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Therefore, the optimal assignment for the shrunken problem, call it S, has a rescaled value of at least this much In terms of the original values, assignment S has a value of at least vi vi vmax n K n vmax n vmax = K vmax K (1 ) n
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The approximability hierarchy
Given any NP-complete optimization problem, we seek the best approximation algorithm possible Failing this, we try to prove lower bounds on the approximation ratios that are achievable in polynomial time (we just carried out such a proof for the general TSP) All told, NP-complete optimization problems are classi ed as follows: Those for which, like the TSP, no nite approximation ratio is possible
Those for which an approximation ratio is possible, but there are limits to how small this can be V ERTEX COVER, k- CLUSTER, and the TSP with triangle inequality belong here (For these problems we have not established limits to their approximability, but these limits do exist, and their proofs constitute some of the most sophisticated results in this eld) Down below we have a more fortunate class of NP-complete problems for which approximability has no limits, and polynomial approximation algorithms with error ratios arbitrarily close to zero exist K NAPSACK resides here 277
Finally, there is another class of problems, between the rst two given here, for which the approximation ratio is about log n SET COVER is an example (A humbling reminder: All this is contingent upon the assumption P = NP Failing this, this hierarchy collapses down to P, and all NP-complete optimization problems can be solved exactly in polynomial time) A nal point on approximation algorithms: often these algorithms, or their variants, perform much better on typical instances than their worst-case approximation ratio would have you believe
93 Local search heuristics
Our next strategy for coping with NP-completeness is inspired by evolution (which is, after all, the world s best-tested optimization procedure) by its incremental process of introducing small mutations, trying them out, and keeping them if they work well This paradigm is called local search and can be applied to any optimization task Here s how it looks for a minimization problem let s be any initial solution while there is some solution s in the neighborhood of s for which cost(s ) < cost(s): replace s by s return s On each iteration, the current solution is replaced by a better one close to it, in its neighborhood This neighborhood structure is something we impose upon the problem and is the central design decision in local search As an illustration, let s revisit the traveling salesman problem
Traveling salesman, once more
Assume we have all interpoint distances between n cities, giving a search space of (n 1)! different tours What is a good notion of neighborhood The most obvious notion is to consider two tours as being close if they differ in just a few edges They can t differ in just one edge (do you see why ), so we will consider differences of two edges We de ne the 2-change neighborhood of tour s as being the set of tours that can be obtained by removing two edges of s and then putting in two other edges Here s an example of a local move:
We now have a well-de ned local search procedure How does it measure up under our two standard criteria for algorithms what is its overall running time, and does it always return the best solution Embarrassingly, neither of these questions has a satisfactory answer Each iteration is certainly fast, because a tour has only O(n 2 ) neighbors However, it is not clear how many 278
iterations will be needed: whether for instance, there might be an exponential number of them Likewise, all we can easily say about the nal tour is that it is locally optimal that is, it is superior to the tours in its immediate neighborhood There might be better solutions further away For instance, the following picture shows a possible nal answer that is clearly suboptimal; the range of local moves is simply too limited to improve upon it
To overcome this, we may try a more generous neighborhood, for instance 3-change, consisting of tours that differ on up to three edges And indeed, the preceding bad case gets xed:
But there is a downside, in that the size of a neighborhood becomes O(n 3 ), making each iteration more expensive Moreover, there may still be suboptimal local minima, although fewer than before To avoid these, we would have to go up to 4-change, or higher In this manner, ef ciency and quality often turn out to be competing considerations in a local search Ef ciency demands neighborhoods that can be searched quickly, but smaller neighborhoods can increase the abundance of low-quality local optima The appropriate compromise is typically determined by experimentation
Figure 97 (a) Nine American cities (b) Local search, starting at a random tour, and using 3-change The traveling salesman tour is found after three moves (a)
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