Paradigms of Combinatorial Optimization by Vangelis Th. Paschos

Author:Vangelis Th. Paschos
Language: eng
Format: epub
Publisher: Wiley
Published: 2013-01-11T16:00:00+00:00

Let us choose an M sufficiently large (for example M > 2s) and, for i = 1, …, s, let us define the weights on the variables z as:

Since we have chosen M sufficiently large, we can ignore the impact of the upper rounding operation on the calculations that will follow. Therefore, for every assignment of truth values T′ on , we define the assignment g(, T′) on φ by the restriction of T′ on the variables that also appear in φ.

So, in every assignment T′ that satisfies , one and only one of the variables z that appear in it can be set to true. Let us therefore consider a feasible solution (assignment) T′ and let us assume that, for an index i ∈ {1, …, s}, the variable zi,b1,…,bk is such a variable (that is the variable zi,b1,…,bk is the only variable set to true in T′). So, m(, T′) = w(zi,b1,…,bk) and, using algebra, we obtain:

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