P and NP

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P and NP

• Irena Pevac
• CS463

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Definitions

• Def Decision problem is a question that has two
  possible answers yes or no.
• Def Problem instance is a combination of the
  problem and a specific input.
• Decision problem has most often two parts
• Instance description part
• Defines the information expected in the input
• Question part
• States the actual yes no question. The question
  contains variables defined in the instance
  description.

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Precise Statement of the Input

• EXAMPLE 1
• Input Instance an undirected graph G(V,E)
• Question Does G contain a clique Ck (complete
  subgraph) of k vertices for some k?
• (?k)(k 1,2,,n , Ck is a subgraph of G )
  ?
• C1 OR C2 OR C3 OR Cn

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• EXAMPLE 2
• Input Instance an undirected graph G (V,E) and
  an integer k
• Question Does G contain a clique Ck (complete
  subgraph) of k vertices ?
• ?
• Ck is a clique and it is a subgraph of G.
• If Question 2 is YES ? Question 1 is YES.
• If Question 1 is YES ? Question 2 does not have
  to be YES.

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Graph Coloring and Chromatic Number Definitions

• Let G(V,E) be undirected graph.
• Coloring is a mapping c V?S where S is a finite
  set of colors and if vw E then c(v)4c(w).
  (Adjacent vertices have different color.)
• The chromatic number of G, denoted ?(G) is the
  smallest number of colors needed to color G.

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Graph Coloring

• Given G(V,E) undirected graph
• OPTIMIZATION PROBLEM Given G, determine ?(G) and
  produce an optimal coloring, that is, one that
  uses only ?(G) colors.
• DECISION PROBLEM Given G, and an integer k, is
  there a coloring of G using at most k colors. (If
  so, G is k colorable).

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Geographic Map Coloring
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Graph Coloring Application

• Geographic map coloring
• Nodes are states
• Edge connects two nodes if they represent two
  adjacent states
• Coloring is used to color the map.

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Graph 4 Coloring of US Map
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Scheduling Application

• Scheduling finals for course-sections
• Nodes are course sections.
• Edges connect two sections that have at least one
  student in common.
• Colors represent time slots for exams.
• PROBLEM Schedule exams into minimal number of
  time slots, so that no student is scheduled more
  than one exam in the same time slot.

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Graph 4 Coloring of Final Exams
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Definitions

• Def A propositional or boolean variable has
  value true or false.
• Def A literal is a propositional variable or its
  negation.
• Def A propositional formula is defined
  inductively
• A constant true or false is propositional
  formula.
• A propositional variable is propositional
  formula.
• If F is propositional formula than NOT F is also
  propositional formula.
• If F and G are two propositional formulas then F
  AND G and F OR G are also propositional formulas.

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Conjunctive Normal Form CNF

• Def Clause is a sequence of literals connected
  with OR operators.
• L1 OR L2 OR L3 OR OR Ln
• Def A propositional formula is in conjunctive
  normal form if it consists of a sequence of
  clauses connected by AND operators.
• Def A truth assignment for a set of
  propositional variables is an assignment of TRUE
  or FALSE for each propositional variable in the
  set.

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Satisfiability

• Def A truth assignment is said to satisfy a
  formula if it makes the value of entire formula
  true.
• A CNF formula is satisfied if each clause
  evaluates to true. A clause evaluates to true if
  at least one of its literals evaluates to true.
• DECISION PROBLEM Given a CNF formula, is there a
  truth assignment that satisfies it?
• DECISION PROBLEM Given a CNF formula, in which
  each clause has at most three literals, is there
  a truth assignment that satisfies it?

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Hamiltonian Cycle and Path

• Def A Hamiltonian cycle in an undirected graph
  is a simple cycle that passes through every
  vertex of the graph exactly once.
• Def A Hamiltonian path in undirected graph is
  simple path that passes through every vertex of
  the graph exactly once.
• Def A Traveling Salesman must visit every city
  once and return to the starting city at the end
  of the tour and travel the shortest possible
  path.
• TSP is used in routing garbage pickups and
  package and mail deliveries.

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Hamiltonian Cycle
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• DECISION PROBLEM Does a given undirected graph
  have a Hamiltonian cycle?
• DECISION PROBLEM Does a given undirected graph
  have a Hamiltonian path?
• OPTIMIZATION PROBLEM Given a complete weighted
  graph, find a minimum weight Hamiltonian cycle.
  (TSP)
• DECISION PROBLEM Given a complete, weighted
  graph and an integer k, is there a Hamiltonian
  cycle with total weight at most k.

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Polynomial Complexity

• Def An algorithm is polynomially bounded if its
  worst case complexity is bounded by a polynomial
  function of the input size.
• Def A problem is polynomially bounded if there
  exist a polynomially bounded algorithm that
  solves the problem.

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Tractable and Intractable Problems

• Tractable problems are those that can be solved
  in polynomial time.
• Problems that cannot be solved in polynomial time
  are called untractable.

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The Class P

• Def The class P is class of decision problems
  that can be solved in polynomial time by
  deterministic algorithm.
• Excluding problems not solvable in polynomial
  time makes sense because of exponentially large
  output in some problems
• Listing all subsets of a given set
• Listing all permutations of n different numbers

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• Some problems that are not decision problems can
  be reduced to a series of decision problems.
• Optimization graph coloring problem (Find
  chromatic number for given graph) can be reduced
  to a series of decision problems.
• Given G, is there a coloring of G using at most
  1 color.
• Given G, is there a coloring of G using at most 2
  colors.
• Given G, is there a coloring of G using at most 3
  colors.
• Given G, is there a coloring of G using at most 4
  colors.
• The first value for which decision problem has
  solution solves the optimization version of the
  graph-coloring problem as well.

