On the Complexity of

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On the Complexity of K-Dimensional-Matching
Elad Hazan, Muli Safra Oded Schwartz
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Maximal Matching in Bipartite Graphs
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Maximal Matching in Bipartite Graphs
Easy problem in P
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3-Dimensional Matching (3-DM)
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3-Dimensional Matching (3-DM)
Matching in a bounded hyper-graph
Bounded Set Packing
NP-hard Karp72
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3-DM Bounded Set-Packing Maximal Matching in
a Hyper-Graph
which is 3-uniform 3-strongly-colorable
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k-DM Bounded Set-Packing Maximal Matching in
a Hyper-Graph
which is k-uniform k-strongly-colorable
Without this this is k-SP
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Main Theorem
Corollary The same holds for k-Set-Packing
and Independent set in k1-claw-free
graphs Some inapproximability factors for small
k-values are also obtained
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Gap-Problems and Inapproximability Maximization
problem A Gap-A-sno, syes
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Gap-Problems and Inapproximability Maximization
problem A Gap-A-sno, syes is
NP-hard. ? Approximating A better than
syes/sno is NP-hard.
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Gap-Problems and Inapproximability Gap-k-DM-
is NP-hard. ? k-DM is NP-hard to
approximate to within
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x1 x2 x3 a1 mod q x7 x4 x2 a2 mod q
x8 x2 x9 an mod q
L-q Input A set of linear equations mod
q Objective Find an assignment satisfying
maximal number of equations App. ratio
1/q Inapp. factor 1/q? Hås97
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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(?)
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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
Can be extended to k-DM
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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
• ? ? H? (V,E)
• We describe hyper edges, then which vertices they
  include.

1st trial
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x1 x2 x3 a1 mod q x7 x4 x2 a2 mod q
x8 x2 x9 an mod q

• 1st trial
• Gap-L-q p Gap-k-SP
• A hyper-edge for each equation and a satisfying
  assignment to it (q2 such assignments).

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x1 x2 x3 a1 mod q x7 x4 x2 a2 mod q
x8 x2 x9 an mod q

• 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

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x1 x2 x3 a1 mod q x7 x4 x2 a2 mod q
x8 x2 x9 an mod q
1st trial Gap-L-q p Gap-k-SP Maximal
matching Consistent assignment
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x1 x2 x3 a1 mod q x7 x4 x2 a2 mod q
x8 x2 x9 an mod q
1st trial Gap-L-q p Gap-k-SP Maximal
matching Consistent assignment Gap-L-q-1/q?,1
- ? ltp Gap-k-SP-1/q?,1- ? What is k ?
k is large !
k ? (cx1cx2cx3) q(q-1)
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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
• Saving a factor of q
• Reuse vertices
• k Still depends on cx1cx2cx3
• which depends on ?

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x1 x2 x3 a1 mod q x7 x4 x2 a2 mod q
x8 x2 x9 an mod q

• 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.

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Which contradicting edges to connect ? A
Connection Scheme for x
Fewer vertices Consistency achieved using
disperser-Like Properties
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DefHSS03 ?-Hyper-Disperser H(V,E) VV1 ?
V2 ? ? Vq E ?V1 V2 Vq ?U
independent set (of the strong sense) ?i, U\Vi
lt ?V If U is large it is concentrated ! This
generalizes standard dispersers
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Lemma HSS03 Existence of ?-Hyper-Disperser ?q
gt1,cgt1 ?1/q2-Hyper-Disperser which is also q
uniform, q strongly-colorable d regular, d
strongly-edge-colorable for d?(q log q)
Proof
Optimality
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DefHSS03 ?-Hyper-Edge-Disperser H(V,E) EE
1 ? E2 ? ? Eq ?M matching ?i, M\Ei lt ?E If
M is large it is concentrated !
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Lemma HSS03 Existence of ?-Hyper-Edge-Disperse
r ?qgt1,cgt1 ?1/q2-Hyper-Edge-Disperser which is
also q regular, q strongly-edge-colorable d
uniform, d strongly-colorable for d?(q log
q)
Jump
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• Constructing the k-SP instance
• ? ? H? (V,E)

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

• E ? for each equation ? and a satisfying
  assignment to it the union of three hyper-edges

? x1 x2 x3 4 A(?)(0,1,3)
e?,(0,1,2)
H? is 3d uniform 3d?(q log q)
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• Completeness
• If ?A satisfying 1-? of ?
• then
• M covering 1-? of V
• (hence of size V/k)
• Proof
• Take all edges corresponding to the satisfying
  assignment. ?

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Soundness If ?A satisfies at most 1/q ? of ?
then ?M covers at most 4/q2 ? of V
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Soundness-Proof Mmaj ? Edges of M that agree
with A Mmin ? M \ Mmaj (Håstad)
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Soundness-Proof
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Soundness-Proof ?
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Gap-L-q-1/q ?,1- ? p Gap-k-SP- O(1/q),1- ?
What is k ? ? Gap-k-SP- is NP-hard.
k3d?(q log q)
?
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Conclusion
Deterministic reduction
This can be extended for k-DM. 4-DM, 5-DM and
6-DM cannot be approximated to within
respectively.
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Open Problems Low-Degree 3-DM,4-DM TSP Stei
ner-Tree Sorting By Reversals
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Open Problems Separating k-IS from k-DM ?
Vis96
HS89
Tre01
HSS03
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THE END
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Optimality of Hyper-Disperser 1/q2-Hyper-Dispers
er Regularity d?(q log q) Restrict hyper
disperser to V1,V2. A bipartite ?-Disperser is of
degree ?(1/? log 1/?) and ? ? 1/q.
Definition
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Existence of Hyper-Disperser Proof random
construction. Random permutations ?ji ?R
Sc j?2,,q, i?d ei,j v1,j, v2,
?2i(j), , vq, ?ki(j) E ei,j
j?2,,q, i?d
Definition
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Proof cont. Candidates bad (minimal)
sets U U U ? V, U 2c/q, U?V1c/q
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Proof cont.
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Proof cont. ?
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Gap-k-SP-O(log k / k), 1-? is NP-hard.

Extending it to k-DM
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Gap-k-DM-O(log k / k), 1-? is NP-hard.

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• 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

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From Asymptotic to Low Degree How to make k as
small as possible ?
E ? ? equation and a satisfying assignment to it
three hyper-edges
? x1 x2 x3 0 A(?)(0,1,1)
e?,(0,1,2),x1
e?,(0,1,2),x2
e?,(0,1,2),x3