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Extra info for Combinatorial Optimization: Exact and Approximate Algorithms
A1,n xn ≤ b1 .. am,1 x1 + . . + am,n xn ≤ bm x1 ≥ 0 .. 2) xn ≥ 0 For every choice of non-negative scaling factors y1 , . . , ym , we can derive the inequality y1 · (a1,1 x1 + . . + a1,n xn ) +··· +yn · (am,1 x1 + . . + am,n xn ) ≤ y1 b1 + · · · ym bm which is true for every feasible solution (x1 , . . 2). 1. THE DUAL OF A LINEAR PROGRAM 43 ≤ y1 b1 + · · · ym bm So we get that a certain linear function of the xi is always at most a certain value, for every feasible (x1 , . . , xn ). The trick is now to choose the yi so that the linear function of the xi for which we get an upper bound is, in turn, an upper bound to the cost function of (x1 , .
Although this is an easy problem on very small instances, it is an NP-hard problem and so it is unlikely to be solvable exactly in polynomial time. In fact, there are bad news also about approximation. 7 Suppose that, for some constant > 0, there is an algorithm that, on input an instance of Set Cover finds a solution whose cost is at most (1 − ) · ln |X| times the optimum; then every problem in NP admits a randomized algorithm running in time nO(log log n) , where n is the size of the input. If, for some constant c, there is a polynomial time c-approximate algorithm, then P = NP.
Xn ). The trick is now to choose the yi so that the linear function of the xi for which we get an upper bound is, in turn, an upper bound to the cost function of (x1 , . . , xn ). We can achieve this if we choose the yi such that c1 ≤ a1,1 y1 + · · · am,1 ym .. 3) cn ≤ a1,n y1 · · · am,n ym Now we see that for every non-negative (y1 , . . 3), and for every (x1 , . . 2), c1 x1 + . . cn xn ≤ (a1,1 y1 + · · · am,1 ym ) · x1 +··· +(a1,n y1 · · · am,n ym ) · xn ≤ y1 b1 + · · · ym bm Clearly, we want to find the non-negative values y1 , .