An improved approximation lower bound for finding almost stable maximum matchings

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  • Information Processing Letters 109 (2009) 10361040

    Contents lists available at ScienceDirect



    A unm

    K i b,a G oto 606b A machi,




    problaveBirs minithin

    lists are of length at most 3. In this paper, we improve this constant to n for any





    00doywords:proximation algorithmse stable marriage problemproximation ratiolynomial-time reduction

    > 0, where n is the number of men in an input. 2009 Elsevier B.V. All rights reserved.


    The stable marriage problem (SM for short), introduced byle and Shapley [3] (see also [7]), is dened as follows:instance consists of the same number n of men and

    omen and each persons preference list. A preference lista totally-ordered list that includes all members of theposite sex in accordance with the owners preference.e problem requires to nd a stable matching, a perfectatching M between men and women in which there ispair of man m and woman w , that are not matched

    gether in M but each prefers the other to ones partnerM . Such a pair (m,w) is called a blocking pair for M .le and Shapley [3] proved that there is at least one sta-e matching in any instance, and proposed an O (n2)-timegorithm to nd one.One possible extension of SM is to allow incompleteeference lists (SMI for short); namely, each person in-udes a subset of the members of the opposite sex in the

    Corresponding author.E-mail addresses: (K. Hamada), (K. Iwama),

    preference list. Those who are included in a person pspreference list are said to be acceptable to p. Now a match-ing is dened as a set of disjoint pairs of mutually accept-able man and woman, and hence is not necessarily perfect.Accordingly, the denition of a blocking pair is extended asfollows: A mutually acceptable pair of man m and womanw is a blocking pair for a matching M if (i) m and w arenot matched together in M , (ii) either m is single or prefersw to his partner in M , and (iii) either w is single or prefersm to her partner in M . There can be many stable match-ings for one instance, but all stable matchings are of thesame size [6].

    However, if we do not care about the stability, therecan be larger matchings in general. So, we may some-times want to obtain larger matchings by sacricing thestability, but even in such a case, it is still natural to seeka matching which is as stable as possible. Related to thisconsideration, Bir et al. [2] dened the following opti-mization problem, called MAX SIZE MIN BP SMI: Givenan SMI instance, nd a matching that minimizes the num-ber of blocking pairs among all the maximum cardinalitymatchings. For integers p and q, MAX SIZE MIN BP (p,q)-SMI is the restriction of MAX SIZE MIN BP SMI so thateach mans preference list is of length at most p, and each

    20-0190/$ see front matter 2009 Elsevier B.V. All rights reserved.i:10.1016/j.ipl.2009.06.008Information Pro


    n improved approximation lower boaximum matchings

    oki Hamada a, Kazuo Iwama a, Shuichi Miyazak

    raduate School of Informatics, Kyoto University, Yoshida-Honmachi, Sakyo-ku, Kycademic Center for Computing and Media Studies, Kyoto University, Yoshida-Hon

    r t i c l e i n f o a b s t r a c t

    ticle history:ceived 12 February 2009ceived in revised form 1 June 2009cepted 19 June 2009ailable online 23 June 2009mmunicated by F.Y.L. Chin

    In the stable marriagefor a given instance hbe larger matchings.matching that containis not approximable wssing Letters


    d for nding almost stable

    -8501, JapanSakyo-ku, Kyoto 606-8501, Japan

    em that allows incomplete preference lists, all stable matchingsthe same size. However, if we ignore the stability, there canet al. dened the problem of nding a maximum cardinalityimum number of blocking pairs. They proved that this problemsome constant > 1 unless P = NP, even when all preference


  • K. Hamada et al. / Information Processing Letters 109 (2009) 10361040 1037



































    Aomans preference list is of length at most q. p = q = means that the lengths of preference lists arebounded. Bir et al. [2] showed the following results;) MAX SIZE MIN BP (,)-SMI is NP-hard and cannotapproximated within the ratio of n1 for any constant

    > 0, unless P = NP; (2) MAX SIZE MIN BP (3,3)-SMI isPX-hard and cannot be approximated within the ratio

    35573556 1.00028 unless P = NP; (3) MAX SIZE MIN BP

    ,)-SMI is solvable in O (n3) time.In this paper, we improve the hardness of the above (2),mely, we improve the constant 35573556 to n

