By G. Ausiello, P. Crescenzi, V. Kann, Marchetti-sp, Giorgio Gambosi, Alberto M. Spaccamela

This e-book files the cutting-edge in combinatorial optimization, featuring approximate suggestions of just about all suitable periods of NP-hard optimization difficulties. The wealth of difficulties, algorithms, effects, and strategies make it an indispensible resource of reference for execs. The textual content easily integrates a number of illustrations, examples, and exercises.

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**Additional info for Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties**

**Example text**

3 How to use a Karp reduction answer As a particular case, we define the polynomial-time Karp-reducibilityby saying that PI is polynomially Karp-reducible to P2 if and only if P1 is Karp-reducible to P2 and the corresponding reduction R is a polynomialtime algorithm. In such a case we write P1

5 in the lexicographic order. y) + X(y). y2). Therefore there exists a unique optimal feasible solution y* (x) in SOL (x). y2), thus implying that yp,(x) E SOLp(x). The optimal solution y* (x) can be easily derived in polynomial time by means of an oracle for PE, since, given m* (x), the position of y*, (x) in the lexicographic order can be derived by computing the remainder of the division between m* (x) and 2P(Fxl)+±. We know that an oracle for AD can be used to simulate PE in polynomial time.

Decision Problem (PD) - Given an instance x E I and a positive integer K E Z+, decide whether m*(x) > K (if goal = MAX) or whether m*(x) < K (if goal = MIN). K) I x E IAm*(x) > K} (or {(xK) I x E IAm*(x) < K} if goal = MIN) is called the underlying language of P. 7. 12). Notice that, for any optimization problem P, the corresponding decision problem PD is not harder than the constructive problem Pc. ,y*(x)); then, it is sufficient to check whether m(x. y*(x)) > K, in the maximization case. 8. An instance of this problem can also be represented by a complete graph G = (VE) of n vertices with positive weights on the edges (the vertices represent the cities and the weight of an edge is equal to the distance between the corresponding pair of cities).