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Algorithms

Algorithms. NP-Complete Problems. Polynomial time algorithms. Most algorithms we have studied so far are polynomial time in the size of their inputs n. Worst case running time is O(n k ) for some constant k. Problems that are solvable by polynomial time algorithms are said to be tractable.

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Algorithms

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  1. Algorithms NP-Complete Problems

  2. Polynomial time algorithms • Most algorithms we have studied so far are polynomial time in the size of their inputs n. • Worst case running time is O(nk) for some constant k. • Problems that are solvable by polynomial time algorithms are said to be tractable.

  3. What’s so great about polynomial time? 1. Although we would consider a problem that is (n100) intractable, this type of problem is rarely encountered. The exponents on the polynomial are usually small. 2. For most reasonable models of computation, a problem that can be solved in polynomial time in one model can be solved in polynomial time in another.

  4. Closure properties of polynomial algorithms • Closed under addition f(n) + g(n) • Closed under multiplication f(n)*g(n) • Closed under composition f(g(n))

  5. Why is polynomial time an important goal? • An algorithm is efficient (or polynomial time) if there exists a polynomial p(n) such that the algorithm can solve any instance of size n in time O(p(n)). • A problem is tractable if it has an efficient algorithm. • This is valid only if we avoid algorithms : • with extremely large constants • with large values for k O(nk) • with very large operands.

  6. Do all problems have polynomial time solutions? NO! • Some problems are not solvable • Turing’s Halting problem is an example • Cannot be solved by any computer no matter how much time is given • Some problems can be shown to require super-polynomial time to solve. • There is a a large class of important problems for which we do not know the answer. These are the NP complete problems.

  7. Some Classifications • Unsolvable–Turing problem • Solvable • Provably intractable (infeasible) • Probably intractable (NP-complete) • Tractable (feasible) • hard (> n2) • easy (most of the problems we have studied so far)

  8. NP Completeness • The Question!! • Does P = NP? • This is an unanswered but very important question. • What are P, NP, and NP-complete? • Why is the question important? • What do we think the answer is? • What if the answer is what we think it is?

  9. NP Complete Problems • The complexity status of these problems is unknown • There is no established lower bound • But, it is known that if one of these problems can be solved in polynomial time, all of them can. • Very important equivalence class

  10. What Do We Think? • Most computer scientists believe that the NP complete problems are intractable. • Why? • Because there are a lot of these problems • Many of them are very important problems • People have been trying to find efficient algorithms for them for a long time and no one has succeeded.

  11. Many P and NP-Complete Problems are Closely Related • Shortest path problem and longest path problem • Euler tour and Hamiltonian Cycle • Euler tour of connected, directed graph visits each edge exactly once • Hamiltonian cycle begins and ends at same vertex and visits each vertex exactly once

  12. 2-SAT and 3-SAT • 2-Conjunctive Normal Form (CNF) Satisfiability • 3-Conjunctive Normal Form (CNF) Satisfiability

  13. The Satisfiability Problem • Given a Boolean formula containing variables whose value are 0 or 1, connected by the Boolean connectives , , and . • A Boolean formula is satisfiable if there is an assignment of values to variables that causes the formula to evaluate to 1

  14. k-CNF • Informal definition: A formula is in k-conjunctive normal form if it is the AND of clauses of ORs of exactly k variables or their negations. • Example of 2-CNF formula (x1  x2)  (x1  x3)  (x2  x3) • What is a satisfying assignment of values to variables?

  15. CNF-Satisfiability • A polynomial time algorithm exists for to determine if a 2-CNF formula is satisfiable. • 3-CNF is NP-complete

  16. Three Classes of Problems • P (Polynomial) • Problems solvable in polynomial time • Can be solved in time O(nk) where k is a constant • NP (Non-deterministic Polynomial) • Problems that are verifiable in polynomial time • Given a “certificate” of a solution, we can verify that the solution is correct in polynomial time • For Hamiltonian path the certificate is a sequence of vertices • For 3-SAT, the certificate is an assignment of values to variables • NPC (NP Complete)

  17. NP Complete • Informal definition: A problem is in the class NPC if it is in NP and is as “hard” as any problem in NP. • If any problem in the class NP can be solved in polynomial time, then all can be solved in polynomial time

  18. Proofs of NP-Completeness • Different from other proofs we have done • Rather than show that a problem has an efficient algorithm, we will be demonstrating that it is hard.

