On the Complexity of K-Dimensional-Matching Elad Hazan, Muli Safra & Oded Schwartz
Feb 03, 2016
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 ?