By Lap Chi Lau
ISBN-10: 0521189438
ISBN-13: 9780521189439
ISBN-10: 1107007518
ISBN-13: 9781107007512
With the arrival of approximation algorithms for NP-hard combinatorial optimization difficulties, numerous thoughts from specified optimization corresponding to the primal-dual technique have confirmed their endurance and flexibility. This e-book describes an easy and strong process that's iterative in essence, and equally necessary in various settings for special and approximate optimization. The authors spotlight the commonality and makes use of of this technique to turn out a number of classical polyhedral effects on matchings, timber, matroids, and flows. The presentation kind is common sufficient to be obtainable to an individual with publicity to uncomplicated linear algebra and graph idea, making the e-book compatible for introductory classes in combinatorial optimization on the top undergraduate and starting graduate degrees. Discussions of complicated functions illustrate their strength for destiny software in study in approximation algorithms
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Additional info for Iterative Methods in Combinatorial Optimization
Example text
This can be shown by a simple inductive argument on the number of iterations. Observe that the claim holds trivially before the first iteration. In any iteration, if we assign job j to machine i in Step (ii)b then the cost of F increases by cij and the current linear programming solution decreases by cij xij = cij since xij = 1. Hence, the claim holds. If we remove a constraint in Step (ii)c, then the cost of F remains the same, while the cost of the current linear program can only decrease. Hence, the claim holds in this case as well.
The residual solution xres , x restricted to G \ {e}, is a feasible solution to the linear programming relaxation of the residual problem. By induction, the algorithm returns a matching F ⊆ E(G ) with weight at least the optimal solution to LPbm (G ). Since w(F ) ≥ w · xres = w · x, the induction hypothesis holds in this case. In the other case, if we find an edge e = {u, v} with xe = 1 in Step (ii)(b). of the algorithm, then the residual problem is to find a matching that contains the edge e. 1 Matchings in bipartite graphs 31 we remove the vertices u and v and their incident edges from G.
1 Given any weight function w there exists an integral matching M such that w(M) ≥ w · x where x is an optimal solution to LPbm (G). 1 as a corollary implies the following theorem. 2 The linear programming formulation LPbm (G) is integral. 1, we give a characterization of extreme point solutions of LPbm (G) for which we need a few definitions. For a set F ⊆ E, let χ (F ) denote the vector in R|E| that has a 1 corresponding to each edge e ∈ F , and 0 otherwise. This vector is called the characteristic vector of F .
Iterative Methods in Combinatorial Optimization by Lap Chi Lau
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