    1 for any con-ant > 0. Our reduction uses basically the same ideathe one used in [2] to prove the above (1). In [2],

    me persons need to have preference lists of unboundedngths for two reasons: One is for garbage collection,d the other is to create a large gap on the costs be-een yes-instances and no-instances. We perform an-trivial modication of the construction and demon-rate that such gadgets can be replaced by persons witheference lists of length at most three.Research on nding almost stable matchings is very ac-

    ve recently. Abraham et al. [1] consider the problem ofding a matching with the fewer blocking pairs in theable roommates problem, and proved that in most vari-ts the problem is hard, even to approximate. In addi-on to the number of blocking pairs, there are some othernitions of instability, such as the number of agents in-lved in blocking pairs and the number of blocking pairslative to the size of the matching (see [4,5] for example).

    Main result

    eorem 2.1.MAX SIZE MIN BP (3,3)-SMI is not approximableithin n1 where n is the number of men in a given instance,r any > 0, unless P = NP.

    oof. We demonstrate a polynomial-time reduction frome same problem as [2], EXACT Maximal Matching (EXACT-M) restricted to subdivision graphs of cubic graphs,hich is NP-complete [8]. A graph G is a subdivision graphit is obtained from another graph H by replacing eachge (u, v) of H by two edges (u,w) and (w, v) whereis a new vertex. In this problem, we are given a graphwhich is a subdivision graph of some cubic graph, asell as a positive integer K , and asked if G contains aaximal matching of size exactly K . Hereafter, we sim-y say EXACT-MM to mean EXACT-MM with the abovestrictions.Given an instance (G, K ) of EXACT-MM, we constructinstance I of MAX SIZE MIN BP (3,3)-SMI in such a wayat (i) I has a perfect matching, (ii) if (G, K ) is a yes-stance of EXACT-MM, then I has a perfect matching withall number of blocking pairs, and (iii) if (G, K ) is a no-stance of EXACT-MM, then any perfect matching of I hasany blocking pairs.

    )-gadget. Before going to the main body of the reduc-

    on, we rst introduce the(mr

    )-gadget. This gadget plays a

    le of garbage collection, just as X and Y in the proof ofeorem 1 of [2].1i : di1 b2i b1i b1i : a1i xi2i : di2 b3i b2i b2i : a2i a1i3i : di3 b4i b3i b3i : a3i a2i



    .r1i : dir1 bri br1i br1i : ar1i ar2iri : dir bri bri : ari ar1i

    d1j : c2j a j12j : d2j d1j d2j : c3j c2j a j23j : d3j d2j d3j : c4j c3j a j34j : d4j d3j d4j : c5j c4j a j4



    .m1j : dm1j dm2j dm1j : cmj cm1j a jm1mj : dmj dm1j dmj : cmj a jm

    Fig. 1. Preference lists of C(X, r).

    Let X be a set of men of size m where X = {x1, . . . , xm},d r (0 < r m) be an integer. The


    )-gadget (with

    spect to X and r), denoted C(X, r), consists of the fol-wing 2mr r men (1im Ai) (

    1 jr C j) and 2mr

    omen (

    1im Bi) (

    1 jr D j).

    i ={a ji : 1 j r


    i ={b ji : 1 j r

    }(1 i m),

    j ={cij: 2 i m


    j ={dij: 1 i m

    }(1 j r).

    Each persons preference list is dened in Fig. 1. A per-n ps preference list p : a b c means that p prefers a, b,d c in this order. For each xi X , the unique woman b1iC(X, r) who includes xi in her preference list is referredas C(X, r)[xi].The role of the gadget C(X, r) is to receive any subset

    X such that |X | = r without creating many blockingirs, as formally stated in the following lemmas. In thellowing lemmas, we assume that each man xi X in-udes the woman C(X, r)[xi] (= b1i ) in his preference list.

    mma 2.2. Let X be a set of men and r be an integer suchat 0 < r |X |. Then, for any X X such that |X | = r, therea matching M for X and C(X, r) such that (i) all members of(X, r) are matched, (ii) all men in X are matched with women

    C(X, r) and all men in X \ X are single, and (iii) no personX is included in a blocking pair, and the number of blockingirs for M is at most r.

    oof. Let X = {xi1 , xi2 , . . . , xir } (1 i1 < i2 < < ir ). We construct the matching M as follows. For each j j r), add the following pairs to M: (aki j ,b

    k+1i j

    ) for

    = 1, . . . , j 1; (aki j ,bki j ) for k = j + 1, . . . , r; (aji j,d

    i jj );

    k+1j ,d

    kj) for k = 1, . . . , i j 1; (ckj,dkj) for k = i j +1, . . . ,m;

    d (xi j ,b1i j). (Fig. 2 gives an example for a specic i j .)

    lso, for each i such that xi X \ X , add (aki ,bki ) for

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    LeMUFig. 2. A part of the matching described in the proof of Lemma 2.2.

