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On the Complexity of K-Dimensional-Matching

On the Complexity of K-Dimensional-Matching. Elad Hazan, Muli Safra & Oded Schwartz. Maximal Matching in Bipartite Graphs. Maximal Matching in Bipartite Graphs. Easy problem: in P. 3-Dimensional Matching (3-DM). 3-Dimensional Matching (3-DM). Matching in a bounded hyper-graph.

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On the Complexity of K-Dimensional-Matching

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  1. On the Complexity of K-Dimensional-Matching Elad Hazan, Muli Safra & Oded Schwartz

  2. Maximal Matching in Bipartite Graphs

  3. Maximal Matching in Bipartite Graphs Easy problem: in P

  4. 3-Dimensional Matching (3-DM)

  5. 3-Dimensional Matching (3-DM) Matching in a bounded hyper-graph Bounded Set Packing NP-hard [Karp72]

  6. Bounded variant: App. : [HS89] Inapp. : [CC03] Set-Packing: [BH92] [Hås99] 3-DM: Bounded Set-Packing Maximal Matching in a Hyper-Graph which is 3-uniform & 3-strongly-colorable

  7. K

  8. K

  9. Bounded variant: App. : [HS89] Inapp. : [Tre01] Set-Packing: [BH92] [Hås99] k-DM: Bounded Set-Packing Maximal Matching in a Hyper-Graph which is k-uniform & k-strongly-colorable Without this this is k-SP

  10. Unless P=NP, k-DM cannot be approximated to within 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

  11. Gap-Problems and Inapproximability Maximization problem A Gap-A-[sno, syes]

  12. Gap-Problems and Inapproximability Maximization problem A Gap-A-[sno, syes] is NP-hard.  Approximating A better than syes/sno is NP-hard.

  13. Gap-Problems and Inapproximability Gap-k-DM-[ ] is NP-hard.  k-DM is NP-hard to approximate to within

  14. 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]

  15. 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()

  16. 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

  17. 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:

  18. 1 : x1 + x2 + x3 = 0 mod 3 A(1)=(0,1,2) 2 : x7 + x4 + x2 = 1 mod 3 A(2)=(1,0,0) 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).

  19. 1 : x1 + x2 + x3 = 0 mod 3 A(1)=(0,1,2) 2 : x7 + x4 + x2 = 1 mod 3 A(2)=(1,0,0) x2:(1,0) 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

  20. 1 : x1 + x2 + x3 = 0 mod 3 A(1)=(0,1,2) 2 : x7 + x4 + x2 = 1 mod 3 A(2)=(1,0,0) x2:(1,0) 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

  21. 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- ] <p Gap-k-SP-[1/q+,1- ] What is k ? k is large ! k  (cx1+cx2+cx3)q(q-1)

  22. x2=0 x2=1 x2=2 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 cx1+cx2+cx3 • which depends on 

  23. 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 somesubsets of contradicting edges • - using a connection scheme.

  24. cx q Which contradicting edges to connect ? A Connection Scheme for x Fewer vertices: Consistency achieved using disperser-Like Properties

  25. Def:[HSS03] -Hyper-Disperser H=(V,E) V=V1 V2 … Vq E V1 × V2 × … × Vq Uindependent set (of the strong sense) i, |U\Vi| < |V| If U is large it is concentrated ! This generalizes standard dispersers

  26. Lemma [HSS03]: Existence of -Hyper-Disperser q>1,c>1 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…

  27. Def:[HSS03] -Hyper-Edge-Disperser H=(V,E) E=E1 E2 … Eq M matching i, |M\Ei| < |E| If M is large it is concentrated !

  28. Lemma [HSS03]: Existence of -Hyper-Edge-Disperser q>1,c>1 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…

  29. (c=cx). •  x - a copy of • V  the vertices of all • Constructing the k-SP instance •   H =(V,E)

  30. 1 X1 X2 0 3 X3 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)

  31. 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. ڤ

  32. Soundness: If A satisfies at most 1/q + of  then M covers at most 4/q2 +  of V

  33. A  most popular values of each Soundness-Proof: Mmaj  Edges of M that agree with A Mmin  M \ Mmaj (Håstad)

  34. Every edge of Mmin is a minority in at least one Soundness-Proof:

  35. Soundness-Proof:

  36. Unless P=NP, k-SP cannot be approximated to within Gap-L-q-[1/q+ ,1- ] ≤p Gap-k-SP- [O(1/q),1- ] What is k ?  Gap-k-SP-[ ] is NP-hard. k=3d=(q log q) 

  37. Unless P=NP, k-SP cannot be approximated to within Conclusion Deterministic reduction This can be extended for k-DM. 4-DM, 5-DM and 6-DM cannot be approximated to within respectively.

  38. Open Problems Low-Degree: 3-DM,4-DM… TSP Steiner-Tree Sorting By Reversals

  39. Open Problems Separating k-IS from k-DM ? [HS89] [Vis96] [HSS03] [Tre01]

  40. THE END

  41. Optimality of Hyper-Disperser: 1/q2-Hyper-Disperser Regularity: d=(q log q) Restrict hyper disperser to V1,V2. A bipartite -Disperser is of degree (1/ log 1/) and   1/q. Definition…

  42. Existence of Hyper-Disperser Proof: random construction. Random permutations: ji R Sc j{2,…,q}, i[d] e[i,j] = { v[1,j], v[2, 2i(j)], …, v[q, ki(j)] } E = {e[i,j] | j{2,…,q}, i[d] } Definition…

  43. Proof – cont. Candidates: ‘bad’ (minimal) sets: U = { U | U  V, |U| = 2c/q, |UV1|=c/q}

  44. Proof – cont.

  45. Proof – cont.

  46. Gap-k-SP-[O(log k / k),1-] is NP-hard. Extending it to k-DM

  47. Use a for each location of a variable. Gap-k-DM-[O(log k / k),1-] is NP-hard.

  48. 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

  49. X2 X1 X3 From Asymptotic to Low Degree – How to make k as small as possible ? • E   equation and a satisfying assignment to it –three hyper-edges e,(0,1,2),x1 : x1 + x2 + x3 = 0 A()=(0,1,1) e,(0,1,2),x2 e,(0,1,2),x3

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