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# 555 Class Note for CSE 565 at PSU

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COURSE
PROF.
No professor available
TYPE
Class Notes
PAGES
3
WORDS
KARMA
25 ?

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This 3 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 21 views.

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Date Created: 02/06/15
Algorithm Design and Analysis Tractability Finding Small Vertex Covers Solving NPhard Graph Problems on Trees m LECTURES 38 39 Extending the Limits of Sofya Raskhodnikova quotquotm quot s germinate baredonrlui em Weyne Coping With NPCornpieteness Q Suppose I need to solve an NPecoinpiete probiein what shouid 1 dov A Theory says you re unlikely to tind poiyetiine aigorithin Must sacritice one ot three desired features Solve probiein to optimality Solve probiein in polynomial time Solve arbitrary instances ot the probiein This iecture Solve soine speciai cases ot Npecoinpiete probieins that arise in practice Finding Small Vertex Covers Vertex Cover A vertex cover of a graph 6 v E is a subset of vertices s g v such that for each edge u v either u e s or v e s or both Problem Given a graph 5 v E and an integer k find avertex cover of size S k if it exists k4 S36710 Finding Small Vertex Covers Q What if k is small7 Brute torce ote nm Try aii Cn e otnt subsets ot size e Takes oten time to chech whether asubset is avertex cover Goal Limit exponential dependency on k e g to 02k n Ex n 1000 k 10 Brute force knk 11034 2 infeasible Better 2k n 10A 2 feasible Remark It e is a constant aigorithin is poiyetime it e is a sinaii constant then it s also practical Important The aigorithin is stiii exponential and hence scaies badly e g consider k40 However it is better than brute torce Finding Small Vertex Covers Clalml Let uev be an edge ot s s has avertex cover ot size s e itt at ieast one ot s etu ands em has avertex cover ot size s eel delete v and all incident edges Pt 2 Suppose s has avertex cover 5 ot size s e S contains either u or v or both WL06 assume it contains u s etu is avertex cover on etu Pt e Suppose s is avertex cover on etu ot size s n1 Then 5 0 u is avertex cover on Finding Small VIItux Conn Algorithm 0mm 2 The followmg algomvhm nd avirvzx cover of me s x m s m H mm md runs m o2 H mm Pf CCI IIZMI folbvt quotan 3le l 7hr 11112 le nu nunan in m muna uh am quotI 1 Finding Small Var Cm amnion Tm Solving NP Hard Graph Problems on Treas Wm 5n IITI Independent m on was Gwen a wee mm a maximum cwdmuhvy subszv of nodes such mm no we share an edgz Fan A quot22 on av 42cm vwo mdzs has at lzast vwo leafmdes deg39eezl Key obseryamn In is a um mm zxms a maximum SIZE mdepBYvdem sew cmvarmgy w exchaga mumpm mmmuchquot auntquot ms Guianau xuuSudnsnm5u nmucpummasmmam m 5am 5lm5vv uu mom M 5m ulna M Al rmn Theorem The followmg greedy algomhm mas amaxmum cardlmlny mdepzndzm 52 m fovuv ma hence Vreu w Correcmess follows from m mymus key obseryanon Remark Can mpiemzm m 00mm by comamg nodes m poslordzr MIMIummmnum delnod 1 ll an Tn wigmea iniqude m a in lm m u nod might uv gt0 nd a indqmdan m 5 Mm mainlxa 2W5 IV on mm xv u v u an a mi nm N inf ands n mm on inchdo u n mm u loaf pd amidm m in Dynamic PFWi quotMin Ilak un nod r u OM not MJ imudglmdQlvm immdum gu UT M IWM Juarmdng mm d scapulay Indqendm39 5 an Treu Algorith Theorem The dynammprogr anmmgabgomhm mas amaxmum weighvzd mdepkndem 3239 m vrus m omva a m L M 39 the a 3 2 um I WI Ellquot Exzrmsz Recover m2 mdzpzndem 2 of maxmmm wetghv by mung back w Tens om Mme smcz we vrsw nodes In posvwder and ixmvme mm Bdgz many om Can39t Indepmdem 52v on ms ms svmcmred spima case 5 ma a becmse we cm ma anode may um um m munmln among m2 subpvob zms m dwffzrzm subvrzzs see Chamer 10 4 vaphs of bounded rm mm E egam generahzanon of mg mar cwmru mm dos of 9739th mm arm n prochce 2mm dnompoxmon mo mdependcm Wm

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