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# 635 Class Note for CSE 565 at PSU

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COURSE
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KARMA
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This 2 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at Pennsylvania State University taught by a professor in Fall. Since its upload, it has received 29 views.

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Date Created: 02/06/15
Algorithm Design and Analysis LECTURE 41 Approximation Algorithms The Pricing Method VerteX Cover Sofya Raskhodnikova 1272007 S Raskhodnikova based on slides by K Wayne Weighted Vertex Cover Weighted vertex cover Given a graph 6 with vertex weights find a vertex cover of minimum weight weight 2 2 48 weight 2911 Pricing Method Idea Each edge must be covered by some vertex Edge e pays price pg 2 O to get covered Fairness Edges incident to vertex i should pay S wi in total for each vertex i 2 pg S w eltijgt Fairness Lemma For any vertex cover 5 and any fair prices pg 28 p8 S wS Proof Zpe S 2 Zpe S Zwi eeE ieS eij ieS each edge e covered by sum fairness inequalities at least one node in S for each node in S Pricing Method Pricing method Set prices and find vertex cover simultaneously Weighted Vartex Cover Approx G VE w foreach e in E pe lt 0 foreach v in V 10 wi totalv 0 1 while El edge ij such that neither i nor j is tight select such an edge e give pe maximum increase without violating fairness increase max wi totaliwj totalj pe pe increase totali totali increase totalI totalI increase S e set of all tight nodes return S Pricing Method Figure 118 Pricing Method Analysis Theorem Pricing method gives a 2approximation Pf Algorithm terminates since at least one new node becomes tight after each iteration of while loop Let S set of a tight nodes upon termination of algorithm 5 is a vertex cover if some edge iJ is uncovered then neither i norJ is tight But then while loop would not terminate Let 5 be an optimal vertex cover We show wS S 2wS wS Zwi Z Zpe S 2 Zpe 22196 S 2wS ieS ieS eij ieV eij eeE T i fairnIess lemma all nodes in S are tight 5 g V each edge counted twuce prices 2 0 Running time Om deved Van Caul Theorem zappmx mavm a gomhm for wmghved vervzx cover Theorem bmungaframm If P NP then no pwoxwmanon for plt1360722n wuh um wzwghrs 106 721 Open research promzm dose m gap

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