Welcome to the course Concrete Complexity Theory

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Welcome to the courseConcrete Complexity Theory

• Winter Term 2008/09
• Lecture 6, 5.12.08
• Friedhelm Meyer auf der Heide

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Organization

• Please register for the oral exam until Dec.
  11th.
• (Master online LSF, DPO4 Prüfungsekretariat)
• Dates Febr. 9. 20.
• March 23. April. 3.

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• Part 1
• Boolean Circuits

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• Monotone Circuits

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Monotone functions

• f 2 Bn is called monotone, iff for all x, y 2
  0,1n,
• It holds that xy implies f(x) f(y).
• (xy , xi yi for i 1,, n)
• MBnf2 Bn, f monotone
• Examples Majority and all threshold-functions
  Tnk,
• (Tnk (x)1 , ? xi k), k-Clique.
• Counterexamples Parity, Enk,
• (Enk (x)1 , ? xi k)

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Monotone cicuits

• are circuits over the basis M2Æ, Ç.
• Remark Monotone circuits can compute exactly all
  monotone functions.
• Remark The log(n) term can be removed.

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Monotone circuits for Majority

• 1. (For those who have attended PuK)
• Sorting networks are essentially monotone
  circuits. Thus we get circuits of size
  (O(nlog(n)2) and depth O(log(n)2) (Batchers
  sorting network) and even of size (O(nlog(n)) and
  depth O(log(n)) (Ajtai, Komlos, Szemeredi).
• 2. Depth O(log(n)2) is not too hard to achieve.
  (homework)
• 3. Theorem (Valiant) DM2(Majorityn) 5.3
  log(n)

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Valiants circuit for Majority

• Theorem (Valiant) DM2(Majorityn) 5.3 log(n)
• Proof (idea)
• Take an alternating tree
• of depth 5.3log(n).

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Monotone lower bounds for the Clique function

• Cliquek(n) receives n(n-1)/2 inputs xi,j,
  1iltjn, describing the adjacency matrix of a
  graph G. Cliquek(n) outputs 1, iff G contains a
  k-clique.
• A breakthrough result by Razbarov from 1985
  (earlier lower bounds for monotone circuit
  complexity were at most quadratic)
• This lower bound over a complete basis would
  imply NP ? P!

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Monotone lower bounds for the Clique function

• We prove a special case in order to demonstrate
  the proof technique
• Theorem
• Remark CM2(Cliquek(n))O(n3).

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Proof The method of approximation

• Consider a monotone circuit for Clique3(n). Let
  f1, ft denote a topological ordering of its
  inputs and gates.
• We will approximate each gate fj by some
  function cj. cj is a constant or a disjunction of
  at most m variables. (m will be specified later.)
• Variable xi,j is approximated by xi,j
• Let the parents of gate g be approximated by
  and , resp.
• Then g is approximated by
• 
  if g
  Ç, and by
• 
  if g Æ
• We define 0 for A ? .
• Note Each cj is a disjunction of at most m
  variables.

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Proof The method of approximation

• - Variable xi,j is approximated by xi,j
• Let the parents of gate g be approximated by
  and , resp.
• Then is approximated by
• 
  
  if g Ç, and by
• 
  
  if g Æ
• Claim

Lemma 1
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Proof The method of approximation

• Negative test graph defined by a set W µ V. Its
  edges are all those connecting W with V \ W.
  (Negative test graphs contain no 3-cliques.)
  There are 2n such graphs.
• (Note For convenience, we count each graph
  twice, once defined by W, once defined by V \ W.)
• Positive test graph consists of one triangle.
  (Positive test graphs contain a 3-clique.) There
  are such graphs.
• Lemma 2 Consider . c is
  either 0 on all positive
• test graphs, or 1 on at least half of the
  negative test graphs.

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Thank you for your attention!
Friedhelm Meyer auf der Heide Heinz Nixdorf
Institute Computer Science Department University
of Paderborn Fürstenallee 11 33102 Paderborn,
Germany Tel. [phone] Fax
49 (0) 52 51/62 64 82 E-Mail fmadh_at_upb.de http/
/www.upb.de/cs/ag-madh