On the Complexity of K-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-graph Bounded Set Packing

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

K

K

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

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

Gap-Problems and Inapproximability Maximization problem A Gap-A-[s no, s yes ]

Gap-Problems and Inapproximability Maximization problem A Gap-A-[s no, s yes ] is NP-hard.  Approximating A better than s yes /s no is NP-hard.

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

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] x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q

Thm [Hås97]: Gap-L-q-[1/q+ , 1-  ] is NP-hard. Even if each variable x occurs a constant number of times, c x = c x (  ) x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q

Gap-L-q ≤ p Gap-k-SP x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n 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. x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q 1 st trial:

Gap-L-q ≤ p Gap-k-SP A hyper-edge for each equation and a satisfying assignment to it (q 2 such assignments). x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q  1 : x 1 + x 2 + x 3 = 0 mod 3 A(  1 )=(0,1,2)  2 : x 7 + x 4 + x 2 = 1 mod 3 A(  2 )=(1,0,0)

1 st 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 x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q  1 : x 1 + x 2 + x 3 = 0 mod 3 A(  1 )=(0,1,2)  2 : x 7 + x 4 + x 2 = 1 mod 3 A(  2 )=(1,0,0) x 2 :(1,0)

1 st trial: Gap-L-q ≤ p Gap-k-SP Maximal matching Consistent assignment x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q  1 : x 1 + x 2 + x 3 = 0 mod 3 A(  1 )=(0,1,2)  2 : x 7 + x 4 + x 2 = 1 mod 3 A(  2 )=(1,0,0) x 2 :(1,0)

1 st 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 ? x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q k is large ! k  (c x1 +c x2 +c x3 ) q(q-1)

Gap-L-q ≤ p Gap-k-SP Saving a factor of q: Reuse vertices k Still depends on c x1 +c x2 +c x3 which depends on  x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q x 2 =1 x 2 =2 x 2 =0

2 nd 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. x 1 + x 2 + x 3 = a 1 mod q x 7 + x 4 + x 2 = a 2 mod q … x 8 + x 2 + x 9 = a n mod q

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

Def:[HSS03]  -Hyper-Disperser H=(V,E) V=V 1  V 2  …  V q E  V 1 × V 2 × … × V q  U independent set (of the strong sense)  i, |U\V i | <  |V| If U is large it is concentrated ! This generalizes standard dispersers

Lemma [HSS03]: Existence of  -Hyper-Disperser  q>1,c>1  1/q 2 -Hyper-Disperser which is also q uniform, q strongly-colorable d regular, d strongly-edge-colorable for d=  (q log q) Proof … Optimality …

Def:[HSS03]  -Hyper-Edge-Disperser H=(V,E) E=E 1  E 2  …  E q  M matching  i, |M\E i | <  |E| If M is large it is concentrated !

Lemma [HSS03]: Existence of  -Hyper-Edge-Disperser  q>1,c>1  1/q 2 -Hyper-Edge-Disperser which is also q regular, q strongly-edge-colorable d uniform, d strongly-colorable for d=  (q log q) Jump …

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

E  for each equation  and a satisfying assignment to it – the union of three hyper-edges  : x 1 + x 2 + x 3 = 4 A(  )=(0,1,3) X1X1 X3X3 X2X2 e ,(0,1,2) Constructing the k-SP instance   H  =(V,E) H  is 3d uniform 3d=  (q log q)

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

Soundness: If  A satisfies at most 1/q +  of  then  M covers at most 4/q 2 +  of V

Soundness-Proof: M maj  Edges of M that agree with A M min  M \ M maj (Håstad) A  most popular values of each

Soundness-Proof: Every edge of M min is a minority in at least one

Soundness-Proof: 

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 k=3d=  (q log q)

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

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

Open Problems Separating k-IS from k-DM ? k-DMk-IS App. ratio Innap. factor [Vis96] [Tre01][HSS03] [HS89]

THE END

Optimality of Hyper-Disperser: 1/q2-Hyper-Disperser Regularity: d=  (q log q) Restrict hyper disperser to V 1,V 2. A bipartite  -Disperser is of degree  (1/  log 1/  ) and   1/q. Definition …

Existence of Hyper-Disperser Proof: random construction. Random permutations:  j i  R S c j  {2,…,q}, i  [d] e[i,j] = { v[1,j], v[2,  2 i (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, |U  V 1 |=c/q}

Proof – cont.



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  : x 1 + x 2 + x 3 = 0 A(  )=(0,1,1) X1X1 X3X3 X2X2 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 ?