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Undecidable Problems

• QUESTION Can every decision problem be solved in
  polynomial time?
• ANSWER No! For some decision problems time is
  exponential.
• For other decision problems no algorithm exists
  to solve it. This are undecidable algorithms.

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Alan Turings Halting Problem

• Alan Turings Halting Problem
• Given a computer program and an input to it,
  determine whether the program will halt on that
  input or continue to work for ever.
• Turings halting problem was one of the most
  important contributions to computer science made
  in 1936. Proof that problem is undecidable (i.e.
  no algorithm exists to solve the problem) is done
  by way of contradiction.

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• PROOF
• Assume that A is an algorithm to solve the
  halting problem. That would mean that for any
  program P and any input I,
• if program P halts on I
  
• A(P,I)
• if program P
  doesnt halt on I
  
• We can consider program P as an input to itself
  and use the output of algorithm A for pair (P, P)
  to construct a program Q as follows

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• if A(P,P)0 i.e. P
  doesnt halt on I.
• Q(P,I)
• if
  A(P,P)1i.e. P halts on I.
• On substituting Q for P we get
• if Q(A,A)0,
  i.e. Q doesnt halt on I.
  
• Q(Q)
• if A(Q,Q)1, if
  Q halts on input I.
• This is a contradiction.

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Decidable Problems

• We proved that halting problem is undecidable.
• Are there decidable but untractable problems?
  Yes. There are.
• There is a large number of problems for which no
  polynomial time algorithm exists, nor has the
  impossibility of such algorithm been proved.

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Examples

• Some such examples were mentioned earlier
• Hamiltonian Circuit, Hamiltonian Path
• Traveling Salesman
• Graph Coloring
• Some other famous examples
• Partition Problem Given n positive integers,
  determine if it is possible to partition them
  into two disjoint sets with the same sum.
• Bin Packing Given n items whose values are
  positive rational numbers not larger than 1, put
  them into the smallest number of bins of size 1.

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Exponential v.s. Polynomial Time

• All those examples have an exponential (or worse)
  growth of choices, as a function of input size.
• Some of examples can be solved in polynomial
  time.
• Eulerian circuit problem- problem of existence of
  a cycle that traverses all the edges of the given
  graph only once (Eulerian circuits) can be solved
  in O(n2).

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Checking the Solution in Polynomial Time

• Most of decision problems are expensive to solve,
  but testing whether a proposed solution solves
  the problem is easier (can be done in polynomial
  time).
• Example To check whether a list of vertices is
  Hamiltonian circuit for a given graph G with n
  vertices we have to check
• That each vertex on the lists belongs to V,
• That first and last are the same, and that first
  n are all different.
• That every two consecutive in the list are
  adjacent vertices in the graph (are connected
  with an edge).

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Nondeterministic Algorithms

• Def A nondeterministic algorithm is a two stage
  algorithm that takes as its input an instance I
  of a decision problem and does the following
• Nondeterministic guessing phase. An arbitrary
  string of characters S is generated at the
  beginning. S may be considered as a guess of a
  solution for I but it may be a gibberish as well.
• The deterministic verifying phase takes both I
  and S and outputs yes if S is solution for I,
  otherwise it either returns No or it may not
  halt at all.

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Example

• Nondeterministic Graph Coloring
• Problem Determine if an undirected graph G is
  k-colorable.
• The first phase of nondeterministic algorithm
  will write some string s. Second phase will
  interpret s as a coloring for G. To verify that
  the coloring is valid it verifies
• 1) Check that there are n colors listed.
• 2) Check that each ci is in range 1,,k.
• 3) Scan the list of edges in G and check that for
  each edge uv cunot equal cv.
• If all the tests are true, return yes. If s does
  not satisfy all three conditions, the verifier
  may return false or go into an infinite loop, and
  the algorithm produces no output for this
  particular execution.

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G( V1,2,3,4,5, E(1,2)(1,4)(2,4)(2,3)(3,5)(
2,5)(3,4)(4,5))

• Input instance is graph G and k4
• Problem Is graph G 4-colorable?
• We denote colors by letters
• B(blue), R(red), G(green), Y(yellow), O(orange)
  rather than integers 1,5.
• The table that follows gives a few possible
  strings s and values returned by the verifier.

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• Since there exists one possible input for which
  verifier returns true, the answer of the
  nondeterministic algorithm for the input G(
  V1,2,3,4,5, E(1,2) (1,4) (2,4) (2,3) (3,5)
  (2,5) (3,4) (4,5) ) and k 4 is yes.

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NP Class of Algorithms

• We say that nondeterministic algorithm solves a
  decision problem if and only if for every yes
  instance of the problem it returns yes on some
  execution.
• A nondeterministic algorithm should be able to
  guess a solution at least once and be able to
  test its validity.
• DEF NP Class is the class of algorithms that can
  be solved by nondeterministic polynomial
  algortihms.

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P and NP

• Most decision problems are in NP class.
• Class P is subset of class NP
• P?NP
• If a problem is in P, we can use the
  deterministic polynomial-time algorithm that
  solves it in the verification stage of a
  nondeterministic algorithm and ignore string S
  from the deterministic stage.

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NP Algorithms

• Hamiltonian circuit, partition problem, decision
  versions of TSP, Graph coloring, the knapsack are
  in NP.
• The Turing halting problem is not in NP.

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Open Question

• One of the most important open questions of
  theoretical computer science is
• Is P a proper subset of NP, or are these two
  classes the same.
• ?
• P NP

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PNP

• If PNP it would follow that each of the many
  hundreds of difficult combinatorial decision
  problems can be solved by a polynomial time
  algorithm, despite the fact that computer
  scientists did not find the algorithms.
• It is believed that
• but there is no proof yet.

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Most Scientist Believe

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Theoretical Possibilities