    = 1, . . . , r to M . It is easy to see that (i) and (ii) are satis-d. Also, it is straightforward to check that blocking pairs

    e only (ci jj ,d

    i jj ) (1 j r, i j = 1), and hence there are at

    ost r blocking pairs. mma 2.3. Let X be a set of men and r be an integer suchat 0 < r |X |. Let M be any matching for X and C(X, r) thatatches all members of C(X, r). Then the number of single menX is |X | r.

    oof. This is obvious because any member in C(X, r) in-udes only persons in C(X, r) and X in the preference list,d there are r more women than men in C(X, r). When X is a set of women, we similarly dene the



    dget by exchanging the roles of men and women.

    ain part of the reduction. Let I = (G, K ) be an instanceEXACT-MM, where G is a subdivision graph of somebic graph and K is a positive integer. Since G is a bi-rtite graph, we can write it as G = (U ,W , E) such that= {u1, . . . ,un1 } and W = {w1, . . . ,wn2 }, where each ver-x in U (W , respectively) has degree exactly 2 (3, re-ectively). (Hence n1 and n2 are related as 2n1 = 3n2.)1i : z1i w

    i,pipi z

    1i : u1i u2i

    2i : z1i z2i w

    i,qiqi z

    2i : u2i g1i,1

    1i,1 : z2i h1pi ,i,pi e

    1i,1 e

    1i,1 : g1i,1 g2i,1

    2i,1 : e1i,1 h2pi ,i,pi e

    2i,1 e

    2i,1 : g2i,1 g3i,1

    3i,1 : e2i,1 h3pi ,i,pi e

    3i,1 e

    3i,1 : g3i,1 g4i,1




    C1i,1 : eC2i,1 hC1pi ,i,pi e

    C1i,1 e

    C1i,1 : gC1i,1 gCi,1

    Ci,1 : eC1i,1 hCpi ,i,pi e

    Ci,1 e

    Ci,1 : gCi,1 g1i,2

    1i,2 : eCi,1 h1qi ,i,qi e

    1i,2 e

    1i,2 : g1i,2 g2i,2

    2i,2 : e1i,2 h2qi ,i,qi e

    2i,2 e

    2i,2 : g2i,2 g3i,2

    3i,2 : e2i,2 h3qi ,i,qi e

    3i,2 e

    3i,2 : g3i,2 g4i,2




    C1i,2 : eC2i,2 hC1qi ,i,qi e

    C1i,2 e

    C1i,2 : gC1i,2 gCi,2

    Ci,2 : eC1i,2 hCqi ,i,qi e

    Ci,2 e

    Ci,2 : gCi,2 u0i

    0i : eCi,2 C

    (U0,n1 K



    Fig. 3. Preference lists of U(ui).

    ithout loss of generality, we may assume that K 0, dene

    = 3/ and C = (n1 + n2)B+1.For each vertex ui U , we construct 2C + 3 men and+ 2 women, whose preference lists are given in Fig. 3,

    here mens lists are given in the left and womens listse given in the right of the gure. We denote U(ui)e set of these men and women. Dene the set U0 =01, . . . ,u

    0n1 } (consisting of men, one from each U(ui) (1

    n1)). We then construct( n1n1K

    )-gadget C(U0,n1 K ).

