4/22/12. NP and NP completeness. Efficient Certification. Decision Problems. Definition of P

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1 Efficient Certification and completeness There is a big difference between FINDING a solution and CHECKING a solution Independent set problem: in graph G, is there an independent set S of size at least k v If I give you such a set S, then you can check in polynomial time, that S is independent: check all pairs in S Copyright 990, Matt Groening 3-SAT: is there a satisfying assignment for 3-SAT CNF instance Φ v If I give you a truth assignment, you can check in polynomial time that it satisfies Φ: evaluate the expression. Decision Problems Definition of P Identify the instances with a yes answer of) a Decision Problem with a set of binary strings X Instance: string s. Algorithm A solves problem X: As) = yes iff s X. Polynomial time. Algorithm A runs in poly-time if for every string s, As) terminates in at most p s ) "steps", where p ) is some polynomial. s is length of s in bits.) The class P. Decision problems for which there is a polynomial-time algorithm. Problem Description Algorithm Yes No MULTIPLE Is x a multiple of y Grade school division 5, 7 5, 6 RELPRIME Are x and y relatively prime Euclid 300 BCE) 34, 39 34, 5 PRIMES Is x prime AKS 00) 53 5 PRIMES: X = {, 3, 5, 7,, 3, 7, 3, 9, 3, 37,. } Algorithm: [Agrawal-Kayal-Saxena, 00] p s ) = s. Later improved to s 6 [Pomerance, Lenstra] See wikipedia: AKS primality test) EDIT- DISTANCE LSOLVE Is the edit distance between x and y less than 5 Is there a vector x that satisfies Ax = b Dynamic programming Gauss-Edmonds elimination niether neither # 0 & % % 4 $ % 0 3 5, # 4& % % $ % 36 acgggt ttttta " 0 0% $ $ # $ 0 &, " % $ $ # $ & 3 4 Checking correctness of a solution is easier than finding it yourself : the class of problems that can be checked efficiently Certification algorithms. Certifier doesnt determine whether s X on its own; rather, it checks a proof t that s X. Definition. Algorithm Cs, t) is a certifier for problem X iff for every string s, s X there exists a string t such that Cs, t) = yes. t is called a certificate or witness.. Decision problems for which there exists a polynomial-time certifier: Cs, t) is a poly-time algorithm with t p s ) for some polynomial p ). 5 6

2 Why is called stands for Nondeterministic Polynomial Think of the certifier as performing a nondeterministic search over all possible certificates, at each step guessing a part of the solution. This brings the search time in an exponential sized search space down to polynomial. Certifiers and Certificates: COMPOSITES COMPOSITES. Given an integer s, is s composite Certificate. A nontrivial factor t of s. Note that such a certificate exists iff s is composite. Moreover t s. Certifier. boolean Cs, t) { if t or t s) return false else if s is a multiple of t) return true else return false } Instance. s = 437,669. Certificate. t = 54 or ,669 = Conclusion. COMPOSITES is in. 7 8 Certifiers and Certificates: 3-Satisfiability SAT. Given a Conjunctive Normal Form formula Φ ands of ors), is there a satisfying assignment 3-SAT. 3 variables in each clause Certificate. An assignment of truth values to the n Boolean variables. Certifier. Check that each clause in Φ has at least one true literal. Example: instance: x x x 3 ) x x x 3 ) x x x 4 ) x x 3 x 4 ) Certifiers and Certificates: Hamiltonian Cycle HAM-CYCLE. Given an undirected graph G = V, E), does there exist a simple cycle C that visits every node Certificate. A permutation of the n nodes. Certifier. Check that the permutation contains each node in V exactly once, and that there is an edge between each pair of adjacent nodes in the permutation. Conclusion. HAM-CYCLE is in. Certificate x =, x =, x 3 = 0, x 4 = Conclusion. SAT is in. instance s certificate t 9 0 P,, EXP P. Decision problems for which there is a polynomial-time algorithm. EXP. Decision problems for which there is an exponential-time algorithm.. Decision problems for which there is a polynomial-time certifier. Claim. P. Proof. Consider any problem X in P. By definition, there exists a polynomial-time algorithm As) that solves X. Certificate: t = ε, certifier Cs, t) = As). P,, EXP P. Decision problems for which there is a polynomial-time algorithm. EXP. Decision problems for which there is an exponential-time algorithm.. Decision problems for which there is a polynomial-time certifier. Claim. EXP. Proof. Consider any problem X in. By definition, there exists a poly-time certifier Cs, t) for X. To solve input s, run Cs, t) on all strings t with t p s ). Return yes, if Cs, t) returns yes for any of these.

