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Bazgan and Fernandez de la Vega [BAZ 99] initiated the systematic study of dense instances of the minimization versions of satisfiability problems with the M IN E2L IN 2 D ELETION problem. More exactly, they showed [BAZ 99] that the everywhere dense instances of M IN E2-L IN 2 D ELETION have a polynomial time approximation scheme. In [BAZ 02, BAZ 03] Bazgan, Fernandez de la Vega and Karpinski have generalized the result obtained for M IN E2-L IN 2 D ELETION to the two problems, M IN kC ONJ D ELETION, k 2, and M IN Ek-L IN 2 D ELETION, k 3, that belong to M IN kSAT D ELETION (F ).

Certain optimization problems do not allow a polynomial time approximation scheme on everywhere dense instances. An example of such a problem is M IN 2S AT D ELETION. In fact, we can render the instances of M IN 2S AT D ELETION everywhere dense, without changing the value of the optimum, by adding disjoint copies of the 24 Combinatorial Optimization 2 original variables, then by adding all the conjunctions that have exactly one original variable and one copy. Since M IN 2S AT D ELETION does not have a polynomial time approximation scheme, the everywhere dense instances of M IN 2S AT D ELETION do not have a polynomial time approximation scheme.

Xn = bn has the value equal to E(W |x1 = b1 , . . , xn = bn ) E(W ) opt . 2 Using the random rounding method, Goemans and Williamson [GOE 94] have improved the previous result. 632. Proof. Let I be an instance of M AX S AT with m clauses C1 , . . , Cm over n variables x1 , . . , xn . The algorithm is the following: 1) Formulate M AX S AT as a linear program in 0–1 variables. With each Boolean variable xi we associate a 0–1 variable yi , and with each clause Cj a variable zj such Optimal Satisfiability 17 that zj will take the value 1 if and only if Cj is satisfied.

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A theorem of arithmetic and its proof by Euler L.


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