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

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

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

Definition

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

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

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

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

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

Knapsack

Definition

Theorem Proof.

Theorem

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

Proof.

Time

MAX3SAT

Theorem

This an important result proved using PCP system. Theorem

Class MAX SNP (APX?)

L-reduction

VC-b Theorem

12 3 4 5 12 3 4 5 GG’ v

Properties (P1) (P2)

PTAS MAX SNP

MAX SNP-complete (APX-complete) Theorem

MAX3SAT-3 Theorem

VC-4 is MAX SNP-complete Proof.

Theorem Proof.

Theorem Proved using PCP system

Theorem MCDS

CLIQUE Theorem Proved with PCP system.

1 2 Exercises

3

hint

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

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