PDalgs - 15-854: Approximations AlgorithmsLecturer:R....

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Unformatted text preview: 15-854: Approximations AlgorithmsLecturer:R. RaviTopic:Primal-Dual AlgorithmsDate:Nov. 21, 2005Scribe:Daniel Golovin21.1Primal-Dual AlgorithmsSo far, we have seen many algorithms based on linear program (LP) relaxations, typically involvingrounding a given fractional LP solution to an integral solution of approximately the same objectivevalue. In this lecture, we will look at another approach to LP relaxations, in which we will constructa feasible integral solution to the LP from scratch, using a related LP to guide our decisions. OurLP will be called thePrimal LP, and the guiding LP will be called theDual LP.As we shall see, the PD method is quite powerful. Often, we can use the Primal-Dual (PD) methodto obtain a good approximation algorithm, and then extract a good combinatorial algorithm fromit. Conversely, sometimes we can use the PD method to prove good performance for combinatorialalgorithms, simply by reinterpreting them as PD algorithms. So without further ado...21.2Every Primal has a DualWe begin with a generic covering LP, and illustrate the ideas later with Vertex Cover as an example.Let [k] :={1,2,...,k}. Suppose we have matrixARmnand vectorscRn,bRm. We canrepresent the primal LP asminjcjxjsubject tojaijxjbii[m]xjj[n](Primal)Now suppose we want to develop a lower bound on the optimal value of this LP. One way to dothis is to find constraints that look likejcjxjZ, for someZ, using the constraints in the LP.To do this, note that any convex combination of constraints from the LP is also a valid constraint.Therefore, if we have non-negative multipliersyion the constraints, we get a new constraint whichis satisfied by all feasible solutions to the primal LP. That is, if for alli,jaijxjbi, thenXiyiXjaijxjXiyibi(21.2.1)Note that we require theyis to be non-negative, because multiplying an inequality (in this casejaijxjbi) by a negative number switches the sign of the inequality. (If a constraint has theformjaijxj=bi, then its multiplieryican be any real number.) Consider equation 21.2.1. If weensureiyijaijxjjcjxj, we will obtain a lower bound ofiyibion the optimal valueof the primal LP. Switching the order of summation, we getiyijaijxj=j(iyiaij)xj,1and can ensure this sum is at mostjcjxjby requiring theyis to satisfyXiyiaijcjj[n](21.2.2)(Note that in the previous step we rely on the fact that thexjs are non-negative.)Putting it all together, if theyis are non-negative and satisfy constraint 21.2.2, thenXiyibiXiyiXjaijxj=XjXiyiaij!xjXjcjxj(21.2.3)Note that the constraints on theyis are linear, as is the lower bound we obtain. Thus we can writedown an LP to find theyis giving the best lower bound. This is thedual LP....
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PDalgs - 15-854: Approximations AlgorithmsLecturer:R....

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