Disposiciones generales

master lists of women and men denoted PW and PM as follows. Let

PW = hz1 z2 ... zm P0 P1 ... Pn−1 t11 t21 t12 t22 ... t1m t2mi

where Pi = hy4i c(x4i) y4i+1 c(x4i+1) c(x4i+3) y4i+3 c(x4i+2) y4i+2i

It is easy to see how all the men’s preference lists from Figure 5.5 can be derived from PW.

PM = hp11 p21 p31 p12 p22 p32 ... p1m p2m p3m x0 x1 ... xn−1

q₁1 q₁2 q3₁ q₂1 q₂2 q₂3 ..., q1_m q_m2 q3_mi

It is easy to see how all the women’s preference lists from Figure 5.5 can be derived from PM. The social network graph G is constructed as follows: the vertices are X ∪C0,

and the edges are {xk, c(xk)} (0 ≤ k ≤ 4n − 1).

We claim that B is satisfiable if and only if I admits a complete socially stable matching. Similar arguments to those used in the proof of Theorem 5.3.1 hold for the proof of this claim. The only difference is the construction of the complete socially stable matching from a satisfying truth assignment. In the proof of Theorem 5.3.3, we had (qj, csj) ∈ M

for each cj ∈ C where cj contained a literal at position s ∈ {1, 2, 3} that is true. Here

we add (qs

j, csj) to M . For the other two men qs

0

j and qs

00

j (where {s, s

0_{, s}00_{} = {1, 2, 3}),}

we assume without loss of generality that s0 < s00. We add (qs0

j , t1j) and (qs

00

j , t2j) to M ,

thus ensuring that M is a complete socially stable matching. The rest of the arguments presented in Theorem 5.3.1 follow naturally.

5.4

Minimum socially stable matchings

In this section we investigate the problem min smiss-d which is defined as follows. Given an smiss instance (I, G) and an integer k, decide whether I admits a socially stable matching of size at most k. We reduce from the minimum maximal matching problem which is defined as follows. Given a graph G and a positive integer k, decide whether G admits a maximal matching M such that |M |≤ k. minimum maximal matching is known to be NP-complete even for bipartite graphs where every left hand vertex has degree 2 and every right hand vertex has degree of exactly 3 [41]. Theorem 5.4.1. min smiss-d is NP-complete.

Proof. Let (G, k) be an instance of minimum maximal matching where every left hand vertex has degree 2 and every right hand vertex has degree at most 3. We define the sets of left hand and right hand side vertices in G as U = {u1, u2, ..., un1} and

5.4. Minimum socially stable matchings 87

two neighbours of ui in W . Without loss of generality if wi,1 = wr and wi,2 = ws,

suppose r < s. For each vertex wj ∈ W , let uj,1, uj,2 and uj,3 be the three neighbours

of wj in W . Without loss of generality if uj,1 = ur, uj,2 = us and uj,3 = ut, suppose

r < s < t. The idea is to construct an instance (I, G0_{) of min smiss from G such that} the preference lists in I would be determined (in part) by the edges in G. We can then show that, for a specific value k0, G has a maximal matching of size ≤ k if and only if (I, G0) has a socially stable matching of size ≤ k0.

We construct (I, G0) from G as follows. The set of men in I is U0 ∪ Z where U0 ₌

{u1

i, u2i : 1 ≤ i ≤ n1} and Z = {zj1, z2j, z3j : 1 ≤ j ≤ n2}. The set of women in I

is W0 ∪ X ∪ Y where W0 _{= {w}1

j, w2j, w3j : 1 ≤ j ≤ n2}, X = {x1, x2, ..., xn2} and

Y = {y1, y2, ..., yn1}. We define relationships between men in U

0 _{and women in W}0

based on the corresponding vertices in G as follows. Given r ∈ {1, 2} and p ∈ {1, 2, 3}, we denote by w(ur_i) the woman w_jp where wj = wi,r and ui = uj,p. Also given q ∈ {1, 2}

and s ∈ {1, 2, 3}, we denote by u(ws

j) the man u q

i where ui = uj,s and wj = wi,q. The

preference lists of the men and women are shown below.

ur_i : yi w(uri) ... (1 ≤ i ≤ n1∧ 1 ≤ r ≤ 2) z_js: xj wsj (1 ≤ j ≤ n2∧ 1 ≤ s ≤ 3) w_js: z_js u(ws_j) ... (1 ≤ j ≤ n2∧ 1 ≤ s ≤ 3) xj : zj1 z 2 j z 3 j (1 ≤ j ≤ n2) yi : u1i u 2 i (1 ≤ i ≤ n1)

In the preference list of ur_i, the symbol ... means everyone in the set {w_a1, w_a2, w_a3, w_b1, w2

b, wb3}\{w(uri)}, where wa = wi,1and wb = wi,2, listed in arbitrary strict order. Also in

the preference list of ws

j, the symbol ... means everyone in the set {u1a, u2a, u1b, u2b, u1c, u2c}

\{u(ws

j)}, where ua = uj,1, ub = uj,2 and uc= uj,3, listed in arbitrary strict order. We

define the set of unacquainted pairs to be

U = {(u1_i, yi) : 1 ≤ i ≤ n1} ∪ {(z1j, xj), (z2j, xj) : 1 ≤ j ≤ n2}

All other acceptable pairs are acquainted. Let k0 = k + n1+ 3n2. We claim that G has

a maximal matching of size ≤ k if and only if I has a socially stable matching of size ≤ k0_.

