On the Complexity ofK-Dimensional-Matching

Elad Hazan, Muli Safra & Oded Schwartz

Maximal Matching in Bipartite Graphs

Easy problem: in P

Maximal Matching in Bipartite Graphs

3-Dimensional Matching (3-DM)

NP-hard [Karp72]

3-Dimensional Matching (3-DM)

Matching in a bounded hyper-graphBounded Set Packing

3-DM: Bounded Set-PackingMaximal Matching in a Hyper-Graph

which is 3-uniform & 3-strongly-colorable

Set-Packing:

[BH92]

[Hås99]

2( )log

nO

n1( )O n 95

94

Bounded variant:

App. : [HS89]

Inapp. : [CC03]

3

2

K

K

k-DM: Bounded Set-PackingMaximal Matching in a Hyper-Graph

which is k-uniform & k-strongly-colorable

Set-Packing:

[BH92]

[Hås99]

2( )log

nO

n1( )O n

( ln )2O k

k

Bounded variant:

App. : [HS89]

Inapp. : [Tre01]

2

k

Without this this is k-SP

Unless P=NP, k-DM cannot be

approximated to within ( )log

kO

k

Main Theorem:

Corollary: The same holds for

k-Set-Packing and

Independent set in k+1-claw-free graphs

Some inapproximability factors for small k-values are also obtained

Gap-Problems and Inapproximability

Maximization problem A

Gap-A-[sno, syes]

Gap-Problems and Inapproximability

Maximization problem A

Gap-A-[sno, syes] is NP-hard.

Approximating A better than syes/sno

is NP-hard.

Gap-Problems and Inapproximability

Gap-k-DM-[ ] is NP-hard.

k-DM is NP-hard to approximate to within ( )

log

kO

k

log( 1),

kO

k

L-q:

Input: A set of linear equations mod q

Objective: Find an assignment satisfying maximal number of equationsApp. ratio: 1/q

Inapp. factor: 1/q+ [Hås97]

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

Thm [Hås97]:

Gap-L-q-[1/q+, 1-] is NP-hard.

Even if each variable x occurs a constant number of times, cx = cx()

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

Gap-L-q ≤p Gap-k-SP

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

Can be extended to

k-DM

Gap-L-q ≤p Gap-k-SP

H = (V,E)

•We describe hyper edges, then which vertices they include.

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

1st trial:

1st trial: Gap-L-q ≤p Gap-k-SP

•A hyper-edge for each equation and a satisfying assignment to it (q2 such assignments).

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

1 : x1 + x2 + x3 = 0 mod 3A(1)=(0,1,2)

2 : x7 + x4 + x2 = 1 mod 3A(2)=(1,0,0)

1st trial: Gap-L-q ≤p Gap-k-SP

•A hyper-edge for each equation and a satisfying assignment to it•A common vertex for each two contradicting edges

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

1 : x1 + x2 + x3 = 0 mod 3A(1)=(0,1,2)

2 : x7 + x4 + x2 = 1 mod 3A(2)=(1,0,0)

x2:(1,0)

1st trial: Gap-L-q ≤p Gap-k-SP

Maximal matching Consistent assignment

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

1 : x1 + x2 + x3 = 0 mod 3A(1)=(0,1,2)

2 : x7 + x4 + x2 = 1 mod 3A(2)=(1,0,0)

x2:(1,0)

1st trial: Gap-L-q ≤p Gap-k-SP

Maximal matching Consistent assignment

Gap-L-q-[1/q+,1- ] <p Gap-k-SP-[1/q+,1- ]

What is k ?

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

k is large !

k (cx1+cx2+cx3) q(q-1)

Gap-L-q ≤p Gap-k-SP

Saving a factor of q:

•Reuse vertices

•k Still depends on cx1+cx2+cx3

which depends on

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

x2=1x

2=2

x 2=0

2nd trial:Gap-L-q ≤p Gap-k-SP

Allow pluralism:•A (few) contradicting edges may reside in a matching•Common vertices for only some subsets of contradicting edges

- using a connection scheme.

x1 + x2 + x3 = a1 mod q

x7 + x4 + x2 = a2 mod q

…

x8 + x2 + x9 = an mod q

Which contradicting edges to connect ?A Connection Scheme for x

cx

q

Fewer vertices:Consistency achieved using disperser-Like Properties

Def:[HSS03] -Hyper-Disperser

H=(V,E)V=V1 V2 … Vq

E V1 × V2 × … × Vq

U independent set (of the strong sense)i, |U\Vi| < |V|

If U is large it is concentrated !This generalizes standard dispersers

Lemma [HSS03]: Existence of -Hyper-Disperser

q>1,c>1 1/q2-Hyper-Disperserwhich is also

q uniform, q strongly-colorabled regular, d strongly-edge-colorable

for d=(q log q)

Proof… Optimality…

Def:[HSS03] -Hyper-Edge-Disperser

H=(V,E)E=E1 E2 … Eq

M matchingi, |M\Ei| < |E|

If M is large it is concentrated !

