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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 IC02044 office hour: Wednesday 10:30-11:30 slides: http://www.ist.tugraz.at/pact16.html Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 1 Institute for Software Technology Graz University of Technology Last Time finished Space Complexity • showed that TQBF is PSPACE-complete • made some space-summary discussed Questions Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 2 Institute for Software Technology Graz University of Technology Last Time started Optimization and Approximation • Discussed Decision vs. Optimization • Formalized the 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 • Defined complexity classes for optimization problems: PO, NPO, APX(r(n)), APX, APX∗ Question:• ConsideredHow the are following these classes optimization defined? problems: Question:MIN VERTEXWhat COVER, are their containmentMAX (k-DEG.) relations? INDEP. SET Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 3-6 Institute for Software Technology Graz University of Technology Last Time started Optimization and Approximation • Discussed Decision vs. Optimization • Formalized the 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 • Defined complexity classes for optimization problems: PO, NPO, APX(r(n)), APX, APX∗ • ConsideredShowed that the for following some NP optimization-hard problems, problems: the MINoptimization VERTEX problems COVER, can MAX be efficiently (k-DEG.) approximated INDEP. SET • For some factors, approximation may still be NP-hard Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 3-7 Institute for Software Technology Graz University of Technology Last Time started Optimization and Approximation • Discussed Decision vs. Optimization • Formalized the 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 • Defined complexity classes for optimization problems: PO, NPO, APX(r(n)), APX, APX∗ • Considered the following optimization problems: MIN VERTEX COVER, MAX (k-DEG.) INDEP. SET Question: What did we show how for these problems? Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 3-10 Institute for Software Technology Graz University of Technology This Time / Next Time continue Optimization and Approximation • Repeat some essentials • Consider MIN TSP, MIN TSP∆ and MAX KNAPSACK • Introduce problems with small solution values • Reductions for approximation problems • New concept / class PCP: Probabilistically Checkable Proofs • Topics for Presentations Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 4 Institute for Software Technology Graz University of Technology Optimization Problems Definition: An optimization problem A can be defined as a 4-tuple A = (I; S; v; d) where • I is the set of instances of A. • S(x): x2I is the set of solutions for instance x. • v(x; s): s2S(x) is the value of soution s for instance x. • d2fmin; maxg is the optimization direction. Definition: The class of all optimization problems whose decision variant is in NP is called NPO. ) x2I decidable in polynomial time. ) y 2S(x) decidable in polynomial time and jyj ≤ p(jxj). ) v(x; s) computable in polynomial time. Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 5 Institute for Software Technology Graz University of Technology Measuring Approximation Definition: For an optimization problem A = (I; S; v; d), an instance x2I, and a solution y 2S(x), the approximation ratio of y with respect to x is defined as n v(x; y) opt(x) o R(x; y) := max ; ≥ 1 opt(x) v(x; y) | {z } | {z } d=min d=max Definition: Let r : lN ! [1; 1) with r(n + 1) ≥ r(n). M is a r(n)-approximation algorithm for A if 8x 2 I : M(x) 2 S(x) and R(x; M(x)) ≤ r(jxj): The worst case approximation ratio rM (n) of M is defined as rM (n) := supfR(x; M(x)) : jxj ≤ ng. Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 6 Institute for Software Technology Graz University of Technology The Class APX Definition: Let r : lN ! [1; 1) with r(n + 1) ≥ r(n). • The complexity class APX(r(n)) contains all optimization problems which can be approximated by a polynomial-time algorithm M with worst-case approximation ratio rM (n) ≤ r(n). S • APX = c≥1 APX(c) is the class of optimization problems admitting a polynomial-time constant factor approximation algorithm for some constant c ≥ 1. ∗ T • APX = c>1 APX(c) is the class of optimization problems admitting a polynomial-time constant factor approximation algorithm for any constant c > 1. Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 7 Institute for Software Technology Graz University of Technology Optimization Problems • Considered several optimization problems: Problem Instance Parameter APX MIN V.C. graph jvertex coverj MAX CLIQUE graph jcliquej APX?MAX IND. SET graph jindep. setj MAX2SAT 2CNF formula jsat. clausesj APX MIN TSP weighted graph cost of tour MAX KNAPSACK objects, t. weight total value Question: Whatif P is6= theNP relation between APX and APX∗? Question: Which of the problems are in APX, APX∗? Make your guess list! Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 8 Institute for Software Technology Graz University of Technology MIN TSP TSP / MIN TSP: the Travelling Salesman Problem • Problem: How efficient can a salesman make a round trip visiting given locations? • Decision problem: Is it possible to visit all locations of L within time t? • Problem instance: A set L = fl1; : : : lng of locations, with travel costs / times (Ti;j) between them, plus a maximum total time t. Theorem: TSP is (strongly) NP-complete. Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 9-2 Institute for Software Technology Graz University of Technology MIN TSP TSP / MIN TSP: the Travelling Salesman Problem • Problem: How efficient can a salesman make a round trip visiting given locations? • Optimization problem: What is the minimum time t needed for a round trip that visits all locations of L? • Problem instance: A set L = fl1; : : : lng of locations, with travel costs / times (Ti;j) between them, plus a maximum total time t. Theorem: TSP is (strongly) NP-complete. Corollary: MIN TSP is in NPO. Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 9-4 Institute for Software Technology Graz University of Technology MIN TSP and APX Theorem: If P 6= NP then MIN TSP is not in APX. Proof: X • Assume there exists a polynomial time c-approximation algorithm A for MIN TSP for some constant c > 1. • Given a directed graph G = (V; E), we construct an input for A by assigning weights to the edges of the complete graph on V : 1 if (i; j) 2 E T = i;j cjV j if (i; j) 62 E ) If G contains a Hamiltonian cycle then A(G)≤cjV j. Otherwise A(G)≥jV j−1+cjV j>cjV j. ) Algorithm A solves HAMCYCLE in polynomial time. Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 10 Institute for Software Technology Graz University of Technology The Gap Technique Theorem: Given an optimization problem such that 1. for every input x 2 I and every solution s 2 S(x), s does not lie in the interval (a; b), i.e., either v(s; x) ≤ a or v(s; x) ≥ b, and 2. it is NP-hard to determine whether 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. Proof: Exercise. Question: What was the gap in our proof for MIN TSP 62 APX if P 6= NP? Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 11-3 Institute for Software Technology Graz University of Technology The Gap Technique Theorem: Given an optimization problem such that 1. for every input x 2 I and every solution s 2 S(x), s does not lie in the interval (a; b), i.e., either v(s; x) ≤ a or v(s; x) ≥ b, and 2. it is NP-hard to determine whether 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. Proof: Exercise. Remark: The proof for MIN TSP 62 APX if P 6= NP used a gap of (jV j; cjV j + jV j − 1). Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 11-4 Institute for Software Technology Graz University of Technology MIN TSP with ∆-Inequality Just proved: MIN TSP does not have polynomial time constant factor approximation for any constant factor c. Question: Does this imply that route planning is completely intractible? ∆ MIN TSP : Given set L = fl1; : : : lng of locations, with travel times (Ti;j) between them, such that 8li; lj; lk 2 L : Ti;j + Tj;k ≥ Ti;k; ljwhat is the minimum time t needed for a Tj;k Ti;j round trip that visits all locations of L? lk T li i;k Birgit Vogtenhuber Problem Analysis and Complexity Theory, 716.054, summer term 2016 12-4 Institute for Software Technology Graz University of Technology MIN TSP with ∆-Inequality Just proved: MIN TSP does not have polynomial time constant factor approximation for any constant factor c.