By Thomas Jansen (auth.), Ernst W. Mayr, Hans Jürgen Prömel, Angelika Steger (eds.)

ISBN-10: 3540642013

ISBN-13: 9783540642015

During the previous couple of years, we've seen really unbelievable development within the sector of approximation algorithms: for a number of basic optimization difficulties we now really comprehend matching top and reduce bounds for his or her approximability. This textbook-like educational is a coherent and basically self-contained presentation of the big contemporary growth facilitated by way of the interaction among the speculation of probabilistically checkable proofs and aproximation algorithms. the elemental ideas, tools, and effects are awarded in a unified strategy to supply a delicate creation for rookies. those lectures are fairly important for complex classes or analyzing teams at the topic.

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**Extra info for Lectures on Proof Verification and Approximation Algorithms**

**Example text**

The mean of X is m := E[X] = ~ j = l ajp~. The following theorem gives Chernoff bounds on the deviation of X above and below its mean. 5. Let ~ > 0 and let m = E[X] >/0. Then 1. Prob[X > ( 1 + 6)m] < ~ 2. ProbtX < (1- ) , )ml < Proof. We only prove the first statement. The second statement can be proved in a similar way. We apply Markov's inequality to the moment generating function e tx and obtain for each t > 0: Prob[X > (1 + $)m] = Prob[e tx > e t(l+6)m] < e - t ( l + 8 ) m E[etX]. Now we exploit the independence of the Xj.

We will now follow Trevisan [Tre97] and show how AP-reductions can be used to prove MAX3SAT to be complete for APA'. 33. AT~X-complete. 24 Chapter 1. Complexity and Approximation problem A problem B inputs inputs f x f(~) = ~/ approximation with ratio 1 + a( r - 1) (/ \ approximation with ratio r ! = a ( ~ ) -- g solutions solutions Fig. 3. Interplay of an a-AP-reduction from A to B and an r-approximation algorithm for B. Proof. Khanna, Motwani, Sudan, and Vazirani [KMSV94] proved that for any APX-minimization problem A there exists an APX-maximization problem B such that A can be reduced to B by a reduction for optimization problems that is called E-reduction.

Since we have the approximation algorithm MA with performance ratio c t h a t yields a solution & for an input x we know that the value of an optimal solution OPTA(X) is in the interval [VA(~), CVA(~)]. We divide this interval in k := [log d c] subintervals with lower bounds divA(~) and upper bounds di+lvA(5C) for 0 x< i < k. Since A E APX C_ AfPO the problem of deciding whether the value of an optimal solution is at least q belongs to A l P for all constants q. We now have k AlP-decision problems such t h a t the "membership proof" we mentioned in the discussion of Cook's theorem is a solution for our optimization problem A with value at least divA(~).

### Lectures on Proof Verification and Approximation Algorithms by Thomas Jansen (auth.), Ernst W. Mayr, Hans Jürgen Prömel, Angelika Steger (eds.)

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