Lemma [HSS03]: Existence of -Hyper-Edge-Disperser

q>1,c>1 1/q2-Hyper-Edge-Disperserwhich is also

q regular, q strongly-edge-colorabled uniform, d strongly-colorable

for d=(q log q)

Jump…

Constructing the k-SP instance H =(V,E)

x - a copy of (c=cx).

•V the vertices of all

•E for each equation and a satisfying assignment to it – the union of three hyper-edges : x1 + x2 + x3 =

4

A()=(0,1,3)X1

X3

X2

e,(0,1,2)

0

1

3

Constructing the k-SP instance H =(V,E)

H is 3d uniform3d=(q log q)

Completeness:

If A satisfying 1- of thenM covering 1- of V

(hence of size |V|/k)Proof:Take all edges corresponding to the satisfying assignment. ڤ

Soundness:

If A satisfies at most 1/q + of thenM covers at most 4/q2 + of V

Soundness-Proof:

Mmaj Edges of M that agree with A

Mmin M \ Mmaj

(Håstad)

A most popular values of each

1maj

VM

kq

Soundness-Proof:

Every edge of Mmin is a minority in at least one

| ( 2) 2

1( )min x A x

cq cM dispe se

q qqr r

3 min

VM

q k

Soundness-Proof:

4

maj min q

VM M M

k

3 min

VM

q k

1maj

VM

kq

Gap-L-q-[1/q+ ,1- ] ≤p Gap-k-SP- [O(1/q),1- ]

What is k ?

Gap-k-SP-[ ] is NP-hard.

Unless P=NP, k-SP cannot be

approximated to within ( )log

kO

k

k=3d=(q log q)

log( 1),

kO

k

Conclusion

Unless P=NP, k-SP cannot be

approximated to within ( )log

kO

k

This can be extended for k-DM.

4-DM, 5-DM and 6-DM cannot be approximated to within respectively.

54 30 23, &

53 29 22

Deterministic reduction

Open Problems

Low-Degree: 3-DM,4-DM…TSPSteiner-TreeSorting By Reversals

Open Problems

Separating k-IS from k-DM ?

k-DM k-IS

App. ratio

Innap. factor

log log( )

log

k kO

k[Vis96]

( ln )2O k

k

2

k

( )log

kO

k[Tre01][HSS03]

[HS89]

THE END

Optimality of Hyper-Disperser:

1/q2-Hyper-DisperserRegularity: d=(q log q)

Restrict hyper disperser to V1,V2.A bipartite -Disperser is of degree (1/ log 1/) and 1/q.

Definition…

Existence of Hyper-DisperserProof: random construction.Random permutations:

ji R Sc j{2,…,q}, i[d]

e[i,j] = { v[1,j], v[2, 2i(j)], …, v[q, k

i(j)] }

E = {e[i,j] | j{2,…,q}, i[d] }Definition…

Proof – cont.

Candidates: ‘bad’ (minimal) sets:

U = { U | U V, |U| = 2c/q, |UV1|=c/q}

2 3

( 1)

| | ( )c

q

q c c

q q e qc c

q q

U

Proof – cont.

1 1

2

U,

,

Pr [U is 'bad'] (1 )j

q qi j i j

i j i

idU

i j

ddU U U U

cc

i j

dc

q

U

c

e e

e

U

Proof – cont.

22 3U| | Pr [U is 'bad'] ( ) 1

3 ln 2

dcc

q qq e q e

d q q q

UU

Extending it to k-DM

Gap-k-SP-[O(log k / k), 1-] is NP-hard.

Use a for each location of a variable.

Gap-k-DM-[O(log k / k), 1-] is NP-hard.

From Asymptotic to Low Degree – How to make k as small as possible ?

•Minimize d ( = 3) – by minimizing q ( = 2)(a bipartite disperser)

•Avoid union of edges

E equation and a satisfying assignment to it –three hyper-edges

: x1 + x2 + x3 = 0

A()=(0,1,1)

X1

X3

X2

e,(0,1,2),x1

e,(0,1,2),x2

e,(0,1,2),x3

From Asymptotic to Low Degree – How to make k as small as possible ?