    Similarly, for each w j W , we construct 3C + 3 mend 3C + 4 women, whose preference lists are given ing. 4. We denote W(w j) the set of these men and the set W 0 = {w01, . . . ,w0n2 } (consisting of women,e from each W(w j) (1 j n2)), and construct

    ( n2n2K


    dget C(W 0,n2 K ).The reduction is now completed. The resulting instance

    contains the same number n = (2 + 2C + 2n1 2K )n1 ++ 3C + 2n2 2K )n2 + K of men and women. Note thatch persons preference list is of length at most three. Itnot hard to see that the reduction can be performed ine polynomial in the size of I .

    operties of gadgets. Before proceeding to the correctnessoof, we prove useful lemmas:

    mma 2.4. For any edge (ui,w j) E, we can form amatchingrestricted to people in U(ui)W(w j) so that (i) all people in(ui)W(w j) are matched, (ii) M contains at most 2 blocking

  • K. Hamada et al. / Information Processing Letters 109 (2009) 10361040 1039




























    paI ,a





    ( f



















    Kscfocoti1j : w1j w2j w1j : v1j u

    j,r jr j

    2j : w2j w3j w2j : v1j v2j u

    j,s js j

    3j : w3j h1j,1 w3j : v2j v3j u

    j,t jt j

    1j,1 : h1j,1 h2j,1 h1j,1 : v3j g1r j , j,r j f


    2j,1 : h2j,1 h3j,1 h2j,1 : f 1j,1 g2r j , j,r j f


    3j,1 : h3j,1 h4j,1 h3j,1 : f 2j,1 g3r j , j,r j f





    C1j,1 : hC1j,1 hCj,1 hC1j,1 : f C2j,1 gC1r j , j,r j f


    Cj,1 : hCj,1 h1j,2 hCj,1 : f C1j,1 gCr j , j,r j f


    1j,2 : h1j,2 h2j,2 h1j,2 : f Cj,1 g1s j , j,s j f


    2j,2 : h2j,2 h3j,2 h2j,2 : f 1j,2 g2s j , j,s j f


    3j,2 : h3j,2 h4j,2 h3j,2 : f 2j,2 g3s j , j,s j f





    C1j,2 : hC1j,2 hCj,2 hC1j,2 : f C2j,2 gC1s j , j,s j f


    Cj,2 : hCj,2 h1j,3 hCj,2 : f C1j,2 gCs j , j,s j f


    1j,3 : h1j,3 h2j,3 h1j,3 : f Cj,2 g1t j , j,t j f


    2j,3 : h2j,3 h3j,3 h2j,3 : f 1j,3 g2t j , j,t j f


    3j,3 : h3j,3 h4j,3 h3j,3 : f 2j,3 g3t j , j,t j f





    C1j,3 : hC1j,3 hCj,3 hC1j,3 : f C2j,3 gC1t j , j,t j f


    Cj,3 : hCj,3 w0j hCj,3 : f C1j,3 gCt j , j,t j f


    w0j : f Cj,3 C(W 0,n2 K



    Fig. 4. Preference lists of W(w j).

    irs, and (iii) for any extension of M to a complete matching ofno person in U(ui) W(w j) will create a blocking pair withperson not in U(ui) W(w j).

    oof. We construct a matching M as follows. Sincei,w j) E , there are integers k and l such that j,i = kd i, j = l, by the denition of and . We rst addki ,w

    lj) to M . Next, we consider people in U(ui). Add the

    llowing pairs to M: (g1i,1, z2i ); (g

    si,1, e

    s1i,1 ) for s = 2, . . . ,C ;

    1i,2, e

    Ci,1); (g

    si,2, e

    s1i,2 ) for s = 2, . . . ,C ; and (u0i , eCi,2). If

    = 1, then add (u2i , z1i ), otherwise, add (u1i , z1i ). Finally, wensider people in W(w j). Add the following pairs to M:3j ,h

    1j,1); ( f


    s+1j,1 ) for s = 1, . . . ,C 1; ( f Cj,1,h1j,2);


    s+1j,2 ) for s = 1, . . . ,C 1; ( f Cj,2,h1j,3); ( f sj,3,hs+1j,3 ) for

    = 1, . . . ,C 1; and ( f Cj,3,w0j ). If l = 1, add (v1j ,w2j ) and2j ,w

    3j ); if l = 2, add (v1j ,w1j ) and (v2j ,w3j ); if l = 3, add

    1j ,w

    1j ) and (v

    2j ,w

    2j ).

    It is straightforward to verify that condition (i) is satis-d. To see that conditions (ii) and (iii) hold, observe thellowing: In U(ui), all men of the form gsi,t for any t, s,d u0i are matched with their rst choices. Clearly, theseen do not form a blocking pair. Also, women who in-ude only these men in their preference lists cannot formblocking pair. So, only u1i , u



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