Combinatorial optimization: Exact and approximate algorithms by Trevisan L.

By Trevisan L.

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Example text

This is not entirely a coincidence: for each inequality, for example x1 + 2x2 ≤ 1, we can look at the line which is the boundary between the region of points that satisfy the inequality and the region of points that do not, that is, the line x1 + 2x1 = 1 in this example. The points on the line that satisfy the other constraints form a segment (in the example, the segment of the line x1 + 2x2 = 1 such that 0 ≤ x1 ≤ 1/3), and that segment is one of the edges of the polygon of feasible solutions. Although it does not happen in our example, it could also be that if we take one of the inequalities, consider the line which is the boundary of the set of points that satisfy the inequality, and look at which points on the line are feasible for the linear program, we end up with the empty set (for example, suppose that in the above example we also had the inequality x1 + x2 ≥ −1); in this case the inequality does not give rise to an edge of the polygon of feasible solutions.

After skipping Sunnyvale the second time, we have the cycle Palo Alto → Mountain View → Sunnyvale → Santa Clara → Cupertino → Palo Alto whose length is 36 miles. In summary, yesterday’s algorithm finds a solution of 37 miles, and a certificate that the optimum is at least 21. Today’s algorithm finds a solution of 36 miles, and a certificate that the optimum is at least 34. 6 The Minimum Set Cover problem is defined as follows: an input of the problem is a finite set X and a collection of subsets S1 , .

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 , . . , ym such that the above upper bound is as strong as possible, that is we want to minimize b1 y1 + · · · bm ym subject to a1,1 y1 + . . + am,1 ym ≥ c1 .. an,1 y1 + . . + am,n ym ≥ cn y1 ≥ 0 .. 4) ym ≥ 0 So we find out that if we want to find the scaling factors that give us the best possible upper bound to the optimum of a linear program in standard maximization form, we end up with a new linear program, in standard minimization form.

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