1. Non-Approximability Results

2. Summary -Gap technique -Examples: MINIMUM GRAPH COLORING, MINIMUM TSP, MINIMUM BIN PACKING -The PCP theorem -Application: Non-approximability of MAXIMUM 3-SAT

3. The Gap Technique -P 1 : NPO minimization problem (same for maximization) -P 2 : NP-hard decision problem -Function f that maps instances x of P 2 into instances f(x) of P 1 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 P 1 exists with r<(1+g) (unless P=NP)

4. 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 P 2 in polynomial time

5. 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)

6. 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)

7. 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.,  1  k  |V|-1 w(v π[k], v π[k+1] )+w(v π[|V|], v π[1] )

8. Inapproximability of TSP -NP-hard to decide whether a graph contains an Hamiltonian circuit -For any g>0, f(G=(V,E))=(G’=(V,V 2 ),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)

9. The NPO world (unless P=NP) NPO APX MAXIMUM SAT ( ?) MINIMUM VERTEX COVER( ?) MAXIMUM CUT( ?) PTAS MINIMUM PARTITION PO MINIMUM PATH MINIMUM TSP MINIMUM BIN PACKING MINIMUM GRAPH COLORING? Certainly not in PTAS

10. Verifier input es. Boolean formula proof es. truth assignment Yes/No Must read the entire proof Deterministically checkable proofs

11. Verifier input es. Boolean formula proof es. truth assignment Yes/No Trade-off between the number of random bits and the number of bits read? random bits Probabilistically checkable proofs

12. 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)]

13. 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)

14. 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)) 0 -Gap: g not explicitily computed -Theorem: MAXIMUM SAT is not r-approximable for r<1+g

15. The reduction -For any random string R of length O(logn), we construct a Boolean formula C R 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 -C R can be written in CNF -The final formula f(x) is the union of all these formulas C R

16. q2q2 01 q2q2 0 0 0 00 0 1 1 1 1 1 q1q1 q3q3 q3q3 q3q3 q3q3 no yes The reduction (continued) -Consider the decision tree of the verifier corresponding to R and let C R encode the accepting paths (not q 1 and not q 2 and q 3 ) or (not q 1 and q 2 and not q 3 ) or (q 1 and q 2 and q 3 )

17. Proof -Let k be the number of clauses in C R (we may assume it is the same for each R) -We have p(n)=2 hlogn =n h formulas C R, that is, the final formula has c(x)=kp(n) clauses -If x is satisfiable, the verifier accept for any R. Any formula C R is satisfiable: hence, m*(f(x))=c(x) -If x is not satisfiable, less than 1/2 of the formulas C R 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

18. 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

19. Product graphs -Given a graph G=(V,E), define G 2 (V 2,E 2 ) as V 2 ={(u,v) : u,v  V} and E 2 ={((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 G 2 has a clique of k 2 nodes

20. 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 G 2, then compute U 2 = A (G 2 ) and, finally, compute the corresponding U in G -From theorem above, m*(G)=sqrt(m*(G 2 )) and |U|=sqrt(|U 2 |) -Hence, performance ratio of U is at most sqrt(r) -Iterating we can obtain any performance ratio

21. The NPO world if P  NP NPO APX MAXIMUM SAT MINIMUM VERTEX COVER( ?) MAXIMUM CUT( ?) PTAS MINIMUM PARTITION PO MINIMUM PATH MINIMUM TSP MINIMUM BIN PACKING MINIMUM GRAPH COLORING? Certainly not in PTAS MAXIMUM CLIQUE? Either in PTAS or not in APX