Non-Approximability Results
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Non-Approximability Results
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Summary- Gap technique
- Examples: MINIMUM GRAPH COLORING, MINIMUM TSP, MINIMUM BIN PACKING
- The PCP theorem- Application: Non-approximability of MAXIMUM 3-SAT
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The Gap Technique
- P1: NPO minimization problem (same for maximization)
- P2: NP-hard decision problem
- Function f that maps instances x of P2 into instances f(x) of P1 such that:
- If x is a YES-instance, then m*(f(x))=c(x)- If x is a NO-instance, then m*(f(x)) c(x)(1+g)
- Theorem: No r-approximation algorithm for P1 exists with r<(1+g) (unless P=NP)
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Proof- A: r-approximation algorithm with r<(1+g)
- If x is a YES-instance, then m*(f(x))=c(x). Hence, m(f(x),A(f(x))) rm*(f(x))=rc(x)<c(x)(1+g)
- If x is a NO-instance, then m*(f(x)) c(x)(1+g). Hence, m(f(x),A(f(x))) c(x)(1+g)
- A allows to decide P2 in polynomial time
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Inapproximability of graph coloring- NP-hard to decide whether a planar graph can be
colored with 3 colors- Any planar graph is 4-colorable
- f(G)=G where G is a planar graph- If G is 3-colorable, then m*(f(G))=3- If G is not 3-colorable, then m*(f(G))=4=3(1+1/3)- Gap: g=1/3
- Theorem: MINIMUM GRAPH COLORING has no r-approximation algorithm with r<4/3 (unless P=NP)
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Inapproximability of bin packing- NP-hard to decide whether a set of integers I can be
partitioned into two equal sets
- f(I)=(I,B) where B is equal to half the total sum- If I is a YES-instance, then m*(f(I))=2- If G is a NO-instance, then m*(f(G)) 3=2(1+1/2)- Gap: g=1/2
- Theorem: MINIMUM BIN PACKING has no r-approximation algorithm with r<3/2 (unless P=NP)
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MINIMUM TSP- INSTANCE: Complete graph G=(V,E), weight
function on E
- SOLUTION: A tour of all vertices, that is, a permutation π of V
- MEASURE: Cost of the tour, i.e., 1k |V|-1w(vπ[k], vπ[k+1])+w(vπ[|V|], vπ[1])
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Inapproximability of TSP- NP-hard to decide whether a graph contains an
Hamiltonian circuit
- For any g>0, f(G=(V,E))=(G’=(V,V2),w) where w(u,v)=1 if (u,v) is in E, otherwise w(u,v)=1+|V|g- If G has an Hamiltonian circuit, then m*(f(G))=|V|- If G has no Hamiltonian circuit, then
m*(f(G)) |V|-1+1+|V|g=|V|(1+g)- Gap: any g>0
- Theorem: MINIMUM TSP has no r-approximation algorithm with r>1 (unless P=NP)
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The NPO world (unless P=NP)NPO
APXMAXIMUM SAT ( ?)MINIMUM VERTEX COVER( ?)MAXIMUM CUT( ?)
PTAS MINIMUM PARTITION
POMINIMUM PATH
MINIMUM TSP
MINIMUM BIN PACKING
MINIMUM GRAPH COLORING? Certainly not in PTAS
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Verifier
inputes. Boolean formula
proofes. truth assignment
Yes/No
Must read the entire proof
Deterministically checkable proofs
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Verifier
inputes. Boolean formula
proofes. truth assignment
Yes/No
Trade-off between the numberof random bits and the number of bits read?
random bits
Probabilistically checkable proofs
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PCP[r,q]- A decision problem P belongs to PCP[r,q] if it admits
a polynomial-time verifier A such that:- For any input of length n, A uses r(n) bits casuali- For any input of length n, A queries q(n) bits of the proof- For any YES-instance x, there exists a proof such that A
answers Yes with probability 1- For any NO-instance x, for any proof A answers Yes with
probability less than 1/2
- Theorem: NP=PCP[log,O(1)]
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The PCP theorem- Given a class F of functions, PCP(r, F) is the union of
PCP[r,q], for all q F- By definition, NP=PCP(0,poly) where poly is the set
of polynomials
- Theorem: NP=PCP(log,O(1))- Proving that NP includes PCP(log, O(1)) is easy- Proving that NP is included in PCP(log, O(1)) is hard
(complete proof is more than 50 pages: we will see only a part)
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Inapproximability of satisfiability- Gap technique
- Intuitive motivation: gap in acceptance probability corresponds to gap in measure
- Reduction f from SAT such that- If x is satisfiable, m*(f(x))=c(x) where c(x) is the number of
clauses in f(x)- If x is not satisfiable, m*(f(x))<c(x)/(1+g) with g>0- Gap: g not explicitily computed
- Theorem: MAXIMUM SAT is not r-approximable for r<1+g
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The reduction- For any random string R of length O(logn), we
construct a Boolean formula CR of constant size which is satisfiable if and only if there exists a sequence of answers to the queries that make the verifier answer Yes
- CR can be written in CNF
- The final formula f(x) is the union of all these formulas CR
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q2
0 1q2
0 0
0000
1
1 1 1 1
q1
q3 q3q3q3
no no no no noyes yes yes
The reduction (continued)- Consider the decision tree of the verifier
corresponding to R and let CR encode the accepting paths
(not q1 and not q2 and q3) or (not q1 and q2 and not q3) or (q1 and q2 and q3)
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Proof
- Let k be the number of clauses in CR (we may assume it is the same for each R)
- We have p(n)=2hlogn=nh formulas CR, that is, the final formula has c(x)=kp(n) clauses
- If x is satisfiable, the verifier accept for any R. Any formula CR is satisfiable: hence, m*(f(x))=c(x)
- If x is not satisfiable, less than 1/2 of the formulas CR are satisfiable. Hence,
m*(f(x))<c(x)/2+(k-1)p(n)/2=c(x)(1-1/(2k))=c(x)/(1+g)
- g depends on the number of queries
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MAXIMUM CLIQUE- INSTANCE: Graph G=(V,E)
- SOLUTION: A subset U of V such that, for any two vertices u and v in U, (u,v) is in E
- MEASURE: Cardinality of U
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Product graphs- Given a graph G=(V,E), define G2(V2,E2) as
V2={(u,v) : u,v V} andE2={((u,v),(x,y)) : u=x and (v,y) is in E or (u,x) is in E}
- Theorem: G has a clique of k nodes iff G2 has a clique of k2 nodes
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Self-improvability of clique- Theorem: If MAXIMUM CLIQUE belongs to APX,
then MAXIMUM CLIQUE belongs to PTAS- Let A be an r-approximation algorithm for MAXIMUM
CLIQUE- Compute G2, then compute U2=A(G2) and, finally, compute
the corresponding U in G- From theorem above, m*(G)=sqrt(m*(G2)) and |U|=sqrt(|
U2|)- Hence, performance ratio of U is at most sqrt(r)- Iterating we can obtain any performance ratio
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The NPO world if PNPNPO
APXMAXIMUM SATMINIMUM VERTEX COVER( ?)MAXIMUM CUT( ?)
PTAS MINIMUM PARTITION
POMINIMUM PATH
MINIMUM TSP
MINIMUM BIN PACKING
MINIMUM GRAPH COLORING? Certainly not in PTASMAXIMUM CLIQUE? Either in PTAS or not in APX