Welcome to ... Problem Analysis and Complexity Theory 716.054, 3 VU Birgit Vogtenhuber Institute for Software Technology Graz University of Technology email: [email protected] office: Inffeldgasse 16B/II, room IC02116 office hour: by appointment, in webex meeting room office hour: Wednesday 10:30-11:30 slides: http://www.ist.tugraz.at/pact20.html Birgit Vogtenhuber Problem Analysis and Complexity Theory 196 Last Time Optimization and Approximation • Decision vs. Optimization • Notion of Approximation: n v(x;y) opt(x) o ◦ approximation ratio R(x; y) = max opt(x) ; v(x;y) ◦ defined r(n)-approximation algorithm • First complexity classes for optimization problems: PO, NPO, APX(r(n)), APX, APX∗ • First optimization problems: MIN VERTEX COVER, MAX (k-DEG.) INDEP. SET Birgit Vogtenhuber Problem Analysis and Complexity Theory 197 - 7 Last Time Presentations: Organizational Issues • Format of the presentations: pdf (optionally: same as lecture) • Length of each presentation: 30-40 minutes • For each presentation: Discussion, Questions • Presentations will take place on Tuesday, 30.06.2020, via Webex, during the lecture time. • Timing: send presentation and needed time at latest until Friday, 26.06.2020 Birgit Vogtenhuber Problem Analysis and Complexity Theory 198 - 2 Last Time Presentations: Organizational Issues More Optimization and Approximation: • Consider MIN TSP • Prove the Gap Theorem • ConsiderGiven an MIN optimization TSP∆ problem such that 1. 8 inputs x 2 I and 8 solutions s 2 S(x), s 62 (a; b), • Introducethat is,approximation either v(s) ≤ a schemeor v(s) ≥, PTASb, and, and FPTAS • 2.Consider it is NP MAX-hard KNAPSACK to determine if opt(x) ≤ a or opt(x) ≥ b. Then it is NP-hard to obtain a worst-case approximation ratio that is smaller than b=a. Birgit Vogtenhuber Problem Analysis and Complexity Theory 198 - 7 Last Time Presentations: Organizational Issues More Optimization and Approximation: • Consider MIN TSP • Prove the Gap Theorem • Consider MIN TSP∆ • Introduce approximation scheme, PTAS, and FPTAS • Consider MAX KNAPSACK Birgit Vogtenhuber Problem Analysis and Complexity Theory 198 - 10 Last Time Definition: Algorithm M is approximation scheme for optimization problem A = (I; S; v; d): 8x 2 I; 8" > 1: R(x; M(x; ")) ≤ ": Definition: Class PTAS: Optimization problems with polynomial time approximation scheme M: M runs in polynomial time. Definition: Class FPTAS: Optimization problems with fully polynomial time approximation scheme M: M(x; ") runs in time polynomial in jxj and 1=(" − 1)). Birgit Vogtenhuber Problem Analysis and Complexity Theory 198 - 13 Last Time Presentations: Organizational Issues More Optimization and Approximation: • Consider MIN TSP • Prove the Gap Theorem • Consider MIN TSP∆ • Introduce approximation scheme, PTAS, and FPTAS • Consider MAX KNAPSACK Birgit Vogtenhuber Problem Analysis and Complexity Theory 198 - 14 This Time / Next Time Optimization and Approximation: • Learn about problems with small solution values • Consider an intermediate overview of approximation results with respect to small solution values • Learn to make reductions for approximation problems • New concept / class PCP: Probabilistically Checkable Proofs • Maybe some bonus material • Summary Presentations Birgit Vogtenhuber Problem Analysis and Complexity Theory 199 - 8 Problems with Small Solutions • We have just seen that MAX KNAPSACK 2 FPTAS. • We have also seen that MAX KNAPSACK is pseudopolynomial: If the maximum value V is polynomial, then so is the optimum solution, and the problem can be solved in polynomial time. ) What about problems that are also complicated for small solution values? Especially, optimization problems whose decision variants are strongly NP-hard? Definition: An optimization problem is a problem with small solution values if all solutions are positive integers with values polynomially bounded in the input length. Birgit Vogtenhuber Problem Analysis and Complexity Theory 200 - 5 Problems with Small Solutions Theorem: If P6= NP, then no NP-hard problem with small solution values has an FPTAS. Proof: Exercise. • Assume(solutions existence≤ ofp(n an); FPTAS"(n) = 1A=p.