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Chapter 10 Complexity of Approximation (1) L-Reduction Ding-Zhu Du

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Traveling Salesman Given n cities with a distance table, find a minimum total-distance tour to visit each city exactly once.

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Definition

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Proof: Given a graph G=(V,E), define a distance table on V as follows: Theorem

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Contradiction Argument Suppose r-approximation exists. Then we have a polynomial-time algorithm to solve Hamiltonian Cycle as follow: r-approximation solution < r |V| if and only if G has a Hamiltonian cycle

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Special Case Traveling around a minimum spanning tree is a 2-approximation. Theorem

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Minimum spanning tree + minimum-length perfect matching on odd vertices is 1.5- approximation Theorem

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Minimum perfect matching on odd vertices has weight at most 0.5 opt.

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Knapsack

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Definition

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Theorem Proof.

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Theorem

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Classify: for i < m, c i < a= c G, for i > m+1, c i > a. Sort For Algorithm

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Proof.

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Time

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MAX3SAT

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Theorem

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This an important result proved using PCP system. Theorem

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Class MAX SNP (APX?)

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L-reduction

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VC-b Theorem

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12 3 4 5 12 3 4 5 GG’ v

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Properties (P1) (P2)

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PTAS MAX SNP

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MAX SNP-complete (APX-complete) Theorem

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MAX3SAT-3 Theorem

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VC-4 is MAX SNP-complete Proof.

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Theorem Proof.

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Theorem Proved using PCP system

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Theorem MCDS

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CLIQUE Theorem Proved with PCP system.

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1 2 Exercises

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3

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hint

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Min-2-DS is MAX SNP-complete in the case that all given pools have size at most 2. 4 Prove that

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5. Is TSP with triangular inequality MAX SNP-complete?

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