  19. Theory of NP-Completeness • Limited to one type of problem - decision problems • Each element of S (the set of solutions) is an element of {yes,no} (or {0,1}) • Seems like a severe restriction, but it is not • Usually dealing with optimization problems that are easily restated as a decision problem. • Can show that if the decision problem is easy, then the optimization problem is easy.

  20. Optimization Problem vs Decision Problem • Shortest path problem as an optimization problem Given a graph G= (V,E), two vertices u,v  V, what is the shortest path that exists in G between u and v? • Shortest path problem as a decision problem Given a graph G= (V,E), two vertices u,v  V and a nonnegative integer k, does a path exist in G between u and v whose length is at most k?

  21. General Approach To change an optimization problem to a decision problem, we can usually just impose a bound on the value to be optimized. • minimization problem example: • length of shortest path at most k • maximization problem example: • length of longest path at least k

  22. General statements • Usually, if we can solve the optimization problem quickly, we can solve the decision problem quickly. • Also, if we can provide evidence that the decision problem is hard, we also provide evidence that the optimization problem is hard.

  23. The Class P • The complexity class P is the set of concrete decision problems that are solvable in polynomial time.

  24. Nondeterministic Polynomial(NP) • Let an oracle guess an answer to the problem (certificate). • If we can find an algorithm that can verify that the answer is correct in polynomial time, then this problem is in the class NP • P is clearly a subset of NP

  25. P = NP? NP NP NP-complete NP NP-complete P P

  26. Example: HAM-CYCLE Hamiltonian Cycle Problem • A Hamiltonian cycle of an undirected graph G=(V,E) is a simple cycle that contains each vertex in V. • The Hamiltonian-cycle problem asks if a given graph contains a Hamiltonian cycle.

  27. HAM-CYCLE NP • HAM-CYCLE is an element of NP because we can verify an answer • Given a list of vertices, does the list represent a Hamiltonian cycle? • We can find a polynomial time algorithm to determine if this list is a Hamiltonian cycle. • Is the first vertex the same as the last vertex? • Are all vertices visited? • For each vertex in the list, is there an edge to the next vertex in the list? • Is any vertex other than the first repeated

  28. Showing Problems to be NP- Complete • Restate problem as a decision problem • Demonstrate that the decision problem is in the class NP • Show that a problem known to be NP-Complete can be reduced to the current problem in polynomial time.

  29. NP-Completeness and Reducibility • NP-complete problems • informally they are the hardest problems in the class NP • no polynomial time algorithm is known to solve any of them • but every NP-complete problem is reducible to every other NP-complete problem in polynomial time

  30. Reducibility Polynomial time Set of instances of Problem A Set of instances of Problem B Decision problem A is polynomial-time reducible to decision problem B if a polynomial time algorithm can be developed which changes each instance of problem A to an instance of problem B such that if the answer for an instance of B is yes, the answer to the corresponding instance of A is yes.

  31. Algorithm from Reduction for Solving Problem A YES Polynomial-time reduction algorithm Polynomial-time algorithm to decide B   NO

  32. A First NP-Complete Problem • This technique for showing that a problem is in the class NP-Complete requires that we have one NP-Complete problem to begin with • Circuit satisifiability was the first problem to be shown to be in NP-Complete.

  33. Lemma 34.3 • If Q1, Q2 are problems such that Q1 p Q2, then Q2  P implies Q1 P. • Proof: Suppose A1 is a polynomial time algorithm that reduces instances of Q1to instances of Q2 and that A2 is a polynomial time algorithm for problem Q2. Then the concatenation of A1 and A2 is a polynomial time algorithm for Q1.

  34. An Instance of Problem Q1 Reduction (polynomial) An Instance of Problem Q2 Polynomial algorithm for Q1 Polynomial algorithm for Q2 Solution for any Instance (YES,NO)

  35. NP-Complete • A problem that satisfies property 2, but not necessarily property 1 is NP-hard.

  36. Theorem 34.4 • If any NP-complete problem is polynomial-time solvable, then P = NP. • If any problem in NP is not polynomial-time solvable, then all NP-complete problems are not polynomial-time solvable.

