Computational Complexity

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ComputationalComplexity

• Cours Voie C
• Serge Abiteboul

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Computability vs. Complexity

• Decidable is there an algorithm to solve a
  particular problem? No for halting problem
• Computability
• Feasible is there a very fast algorithm to solve
  a particular problem? No for traveling salesman
• Complexity

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Complexity class, e.g., time(nlog(n))

• Model of computation, e.g., multitape Turing
  machine
• Mode of computation, i.e., when is the machine
  accepting deterministic and nondeterministic
• Resource we wish to bound time, space...
• The bounding function, e.g., nlog(n)

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Complexity class

• A complexity class is the set of all languages
  decided by some multitape TM M in the appropriate
  mode, and such that for each input w, M uses ast
  most f(w) of that particular resource
• time, ntime, space, nspace

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More not studied here

• average case complexity
• circuit and parallelism
• Parallel Random Access Machines
• NC polylogarithmic parallel time with
  polynomially many processors
• Can we find a good approximation?
• MAXSNP, e.g. MAX-CUT given a graph G, partition
  nodes between S and V-S with as many edges as
  possible between them

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Nondeterministic TM

• It may have choices, ?(a,p) is not unique
• Many possible behaviors on a given input
• M accepts w if there exists a computation on
  input w that accepts w
• NP nondeterministic TM in time polynomial
• intuition deterministic machine would have to
  try all branches ? exponential time

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Nondeterministic computation acceptance
Accept
Reject
yes
all no
Accept if some branch leads to an accept state
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Main classes

• Inclusion of main classes
• L ? NL ? P ? NP ? PSPACE ? EXP ? ???
• More resources means more problems can be solved
• L space(log(n)) log space NL
• P time(nk) polynomial time NP
• PSPACE space(nk) poly space NPSPACE
• EXP time(2nk) exponential time NEXP

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Some Results on space

• Pour tout f, nspace(f(n)) space(f2(n))
• pspace npspace
• space hierarchy theorem
• space(f(n)) proper subset of space(f(n)log(n))
• ntime(f(n)) ? space(f(n))
• nspace(f(n)) ? time(klog(n)f(n))

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Comparing problems Reduction

• A language L1 is reducible to L2 iff there exists
  a function r from strings to strings computable
  in space O(log(n)) such that for all input x, x ?
  L1 iff r(x) ? L2
• If I can solve L2, I can solve L1 at about the
  same cost
• r is a reduction from L1 to L2

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Comparing problems NP-completeness and others

• circuit SAT given a circuit (and/or/not) is
  there an input s.t. circuit returns true
• circuit SAT is in NP
• circuit SAT is NP-hard
• any NP problem reducible to circuit SAT
• circuit-SAT at least as hard as any NP problem
• 3-SAT is NP-hard
• 3-SAT is in NP

3-SAT is NP-complete
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Circuit SAT is NP-complete

• NP easy
• NP-complete
• show how we can simulate the computation of any
  nondeterministic ptime TM using circuits
• i.e., reduction from an arbitrary TM to circuit
  SAT

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Example of reductionHamiltonian path to SAT

• Hamiltonian path
• input a graph G
• output is there a path that visits every node
  exactly once?
• SAT
• input a boolean formula in CNF a set of
  clauses
• output is there a satisfying assignment of the
  variables?

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Hamiltonian path to SAT

• nodes are 1,2,...,n r(G) has variables xi,j
• represents that j is the ith node on the path
• for each j, x1,j or ... or xn,j j is on the
  path
• for each j, i, k, i ? k, not xi,j or not xk,j j
  is only once on the path
• for each i, xi,1 or ... or xi,n some node is
  ith
• for each i, j, k, j ? k, not xi,j or not xi,k
  there is only one ith node
• for each (i,j) that is not an edge in G, fo each
  k, not xk,i or not x(k1)j

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Hamiltonian path to SAT
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x12 or ... or x52 not x12 or not x22 x41 or ...
or x45 not x41 or not x42 not x12 or not x21
(2,1) is not in G
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5
1
2
try also find a path that visits each edge
exactly once
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Easy part problem is in NP

• Hamiltonian path in NP
• guess a list of n vertices
• check whether it is an Hamiltonian path
• SAT is in NP
• guess a value TRUE/FALSE for each variable
• check whether the formula evaluates to TRUE
• Make a polynomial number of guesses
• For each one, do a polynomial computation

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Some NP-complete problems

• Hamiltonian path
• SAT DNF formulas
• 3SAT clauses have 3 items
• Circuit SAT gates with and/or/not
• Clique G,k, is there a clique of size k
• 3 coloring of a graph no two adjacent vertices
  have the same color

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3-coloring
NO!
YES!
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More NP-complete problems

• Knapsack items have value vi and weight wi.
  Maximize the value under the constraint that the
  weight must be below W
• Traveling salesman n cities and di,j between
  cities. Find the shorter tour of the cities
• Max flow a graph with a source and a sink, a
  capacity for each edge, get the largest possible
  flow on this network

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Intuition for NP-complete problems

• It is very simple to test a solution SAT verify
  that it makes indeed the formula true
• Most useful computational problems are in NP.
  They may be in P or not
• NP-complete not in P unless P NP

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Showing that a problem H is NP-complete

• It is in NP (usually easy)
• guess a certificate for H
• check that it is a certificate
• Show it is NP-hard (a bit more complex)
• choose a problem H known to be NP-hard
• construct a reduction from H to H
• so H is at least as hard as H and thus is NP-hard

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P, NP, co-NP

• co-NP complement of a problem NP
• the formula is satisfiable by a valuation NP
• the formula is satisfied for all valuations co-NP
• NP short certificate
• coNP not the case

co-NP
NP
P