3 The One Million Dollar CS Question: P = The Simpsons: P = P = Is the decision problem as easy as the certification problem Clay mathematics institute $ million prize. EXP P EXP P = If P If P = would break RSA cryptography and potentially collapse economy) If yes: Efficient algorithms for 3-COLOR, TSP, FACTOR, SAT, If no: No efficient algorithms possible for 3-COLOR, TSP, SAT, Copyright 990, Matt Groening Consensus opinion on P = Probably no. 3 4 Futurama: P = -Completeness -complete. A problem Y in with the property that for every problem X in, X p Y. Theorem. Suppose Y is an -complete problem. Then Y is solvable in polynomial-time iff P =. Proof. If P = then Y can be solved in poly-time since Y is in. Suppose Y can be solved in poly-time. Let X be any problem in. Since X p Y, we can solve X in poly-time. This implies P. We already know P. Thus P =. Theorem. CIRCUIT-SAT is -complete. [Cook 97, Levin 973] Copyright 000, Twentieth Century Fox 5 6 yes: 0 Circuit Satisfiability CIRCUIT-SAT. A combinational circuit is a directed acyclic graph built out of AND, OR, and NOT nodes. Given such circuit, is there a way to set the circuit inputs so that the output is output 0 The "First" -Complete Problem Theorem. CIRCUIT-SAT is -complete. [Cook 97, Levin 973] Proof. sketch) Consider some problem X in. It has a poly-time certifier Cs, t). Any algorithm incl. certifier) that takes a fixed number of bits n as input and produces a yes/no answer can be represented by a circuit. Convert Cs, t) into a circuit K. first s bits are hard-coded with s remaining bits represent bits of t Circuit K is satisfiable iff Cs, t) = yes. hard-coded inputs inputs 7 8 3

4 Independent Set p CircSAT Example. Construction below creates a circuit C whose inputs can be set so that C outputs true iff graph G has an independent set of size >=. independent set of size >= v u both endpoints of some edge have been chosen w independent set set of size >= Establishing -Completeness Remark. Once we establish one "natural" -complete problem, others fall like dominoes. Recipe to establish -completeness of problem Y. Step. Show that Y is in. Step. Choose an -complete problem X. Step 3. Prove that X p Y. Justification. If X is an -complete problem, and Y is a problem in with the property that X P Y then Y is complete. G = V, E), n = 3 Try it for {u,v} Try it for {u,w} u-v u-w 0 v-w u v w Proof. Let W be any problem in. Then W P X P Y. By transitivity, W P Y. Hence Y is -complete. by definition of -completeness by assumption hard-coded inputs graph description) n inputs nodes in independent set) SAT is -Complete Theorem. 3-SAT is -complete. Proof. Enough to show that CIRCUIT-SAT P 3-SAT since 3-SAT is in. Let K be any circuit. Create a 3-SAT variable x i for each circuit element i. Make 3-SAT clauses compute values for each circ-sat node, eg.: x = x 3 add clauses: x x 3, x x 3 x = x 4 x 5 add 3 clauses: x x 4, x x 5, x x 4 x 5 x 0 = x x add 3 clauses: x 0 x, x 0 x, x 0 x x Hard-coded input values and output value. x 5 = 0 add clause: x 5 x 0 = add clause: x 0 Final step: turn clauses of length < 3 into clauses of length exactly 3. x 5 x output x 0 x 4 0 x x 3 Interim Summary v Polynomial time reductions X P Y) v v v The class problems with polynomial time certifiers complete problem problem in such that every other problem has a reduction to it. Examples of -complete problems: Circuit-SAT, 3-SAT -Completeness Observation. All problems below are -complete and polynomially reduce to one another! -Completeness Observation. All problems below are -complete and polynomially reduce to one another! CIRCUIT-SAT CIRCUIT-SAT by -completeness of CIRCUIT-SAT 3-SAT 3-SAT reduces to 3-SAT reduces to 3-SAT DIR-HAM-CYCLE GRAPH 3-COLOR SUBSET-SUM VERTEX COVER VERTEX COVER HAM-CYCLE PLANAR 3-COLOR SCHEDULING SET COVER SET COVER TSP 3 4 4

5 Extent and Impact of -Completeness Extent of -completeness. [Papadimitriou 995] Prime intellectual export of CS to other disciplines. 6,000 citations per year Broad applicability and classification power. "Captures vast domains of computational, scientific, mathematical endeavors, and seems to roughly delimit what mathematicians and scientists had been aspiring to compute feasibly." -completeness can guide scientific inquiry. 96: Ising introduces simple model for phase transitions. 944: Onsager solves D case. 9xx: Feynman and other top minds seek 3D solution. 000: Istrail proves 3D problem -complete. More Hard Computational Problems Aerospace engineering: optimal mesh partitioning for finite elements. Biology: protein folding. Chemical engineering: heat exchanger network synthesis. Civil engineering: equilibrium of urban traffic flow. Economics: computation of arbitrage in financial markets with friction. Electrical engineering: VLSI layout. Environmental engineering: optimal placement of contaminant sensors. Financial engineering: find minimum risk portfolio of given return. Game theory: find Nash equilibrium that maximizes social welfare. Genomics: phylogeny reconstruction. Mechanical engineering: structure of turbulence in sheared flows. Medicine: reconstructing 3-D shape from biplane angiocardiogram. Operations research: optimal resource allocation. Physics: partition function of 3-D Ising model in statistical mechanics. Politics: Shapley-Shubik voting power. Pop culture: Minesweeper consistency. Statistics: optimal experimental design Complete Problems Most problems are either known to be in P or -complete. Notable exceptions. Factoring, graph isomorphism, Nash equilibria. image from: 7 5