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CSE 202 - Algorithms

Incomplete notes on NP Completeness. CSE 202 - Algorithms. Decision Problems. Instance: an input, an output, and a size. Problem: a set of instances. Decision Problem: Problem where each instance’s output is “T” or “F”.

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CSE 202 - Algorithms

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  1. Incomplete notes on NP Completeness CSE 202 - Algorithms CSE 202 - NP-complete

  2. Decision Problems • Instance: an input, an output, and a size. • Problem: a set of instances. • Decision Problem: Problem where each instance’s output is “T” or “F”. • “Given graph G and nodes x and y, what is the shortest path from x to y?” is not a decision problem. • “Given G, x, y, and k, is there a path from x to y of length  k?” • Usually, given an algorithm for a decision problem, one can use it to solve the associated optimization problem. • E.g., use binary search: “Is there a path of length  50?”, “ 25?“, “ 37?“, ... . (You know this already.) CSE 202 - NP-complete

  3. Reductions • Given two decision problems A and B, we say that a function f from A to B is a reduction of A to B if, for every instance x in A, x is true if and only if f(x) is true. • We write A B, and say A is reducible to B. f f(x) x an instance of problem A instance of B A B {T,F} CSE 202 - NP-complete

  4. More about reductions • Example: • Reducing a maximum matching problem on bipartite graph to a max flow problem. • Except for 2 trivial problems, every computable function is reducible to every other computable function. • Let bT be a true instance of B & bF a false instance. • Given instance x of problem A, f could figure out the answer, then return bT if x is true, bF otherwise. • Mathematicians use reductions to study uncomputable functions (like the halting problem). “Always True” & “Always False” CSE 202 - NP-complete

  5. Polynomial Time Reducibility • In computer science, we limit how much work f can do. • Typically, f must be a polynomial-time algorithm. • We write A p B, and say “A is polynomial-time reducible to B” or “A is no harder than B”. • If A p B, can we conclude B p A ?? • Isn’t f-1 is a reduction of B to A? (Give two reasons this doesn’t work) • IMPORTANT! To show A p B, we must show: for every instance x of A, how to construct an instance f(x) of B relatively quickly (i.e. in polynomial time), and if f(x) is true, then x is true, and if f(x) is false, then x is false. CSE 202 - NP-complete

  6. Typical Garey&Johnson Style Entry • Actually fromwww.csc.liv.ac.uk/~ped/teachadmin/COMP202/annotated_np.html • Name: 3-Dimensional Matching (3DM) [SP1] 3 • Input: • 3 disjoint sets X, Y, and Z each comprising exactly n elements; • a set M of m triples {(xi, yi, zi): 1  i  m} such that xi is in X, yi in Y, and zi in Z, i.e. M is a subset of X x Y x Z. • Question: Does M contain a matching? • i.e. is there a subset Q of M such that |Q|=n and for all distinct pairs of triples (u,v,w) and (x,y,z) in Q, u  x, v  y and w  z. • Comments: The variant 2-dimensional matching in which 2 disjoint sets X and Y form the basis of a set of pairs, can be solved by a number of fast methods. CSE 202 - NP-complete

  7. Some NP-complete problemsevery literate computer scientist should know ... • Traveling Salesman • Subset Sum • Hamiltonian Cycle • K-Clique • 3-Colorability • Satisfiability • 3-Sat • ... CSE 202 - NP-complete

  8. Given 3-CNF formula (x1 x3 x4)  (x2  x3  x5)  ... 3-SAT p 3-Colorable U T ... F x2 x2 x3 x3 x1 x1 Note that one of x1 and x1 will be T, the other will be colored F. CSE 202 - NP-complete

  9. Given 3-CNF formula (x1 x3 x4)  (x2  x3  x5)  ... 3-SAT p 3-Colorable U T ... F x2 x2 x3 x3 x1 x1 T If x1 is F, this node must be U. If x1 is T, this can be F. CSE 202 - NP-complete

  10. Given 3-CNF formula (x1 x3 x4)  (x2  x3  x5)  ... 3-SAT p 3-Colorable U T ... F x2 x2 x3 x3 x1 x1 T This node is forced to be F unless x1 is T or X3 is F. (Etc.) CSE 202 - NP-complete

  11. Glossary (in case symbols are weird) •        subset  element of infinity  empty set  for all  there exists  intersection  union  big theta  big omega  summation  >=  <=  about equal  not equal  natural numbers(N)  reals(R)  rationals(Q)  integers(Z) CSE 202 - NP-complete

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