Primal-Dual Approximation Algorithms

May 11th, 2015
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We just saw how the primal-dual schema permits sometimes designing efficient combinatorial algorithms for solving certain problems. We will now see an example of how a related technique can sometimes be used to design effecient approximation algorithms

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Primal DualPrimal-Dual Approximation AlgorithmsWe just saw how the primal-dual schema permits sometimes designing efcient combinatorial algorithms for solving certain problems. We will now see an exampleof how a related technique can sometimes be used todesign efcient approximation algorithmsThe major tool that we will use will be theRELAXED Complementary Slackness conditionsThe problem we examine will again beweighted set-cover.1Recall that given canonical primalnminimizecj xjsubject toj =1a x bi,ixj 0,i = 1, . . . , mj = 1, . . . , nthe dual ismmaximizesubject tobiii=1Aj cj ,i 0,j = 1, . . . , ni = 1, . . . , m2Theorem (Complementary Slackness):Let x and respectively be primal and dual feasiblesolutions. Then x and are both optimal if and only ifall of the following conditions are satised.Primal Complementary Slackness conditions1 j n : either xj = 0 or Aj = cjDual Complementary Slackness conditions1 i m : either i = 0 or a x = bii3Theorem (RELAXED Complementary Slackness):Let x and y respectively be primal and dual feasible solutions. Suppose further that for some > 1, x and y satisfy all ofPrimal Complementary Slackness conditions1 j n : either xj = 0 or Aj = cjRELAXED Dual C.S. conditions1 i m : either i = 0 or a x biiThenmncj xj j =1biii=1Proof:j =1a x

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