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By G. Ausiello, M. Lucertini, P. Serafini (eds.)

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Lab. RZ 1016, 1980. [37] HILLIER, CONNORS: Quadratic assignment problem algorithm, and Location of Indivisible facilities. Man. , 1966. [38] HOFRI, JENNY: On the allocation o~ processes in distributed com- puting systems. IBM Zurich Res. Lab. RZ 905, 1978. [391 HOSKEN: Optimum partitioning of the addressing structures. SIAM J. , 1975. [40] HU: Integer programming and network flows. Addison-Wesley, 1970. [41] HU, RUSKEY: Circular cuts in a network. Math. of Op. , 1980. [42] HYAFIL, RIVEST: Graph partitioning and constructing optimal decision trees are polynomial complete problems.

N. ~ ~ i=l problem, the set of constraint (2) can be dropped. S ~ A local search technique leading in most practical cases to good solutions (without any "a priori" guarantee) work as follows. Let be given a feasible solution of the problem and the corresponding objective function. At each step of the algorithm take k nodes in all the possible ways and try to reallocate the k nodes in all the possible way. For each possibility calculate the corresponding value of the objective function. If a better solution is found take it as the new solution and go to the next step.

SUBSET-SUM is a clear example of pseudopolynomial problem. In fact since we may solve the SUBSET-SUM problem in time O(nob) it t~rns out that the pro- blem is not polynomial in the size of the input, but is polynomial in the value appearing in the input. Hence if we bound MAX by a polynomial function we have a polynomially solvable problem. An the other side the problem MAX-CUT is a strongly NP-complete problem. In fact even if we restrict all weights to be equal to one we still remain with an NP-complete optimization problem.

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