Lots of NP Complete Problems When confronted with trying to show a - PDF document

Lot’s of NP Complete Problems • When confronted with trying to show a problem is NP-Complete NP-Complete Problems – There are lots to chose from which to perform a reduction. – Garey and Johnson lists over 300 (and that was printed in 1979. The big 7 The Big Daddy • Basic core of known NP-complete • SATISFIABILITY problems. – INSTANCE – Basis for beginner in chosing a problem for • A set of U boolean variables and a collection C of boolean clauses over U reduction – PROBLEM • Is there a satisfying truth assignment for C? – From Garey and Johnson. – The one that started it all! The other 6 The other 6 • 3-SATISFIABILITY (3-SAT) • 3-DIMENSIONAL MATCHING (3DM) – INSTANCE: – INSTANCE: • A set M ⊆ (W x X x Y) where W, X, Y are disjoint • Collection C = {c 1 ,c 2 , …, c m } of boolean clauses on a finite set of boolean variables U such that each c i sets having the same number of elements contains exactly 3 variables. – PROBLEM – PROBLEM: • Does M contain a matching • Is there a truth assignment for U that satisfies all – A subset M’ ⊆ M such that M’ has the same size as M and clauses of C no 2 elements in M’ agree in any coordinate. 1

The other 6 The other 6 • VERTEX COVER • CLIQUE – INSTANCE: – INSTANCE: • A Graph G = (V,E) and integer k ≤ |V| • A Graph G = (V,E) and integer k ≤ |V| – PROBLEM: – PROBLEM: • Is there a vertex cover of size k or less on G • Does G contain a clique of size k or more. – Subset of vertices V’ such that for each edge {u,v}, at – A subset of vertices, V’ such that every 2 vertices in V’ least one of u and v belongs to V’ are joined by an edge The other 6 The other 6 • HAMILTONIAN CIRCUIT (HC) • PARTITION – INSTANCE: – INSTANCE: • A finite set A and a “size function” s(a) a ∈ A • A Graph G = (V,E) – PROBLEM: – PROBLEM: • Is there a subset A’ ⊆ A such that • Does G contain a hamiltonian circuit ∑ ∑ – An ordering of all vertices <v 1 ,v 2 , …, v n > such that {v i , v j } = ( ) ( ) s a s a is an edge and {v n , v 1 } is an edge. ∈ ∈ − ' ' a A a A A The big 7 Problem Categories • Graph Theory SAT • Network Design • Sets and Partitions, 3-SAT • Storage and Retrieval – Database Related Problems, Data Storage, Sparse 3DM VC Matrix Compression • Sequence and Scheduling HC PARTITION CLIQUE – Multiprocessor Scheduling • Mathematical Programming – Linear Programming 2

Favorite NP-Complete Problems Problem Categories • Algebra and Number Theory • Traveling Salesman • Games and Puzzles – INSTANCE • Logic • A finite set of “cities” C = { c 1 , c 2 , …, c n } • A distance function d: C 2 → N – SAT, 3SAT – d( c i , c j ) = distance between c i and c j • Automata and Languages • Upper bounds B • Program Optimization – PROBLEM – Program Equivalences • Is there a “tour” of all cities such that the distance • Other traveled will be less than B? Some NP-Complete Problems Some NP-Complete Problems • NDFA inequivalence • Traveling Salesman – INSTANCE – PROBLEM • Two nondeterministic finite automata A 1 and A 2 • Said another way, is there an ordering of the cities having the same input language Σ . <c i1 , c i2 , …, c in > such that – PROBLEM − • Do A 1 and A 2 recognize different languages? 1 n ∑ + ≤ ( , ) ( , ) d c c d c c B – NOTE + 1 1 ii ii in i • Can be done in polynomial time if A 1 and A 2 are = i 1 deterministic. Some NP-Complete Problems Some NP-Complete Problems • Inequivalence of Programs with • Inequivalence of Programs with Assignments Assignments – INSTANCE – PROBLEM • Finite set of X variables • Is there an initial assignment of a value from V for each variable such that the two programs will yield • Two programs P 1 and P 2 different final results? – Each a sequence of the form x 0 = (x1 == x 2 ) ?x 3 : x 4 • A value set V which defines possible values for each variable. 3

Some NP-Complete Problems Some NP-Complete Problems • Crossword Puzzle Construction • You want more? – INSTANCE • A finite set W ⊆ Σ * of words – “Computers and Intractability” by Michael R. • An n by n matrix A consisting of 1’s and 0’s Garey and David S. Johnson. – PROBLEM • Can an n by n crossword puzzle be built up including all words from W and blank squares – Questions? consisting of the “0” cells in A? Summary of Complexity Classes Summary of Complexity Classes • P – set of decision problems solvable by deterministic algorithms in polynomial time • NP NP – Set of decision problems solvable by non- deterministic algorithms in polynomial time. P • Certainly P is a subset of NP Summary of Complexity Classes NP-Complete and P • NP-Hard • The (sad) world of NP – Problems L such that any problem in NP reduces to L • NP-Complete NP-complete – Problems both in NP and NP-Hard NP – SAT P • Note that SAT is in P iff P = NP • No one has proven: – SAT is in P Is there something in After 40 years of research…nobody – SAT cannot be in P here? knows 4

Quick road to Theory Fame Other classes • Find a deterministic polynomial algorithm • Co-NP for ANY problem in NP-complete… – Complement of NP – And you’ve found one for all • P/NP SPACE – In fact, it will show that P = NP. – based on amount of space (tape) used by TM • Classes based on Randomization – Allows TM to make random choices • They’ll be naming buildings after you before you are dead! • See Chapter 11 for details or take Algorithms! Where to next? The Theory Hall of Fame • Stephen Cole Kleene • Language Design / Compiler Construction – Regular expressions & FAs – 4003-580 Language Processors • Noam Chomsky – 4003-560 Compiler Construction Lab (if – The grammar guy offered again) • W. Ogden • Complexity & Computation – Pumping Lemma for CFLs • Kurt Godel – 4003-481 Complexity and Computability – Kicked off study of computability – 4003-515 Analysis of Algorithms • Alan Turing – Turing Machines The Theory Hall of Fame The Theory Hall of Fame • Alonzo Church • YOU – Unlimited support for TMs – If you find a polynomial solution to any • Emil Post problem in NP-complete! – Post Correspondence Problem • H.G. Rice – Decision problems for TMs • Stephen Cook – NP-completeness • Aho, Hopcraft, and Ullman – Bell Labs guys – work on programming language 5

So There You Have It! Next Time • Final exam review • I bet you thought I forgot! • Thursday, Nov 18 th : Final exam – What is a language? – 70-3435 – What is a class of languages? – 8:00am – 10:00am – Closed book – 1 page study guide okay. • Thanks for playing. • Now time for Course Evaluations 6