(n) + 1;) • Let jxj = n and p(n) be a polynomial bound for the solution values for x, and Let "(n) := 1=p(n) + 1 ) A(x; "(n)) runs in q(n; 1=("(n) − 1)) = q(n; p(n)) time, which is polynomial in n. ) A(x; "(n)) can't be the optimal solution for all inputs x. ) As all solutions 2 f1; : : : ; p(n)g, every non-optimal solution gives a worst case approximation ratio of at least p(n)=(p(n) − 1) = 1 + 1=(p(n) − 1) > "(n). Birgit Vogtenhuber Problem Analysis and Complexity Theory 200 - 8 Problems with Small Solutions Theorem: If P6= NP, then no NP-hard problem with small solution values has an FPTAS. Proof: Exercise. • Assume existence of an FPTAS A. • Let jxj = n and p(n) be a polynomial bound for the solution values for x, and Let "(n) := 1=p(n) + 1 ) A(x; "(n)) runs in q(n; 1=("(n) − 1)) = q(n; p(n)) time, which is polynomial in n. ) A(x; "(n)) can't be the optimal solution for all inputs x. ) As all solutions 2 f1; : : : ; p(n)g, every non-optimal solution gives a worst case approximation ratio of at least p(n)=(p(n) − 1) = 1 + 1=(p(n) − 1) > "(n). Birgit Vogtenhuber Problem Analysis and Complexity Theory 200 - 10 Approximation Results Approximation results we have seen by now: • MAX KNAPSACK 2 FPTAS • MIN TSP 62 APX if P6= NP: Question: gap-argument with HAMCYCLE What do the insights • MIN TSP∆ 2 APX: about small solution 2-approximation values imply for these • MIN VERTEX COVER 2 APX: problems? 2-approximation • k-DEGREE INDEP. SET 2 APX for k constant: (k + 1)-approximation. Birgit Vogtenhuber Problem Analysis and Complexity Theory 201 - 7 Approximation Results Approximation results we have seen by now: • MAX KNAPSACK 2 FPTAS • MIN TSP 62 APX if P6= NP: gap-argument with HAMCYCLE • MIN TSP∆ 2 APX: 2-approximation • MIN VERTEX COVER 2 APX: 2-approximation • k-DEGREE INDEP. SET 2 APX for k constant: (k + 1)-approximation. 62 FPTAS if P6= NP: small solution values Birgit Vogtenhuber Problem Analysis and Complexity Theory 201 - 8 PTAS Reductions Definition: A PTAS reduction A ≤P T AS B of an opt. problem A = (IA;SA; vA; dA) to B = (IB;SB; vB; dB) is a function-triple (f; g; α) with the following properties: • f : IA ! IB maps instances of A to instances of B and can be computed in polynomial time. • g maps triples (x; s; "), where x 2 IA, s 2 SB(f(x)), + and " 2 Q to solutions g(x; s; ") 2 SA(x) and can be computed in polynomial time • α : Q+ ! Q+ is a surjective polynomial-time computable function. And • if rB(f(x); s) ≤ 1 + α("), then rA(x; g(x; s; ")) ≤ 1 + ". Birgit Vogtenhuber Problem Analysis and Complexity Theory 202 - 5 PTAS Reductions Definition: A PTAS reduction A ≤P T AS B of an opt. problem A = (IA;SA; vA; dA) to B = (IB;SB; vB; dB) is a function-triple (f; g; α) with the following properties: • f : IA ! IB maps instances of A to instances of B and can be computed in polynomial time. • g maps triples (x; s; "), where x 2 IA, s 2 SB(f(x)), + and " 2 Q to solutions g(x; s; ") 2 SA(x) and can be computed in polynomial time • α : Q+ ! Q+ is a surjective polynomial-time computable function. And • if rB(f(x); s) ≤ 1 + α("), then rA(x; g(x; s; ")) ≤ 1 + ". Birgit Vogtenhuber Problem Analysis and Complexity Theory 202 - 6 PTAS Reductions Definition: A PTAS reduction A ≤P T AS B of an opt. problem A = (IA;SA; vA; dA) to B = (IB;SB; vB; dB) is a function-triple (f; g; α) with the following properties: • f : IA ! IB maps instances off A to instances of B IB and can be computedIA in polynomial time. f•, g,maps and α triples (x; s; "), where x 2 IA, s 2 SB(f(x)), + poly-timeand " 2 Q to solutions g(x; s; ") 2 SA(x) and can be computablecomputed in polynomial time SA(x) + + SB(f(x)) • α : Q ! Q is a surjectiveg polynomial-time computable function. And • ifz rB(f(x); s}|) ≤ 1 + α("{), thenz rA(x; g(x;}| s; ")) ≤ 1 +{ ". rA(x; g(x; s; ")) ≤ 1 + " ( rB(f(x); s) ≤ 1 + α(") Birgit Vogtenhuber Problem Analysis and Complexity Theory 202 - 7 PTAS Reductions Lemma: If A ≤P T AS B and B 2 PTAS, then A 2 PTAS. Proof: • PTAS for A: must get input (x; ") • Compute (f(x); α(")) in polynomial time • Use PTAS for B on (f(x); 1 + α(")) to obtain a (1 + α("))-approximation s of B • Compute g(x; s; ") 2 SA(x) in polynomial time. • By the last property, g(x; s; ") is a (1+")-approximation of x for B. Remark: some literature uses the term "-optimal solution.X "-optimal solution == (1 + ")-approximation Birgit Vogtenhuber Problem Analysis and Complexity Theory 202 - 15 PTAS Reductions Lemma: If A ≤P T AS B and B 2 PTAS, then A 2 PTAS. Lemma: If A ≤P T AS B and B 2 APX, then A 2 APXX. Proof: • B 2 APX ) 9 poly-time (1+δ)-approximation algorithm for B for some δ 2 Q+. • α surjective ) 9 " 2 Q+ : α(") = δ. • Combining f, g, and (1+δ)-approximation algorithm for B gives a (1+")-approximation algorithm for A.X Birgit Vogtenhuber Problem Analysis and Complexity Theory 202 - 20 PTAS Reductions Lemma: If A ≤P T AS B and B 2 PTAS, then A 2 PTAS. Lemma: If A ≤P T AS B and B 2 APX, then A 2 APXX. Theorem: MAX 2SAT ≤P T AS MAX INDEP. SET andX MAX INDEP. SET ≡P T AS MAX CLIQUE Proof: Exercise.