  37. Proof of Theorem 34.4 Proof of Statement 1: If any NP-complete problem is polynomial-time solvable, then P = NP. Suppose that Q P and Q NPC. Then, for any Q’ NP, we know that Q’ pQ by property 2 of NP-completeness. By Lemma 34.3, Q’ P. P Q NPC Q NP Q’ P/NP NPC

  38. Proof of Theorem 34.4 Proof of Statement 2: If any problem in NP is not polynomial-time solvable, then all NP-complete problems are not polynomial-time solvable. Suppose that there exists an Q NP such that Q P. Let Q’ NPC be any NP-complete problem and suppose that Q’  P. By Lemma 34.3, Q’ pQ and so Q P. Contradiction NPC Q P Q’ NPC

  39. Importance of Theorem • The theorem make the focus of the P=NP question the NP-complete problems • Most theoretical computer scientists believe that P NP. • But, if a polynomial time algorithm exists for any NPC problem, then P = NP. • But, no one has been able to do this so far.

  40. NP-Completeness Proofs • Suppose that we can show that one problem (Q) is in the class NP-complete. Then we can show that another problem Q’ is in the class NP-complete by showing that • Q’ is in NP • Q can be reduced to Q’ in polynomial time. • Describe an algorithm for the reduction • Prove that the answer to Q’ is yes iff the answer to Q is yes • Prove that the algorithm for the reduction runs in polynomial time

  41. Circuit Satisifiability • Cook proved that the CIRCUIT-SAT problem is NP-complete. • 3-SAT was also shown to be NPC by Cook. • It is quite difficult to show that a particular problem in NP can be reduced to every other problem in NP. • But once one problem is know to be NP-complete, the others are easier. • Many other problems have since been proven to be NP-complete.

  42. Sketch of NP-Completeness Proof for CIRCUIT-SAT • Very long and difficult in the details. • Used to provide the basis for most other proofs of NP-Completeness. • Problem domain is combinatorial circuits.

  43. Combinatorial Circuit Components

  44. Circuit-Satisfiability Problem Given a boolena combinatorial circuit composed of AND, OR, and NOT gates, is it satisfiable?

  45. Demonstrating Circuit Satisfiability is NP Complete • This proof was done from first principles. • It did not depend on the existence of any other NPC problems. • A problem Q is NP-Complete if 1. It is an element of the class NP 2. Q’ pQ for every Q’ in NP

  46. Step 1Prove CIRCUIT-SAT  NP • Develop an algorithm A that can verify a solution in polynomial time. • Inputs for A • A standard representation of a combinatorial circuit • An assignment of input values to the circuit • Algorithm A determines the output of the combinatorial circuit. Output is 0 or 1. • This can be done in polynomial time (linear for a clever algorithm.) • Therefore CIRCUIT-SAT NP

  47. Step 2: Reduction • We need to show that every other problem in NP can be reduced to CIRCUIT-SAT. • The basic argument is: • A combinatorial circuit can be used to implement a computer • Program • Memory • Program counter • Working storage, etc. • You can show that every problem in NP can be mapped onto operations that can be represented as a sequences of states of combinatorial circuits. • You show that this can be done in polynomial time. • It is not simple to show all of this, but in the end you can show that any problem in NP can be reduced to CIRCUIT-SAT in polynomial time

  48. Implications of CIRCUIT-SATNPC • CIRCUIT-SAT is the “seed” problem in NPC • Once we know that one problem is in NPC, we can use it to demonstrate that other problems are in NPC using a simpler procedure.

  49. CIRCUIT-SAT SAT 3-SAT CLIQUE SUBSET-SUM VERTEX-COVER HAM-CYCLE TSP

  50. EXAMPLE • Suppose someone has proven that the problem of determining if a graph G has a Hamiltonian cycle has been shown to be NP-complete. • And we want to show that the Traveling Salesman Problem (TSP) is NP-complete. • Two steps • show that TSP is in the class NP • show that HAM reduces to TSP in polynomial time

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