Lecture 23 Scribble Topics Reductions

Chat Moderators Junyeob Cliques Independent so

Emerson SAT

Two Problems XEpY

Yes4 R yAYµ

Ax

A finding shortest path from 5 tot Djikostoa's

Ay finding all shortest paths Bettman Ford

Ojikstra's E Ballmautford

easy has a

Yes polynomial time4 R s Ay deterministicsolution

No

A hard

X Ep Id X is no harder than't

If X is hard it should also be hardY is atleast as hard as X

Example CliqueyIndependent SetGCU E

eeIndependent set I C V St

Tryno Yu C I are connected

6 eby an edge aod

Clique C E U sat u u EE offor all u v C C e f

Given GCU E and an integer k i

Does G have an II SI 3 K MAX INO SETDoes G have a clique let 3k MAX CLIQUE

Ay Solves MAX CLIQUE

V EGCU E

a a

dd

t.LIe f

yesMIS Ine T R finding the edgeG R Max Clique

9 No complement of theG graphMax IS 0Gt

Maxis Ep MarxClique Max IS p Max Clique

yesX Ep Y

4 R yAYµ

A

If Y is hard is also hard FALSE

If X is known to be hard then Y does not havea efficient algorithm TRUEWWant to prove HW P 4G is hardReduce a hard problem to X

GIS or Clique

If Y has a polynomial time algorithm then it impliesX has a polynomial time algorithm D pendsIf R is polynomial then true

If R is hard the false

X Ep Y implies Yep X FALSE

f X Ep Y and Y Ep Z then X Ep 2 TRUE

no

SAT Problems CNF SAT 3CNF 3 SAT

X x xz Xn V o G A and o

x U zV Iii N xz V x 5 Axs Is there a assignment to

fr the boolean variables etclause the expression is true

conjunctive normal form Iff.ITmaIIIi3CNFx VxzVx a N xioVxzVxzDN

Every boolean expression can be expressed as 3 CNF

aba Ab EUWE r a Ub N auto

i

1

Problem SAT satisfiability NRGiven a CDF formula is there an assignment which results

in the formula as evaluating true

Every CNF expression caru be reduced to 3 CNFif clause 3 literals Ca V love Lavrov e Acc VEX Y are boolean expressions if X V Y are SATVybve U d Caubuz A evolve then X z h Y E are SAT

if clause has 3k literal

aub aub Vu A fav 6 VE

if clause has 1 literala La Vu Vu a Cavivu a Cavuv Native