Suppose G has a maximal matching M such that |M |≤ k. We construct a matching M0 in (I, G0) as follows. Let i (1 ≤ i ≤ n1) be given. Suppose ui is matched and (ui, wj) ∈

M . Suppose wj = wi,1. Add (u1i, w(u1i)) and (u2i, yi) to M0. Similarly if wj = wi,2 add

(u2

5.4. Minimum socially stable matchings 88

M0 (leaving u2

i unmatched in M

0_{). Now let j (1 ≤ j ≤ n}

2) be given. Suppose wsj is

matched in M0 for some s (1 ≤ s ≤ 3). Then wp_j and wq_j are unmatched in M0 where {1, 2, 3} = {p, q, s}. Add (z_jp, wp_j) to M0 and (z_jq, wq_j) to M0. Also add (zs

j, xj) to M0.

Now suppose ws_j is unmatched in M0 for some s (1 ≤ s ≤ 3). Add (z_j1, xj), (zj2, w2j) and

(z3_j, w3_j) to M0 (leaving w_j1unmatched in M0). Then |M0|≤ |M |+n1+3n2 ≤ k+n1+3n2.

Thus |M0|≤ k0_.

We claim that M0 is socially stable in I. Suppose not. The pair (u2

i, yi) cannot socially

block M0 as yi is matched in M0 for all i (1 ≤ i ≤ n1). The pair (zj3, xj) cannot socially

block M0 as xj is matched in M0 for all j (1 ≤ j ≤ n2). So no zsj can be part of

any socially blocking pair as every zs

j is matched in M0 for all j (1 ≤ j ≤ n2) and

s (1 ≤ s ≤ 3). Also xj cannot be part of a socially blocking pair as xj is matched in

M0 for all j (1 ≤ j ≤ n2). Similarly yi cannot be part of a socially blocking pair as

yi is matched in M0 for all i (1 ≤ i ≤ n1). Now suppose (uri, wsj) socially blocks M0.

Then ur

i is unmatched in M

0 _{since u}r

i, if matched in M

0_{, never has worse than his 2}nd

choice. Also ws

j is unmatched in M

0 _{for a similar reason. Thus u}

i is unmatched in M

and wj is unmatched in M and {ui, wj} ∈ E(G), a contradiction to the maximality of

M in G.

Conversely suppose M0 is a socially stable matching such that |M0|≤ k0_{. The following}

facts are easy to establish.

Fact 1. yi is matched in M0 for all i (1 ≤ i ≤ n1) for otherwise (u2i, yi) socially blocks

M0 in I.

Fact 2. xj is matched in M0 for all j (1 ≤ j ≤ n2) for otherwise (z3j, xj) socially blocks

M0 in I. Fact 3. zs

j is matched in M

0 _{for all j (1 ≤ j ≤ n}

2) and s (1 ≤ s ≤ 3) for otherwise

(zs

j, wjs) socially blocks M0 in I.

Suppose (ur

i, wsj) ∈ M

0_{. Add (u}

i, wj) to M . For each i (1 ≤ i ≤ n1) there exists at

most one r (1 ≤ r ≤ 2) such that ur

i is matched to a wsj for some j (1 ≤ j ≤ n2)

and s (1 ≤ s ≤ 3). by Fact 1. Also for each j (1 ≤ j ≤ n2), there exists at most

one s (1 ≤ s ≤ 3) such that ws

j is matched to a uri for some i (1 ≤ i ≤ n1) and

r (1 ≤ r ≤ 2) by Facts 2 and 3. Thus M is a matching in G. Also Facts 1, 2 and 3 imply that |M |= |M0|−n1− 3n2 ≤ k0 − n1 − 3n2. Thus |M |≤ k. Finally suppose

M is not maximal. Then there exists some (ui, wj) ∈ G such that ui and wj are both

unmatched in M . Thus there exists some ur_i that is unmatched in M0 (by Fact 1) and there exists some ws_j that is unmatched in M0 (by Facts 2 and 3). Since ur_i finds w_js acceptable (ur

i, wsj) socially blocks M

0 _{in I, a contradiction to the social stability of M}0