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By Shuhei Hayashi

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Additional resources for A C(1) Make or Break Lemma

Example text

This is the subject of this section. 1 The Convex Hull of Integer Solutions The constraints of a Linear Programme (LP) restrict the feasible solutions to a set which can be represented by a Polytope in multidimensional space. For Integer Programmes the set is further restricted to the lattice of integer points within a Polytope. 2, by the constraints of a 2-Variable IP, which can therefore be represented in 2-dimensional space. 32) are represented by the lines AB, BC and CD respectively and the non-negativity conditions by OA and OD.

In [42] this margin was tightened to n/2^^^^^^ ~\ The approaches to the problem offered include such common combinatorial optimization techniques as sequential greedy heuristics, local search heuristics, methods based on simulated annealing, neural networks, genetic algorithms, tabu search, etc. However, there are also methods utilizing various formulations of the clique problem as a continuous optimization problem. An extensive survey on the maximum clique problem can be found in [5]. The simplest integer formulation of the maximum weight independent set problem is the following so called edge formulation: n maxf(x) = Y2^i^i^ (2-8) subject to Xi + Xj < 1, V (ij) eE, xe {0, l}"".

As the above example demonstrates Constraint Satisfaction models are usually much more compact than IP models. The successive domain reduction of the variables mirrors the Presolve facility of many commercial IP systems as was mentioned previously. However Constraint Satisfaction does not rely on the concept of an LP relaxation which can prove very valuable in reducing a tree search. In fact for some models (eg. the Assignment problem mentioned above) the LP relaxation yields a feasible integer solution immediately.

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A C(1) Make or Break Lemma by Shuhei Hayashi


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