilty result for the problem of finding a moststable matching in an instance of the Hospitals / Residents problem with Couples [3].
The smallest unsolvable instance of 3srti is a 3cycle with cyclic strict preferences. Aside from this, we add k disjoint copies of 3srti instance created in the proof of Theorem 12 (from the same Boolean formula B), for large enough k. In particular we let c = ⌈2/ε⌉ and \(k = {n_{0}^{c}}\), where n_{0} is the number of variables in B. We use m_{0} to denote the number of clauses in B. Let \(\mathcal I\) be the instance of 3srti that has been constructed. Due to the proof of Theorem 12 above, if B is satisfiable then \(bp(\mathcal I)=1\), and if B is not satisfiable then \(bp(\mathcal I)\geq k+1\). Hence a kapproximation algorithm for min bp 3srti could be used to solve (2,2)e3sat in polynomial time.
In Equality (6) can be deduced by inspection of the 3srti instance constructed in the proof of Theorem 12. In step (7) we substitute \(m_{0}=\frac {4n_{0}}{3}\), which follows from the structure of B. We can assume without loss of generality that kn_{0} ≥ 3, which we use in Inequality (8). Finally, in Equality (9) we substitute \(k = {n_{0}^{c}}\).
Inequality (11) is obtained by raising n to the power of each side of Inequality (10). Inequality (12) follows from the bound for n established in Inequalities (6)–(9). Now in Inequality (13) we can assume without loss of generality that n_{0} ≥ 32 and use that \(\frac {c1}{c+1} < 1\). In the last step, we use the definition of k. □
To complete the study of cases of min bpdsrti, we establish a positive result for instances with degree at most 2.
Theorem 17
min bp 2srti is solvable in\(\mathcal {O}(V)\)time.
Proof
For an instance \(\mathcal I\) of min bp 2srti, clearly every component of the underlying graph G is a path or cycle. We claim that \(bp(\mathcal {I})\) equals the number of odd parties in G, where an odd party is a cycle C = 〈v_{1},v_{2},...,v_{k}〉 of odd length, such that v_{i} strictly prefers v_{i+1} to v_{i−1} (addition and subtraction are taken modulo k).
Since an odd party never admits a stable matching, \(bp(\mathcal {I})\) is bounded below by the number of odd parties [24]. This bound is tight: by taking an arbitrary maximum matching in an odd party component, a moststable matching is already reached. Now we show that a stable matching M can be constructed in all other components.
Each component that is not an odd cycle is therefore a bipartite subgraph (indeed either a path or an even cycle). Such a subgraph therefore gives rise to the restriction of srti called the Stable Marriage problem with Ties andIncomplete lists (smti). An instance of smti always admits a stable solution and it can be found in linear time [22]. Thus these components contribute no blocking edge.
Regarding oddlength cycles that are not odd parties, we will show that there is at least one vertex not strictly preferred by either of its adjacent vertices. Leaving this vertex uncovered and adding a perfect matching in the rest of the cycle results in a stable matching.
Assume that every vertex along a cycle C_{k} (where k is an odd number) is strictly preferred by at least one of its neighbours. Since each of the k vertices is strictly preferred by at least one vertex, and a vertex v can prefer at most one other vertex strictly, every vertex along C_{k} has a strictly ordered preference list. Now every vertex can point at its unique firstchoice neighbour. To avoid an odd cycle, there must be a vertex pointed at by both of its neighbours. This implies that there is also a vertex v pointed at by no neighbour, and v is hence ranked second by both of its neighbours. □
4 Egalitarian Stable Matchings in srti

When considering the approximability of egaldsri, we restricted attention to the case of solvable instances, in the knowledge that solvability can be determined in linear time [15]. However in the case of srti, we can no longer assume this, since solvable 3srti is NPcomplete as Theorem 12 shows.

In instances of egaldsri, not all agents are necessarily matched in all stable matchings, but due to Theorem 4.5.2 of [12], which states that the same agents are matched in all stable matchings, we can discard unmatched agents and consider only the remaining agents when reasoning about approximation algorithms. There is no analogue of Theorem 4.5.2 in the case of dsrti (indeed, stable matchings can be of different sizes in a given instance of srti [17]). This means that any approximation algorithm for the problem of finding an egalitarian stable matching in an instance of srti would need to consider the cost of an unmatched agent in a given stable matching, and the choice of value for such a case is not universally agreed upon in the literature. Chen et al. [6] study the fixedparameter tractability of egal srti under different choices of cost value for an unmatched agent, namely 0, some positive constant and the length of its preference list.

Similarly in the case of srti, the choice of value for the rank of an agent a_{j} in a given agent a_{i}’s preference list is again not universally agreed upon – for example if a_{i} has a tie of length 2 at the head of his preference list, followed strictly by a_{j}, then rank(a_{i},a_{j}) could reasonably be defined to be either 2 or 3 depending on the definition adopted. In most competitions, everybody in the tie receives the rank that directly follows the number of agents ranked strictly higher than them, which would be 3 in the previous example. On the other hand, setting the rank to the number of ties (of any cardinality) in the list up to the current tie is the correct way of dealing with this issue in markets where agents rank their possible partners into wellseparated tiers and the cardinalities of these do not matter as much as the tier they end up being matched to – this principle assigns 2 to rank(a_{i},a_{j}) in the example above.
5 Open Questions
Theorems 6, 8 and 10 improve on the best known approximation factor for egaldsri for small d. It remains open to come up with an even better approximation or to establish an inapproximability bound matching our algorithm’s guarantee. A more general direction is to investigate whether the problem of finding a minimum weight stable matching can be approximated within a factor less than 2 for instances of dsri for small d. Finally, the various alternatives regarding the definition of an egalitarian stable matching in instances of srti open the gate to a number of questions.
Notes
Acknowledgements
We thank the anonymous reviewers of this paper and an earlier version of it for their valuable comments, which helped to improve the presentation.
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