Non-Approximability Results

Summary- Gap technique

- Examples: MINIMUM GRAPH COLORING, MINIMUM TSP, MINIMUM BIN PACKING

- The PCP theorem- Application: Non-approximability of MAXIMUM 3-SAT

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)

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

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)

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)

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])

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)

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

Verifier

inputes. Boolean formula

proofes. truth assignment

Yes/No

Must read the entire proof

Deterministically checkable proofs

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

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)]

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)

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

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

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)

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

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

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

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

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