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For every i, define cost(xi ) := 1 ci The intuition for this definition is that, at the step in which we covered xi , we had to “pay” for one set in order to cover ci elements that were previously uncovered. Thus, we can think of each element that we covered at that step as having cost us c1i times the cost of a set. 3. SET COVER VERSUS VERTEX COVER 29 n apx = cost(xi ) i=1 Now, consider the items xi , . . , xn and let us reason about how the optimum solution manages to cover them. Every set in our input covers at most ci of those n − i + 1 items, and it is possible, using the optimal solution, to cover all the items, including the items xi , .

If there is an edge that contains points of arbitrarily large cost, then output “optimum is unbounded” • Else, if there are edges that contain points of cost larger than (a1 , . . , an ), then let (b1 , . . , bn ) be the second endpoint of one of such edges – (a1 , . . , an ) := (b1 , . . , bn ); – go to 2 • Else, output “(a1 , . . , an ) is an optimal solution” This is the outline of an algorithm called the Simplex algorithm. It is not a complete description because: • We haven’t discussed how to find the initial vertex.

In general, how do we find a good choice of scaling factors for the inequalities, and what kind of upper bounds can we prove to the optimum? Suppose that we have a maximization linear program in standard form. maximize c1 x1 + . . cn xn subject to a1,1 x1 + . . + 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 , .

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