Black-box (oracle)
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Transcript of Black-box (oracle)
Black-box (oracle)
Feed me a weighted graph G and I will tell you the weight of the max-weight matching of G.
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Black-box (oracle)
Feed me a weighted graph G and I will tell you the weight of the max-weight matching of G.
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Black-box (oracle)
Feed me a weighted graph G and I will tell you the weight of the max-weight matching of G.
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Black-box (oracle)Feed me a weighted graph G and I will tell you the weight of the max-weight matching of G.
here is a graph G, find the max-weight matching
G
Black-box (oracle)Feed me a weighted graph G and I will tell you the weight of the max-weight matching of G.
here is a graph G, find the max-weight matching
pick a vertex uV(G) for each edge {u,v}E(G) wundefined if oracle(G-u-v) + w(u,v) = oracle (G) then wv if w is undefined then recurse on (G-u) else print({u,w}); recurse on (G-u-v)
3-SATx = variablex = negation of a variable
literalsclause = disjunction of literals x y z x z
3-SATINSTANCE: collection C of clauses, each clause has at most 3 literalsQUESTION: does there exist an assignment of true/false to the variables which satisfies all the clauses in C
3-SAT INSTANCE: collection C of clauses, each clause has at most 3 literalsQUESTION: does there exist an assignment of true/false to the variables which satisfies all the clauses in C
x y zx y zx yx
Independent Set
subset S of vertices such that notwo vertices in S are connected
Independent Set
subset S of vertices such that notwo vertices in S are connected
Independent Set
INSTANCE: graph GSOLUTION: independent set S in GMEASURE: maximize the size of S
INSTANCE: graph G, number KQUESTION: does G have independent set of size K
OPTIMIZATION VERSION:
DECISION VERSION:
Independent Set 3-SAT
“is easier than”
if we have a black-box for 3-SAT then we can solve Independent Setin polynomial time
Independent Set reduces to 3-SAT
Independent Set 3-SATif we have a black-box for 3-SAT then we can solve Independent Set in polynomial time
Give me a 3-SATformula and I willtell you if it is satisfiable
We would like to solve theIndependent Set problemusing the black box in polynomial time.
Independent Set 3-SATGive me a 3-SATformula and I willtell you if it is satisfiable
Graph G, K 3-SAT formula Fefficient transformation (i.e., polynomial – time)G has independent set of size K F is satisfiable
Independent Set 3-SATGive me a 3-SATformula and I willtell you if it is satisfiable
Graph G, K 3-SAT formula FV = {1,...,n} variables x1,....,xnE = edges xi xj for ij E
+ we need to ensure that K ofthe xi are TRUE
3-SAT Independent SetGive me a graph G and a number K and I willtell you if G has independentset of size K
3-SAT formula F graph G, number K
3-SAT Independent SetGive me a graph G and a number K and I willtell you if G has independentset of size K
3-SAT formula F graph G, number Kx y z w y z
xw
y y
zz
3-SAT Independent Set3-SAT formula F graph G, number K
x y z w y z
xw
y y
zz
1) efficiently computable2) F satisfiable IS of size m3) IS of size m F satisfiable
3-SAT Independent SetIndependent Set 3-SAT
if 3-SAT is in P then Independent Set is in Pif Independent Set is in P then 3-SAT is in P
3-SATIndependent Set
Many more reductions
3-SATIndependent Set
CliqueSubset-Sum3-COLPlanar 3-COLHamiltonian path
P and NP
P = decision problems that can be solved in polynomial time.
NP = decision problems for which the YES answer can be certified and this certificate can be verified in polynomial time.
NP = decision problems for which the YES answer can be certified and this certificate can be verified in polynomial time.
3-SAT Independent Set
NOT-3-SAT ?
NP = decision problems for which the YES answer can be certified and this certificate can be verified in polynomial time.
Every problem A NP A 3-SAT
COOK’S THEOREM
NP = decision problems for which the YES answer can be certified and this certificate can be verified in polynomial time.
if every problem A NP A B
B is NP-hard
B is NP-completeif B is NP-hard, andB is in NP
NP
P
NP-complete
NP-hard
3-SATIndependent Set
CliqueSubset-Sum3-COLPlanar 3-COLHamiltonian path
Some NP-complete problems
Clique
subset S of vertices such that everytwo vertices in S are connected
Clique INSTANCE: graph G, number KQUESTION: does G have a clique of size K?
Subset-Sum INSTANCE: numbers a1,...,an,B QUESTIONS: is there S {1,...,n} such that
ai = B iS
3-COL INSTANCE: graph G
QUESTION: can the vertices of G be assigned colors red,green,blue so that no two neighboring vertices have the same color?
3-SAT 3-COL
R G
B
x x
B
x y zG G G
G R
G=true
Planar-3-COL INSTANCE: planar graph G
QUESTION: can the vertices of G be assigned colors red,green,blue so that no two neighboring vertices have the same color?
3-COL Planar-3-COL
4-COL INSTANCE: graph G
QUESTION: can the vertices of G be assigned one of 4 colors so that no two neighboring vertices have the same color?
3-COL 4-COL
3-COL 4-COL
G G
planar 4-COL INSTANCE: planar graph G
QUESTION: can the vertices of G be assigned one of 4 colors so that no two neighboring vertices have the same color?
planar 3-COL planar 4-COL ???
4-COL 3-COL
Thus:
4-COL 3-COL
4-COL NPCook 4-COL 3-SAT3-SAT 3-COL
2-COL 3-COL
2-COL 3-COL
G G
3-COL 2-COL ??? 2-COL in P