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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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**On the Complexity of**K-Dimensional-Matching Elad Hazan, Muli Safra & Oded Schwartz**Maximal Matching in Bipartite Graphs**Easy problem: in P**3-Dimensional Matching (3-DM)**Matching in a bounded hyper-graph Bounded Set Packing NP-hard [Karp72]**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**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**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-[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**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]**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 Can be extended to k-DM**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:**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).**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**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**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)**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 **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.**cx**q Which contradicting edges to connect ? A Connection Scheme for x Fewer vertices: Consistency achieved using disperser-Like Properties**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**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…**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 !**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…**(c=cx).**• x - a copy of • V the vertices of all • Constructing the k-SP instance • H =(V,E)**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)**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/q2 + of V**A most popular values of each**Soundness-Proof: Mmaj Edges of M that agree with A Mmin M \ Mmaj (Håstad)**Every edge of Mmin is**a minority in at least one Soundness-Proof:**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) **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.**Open Problems**Low-Degree: 3-DM,4-DM… TSP Steiner-Tree Sorting By Reversals**Open Problems**Separating k-IS from k-DM ? [HS89] [Vis96] [HSS03] [Tre01]**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…**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…**Proof – cont.**Candidates: ‘bad’ (minimal) sets: U = { U | U V, |U| = 2c/q, |UV1|=c/q}**Gap-k-SP-[O(log k / k),1-] is NP-hard.**Extending it to k-DM**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**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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