On the Complexity of Search Problems George Pierrakos

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On the Complexity of Search ProblemsGeorge
Pierrakos

• Mostly based on
• On the Complexity of the Parity Argument and
  Other Insufficient Proofs of Existence Pap94
• On total functions, existence theorems and
  computational complexity MP91
• How easy is local search? JPY88
• Computational Complexity Pap92
• The complexity of computing a Nash equilibrium
  DGP06
• The complexity of pure Nash equilibria FPT04
• slides and scribe notes from many people

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Outline

• Generally on Search Problems
• The Class TFNP
• Subclasses of TFNP part I PPA, PPAD
• Problems in PPA, PPAD
• Completeness in PPAD
• Subclasses of TFNP part II PPP, PLS
• PPAD-completeness of NASH the complexity of
  computing equilibria in congestion games

3
Outline

• Generally on Search Problems
• The Class TFNP
• Subclasses of TFNP part I PPA, PPAD
• Problems in PPA, PPAD
• Completeness in PPAD
• Subclasses of TFNP part II PPP, PLS
• PPAD-completeness of NASH the complexity of
  computing equilibria in congestion games

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Decision Problems vs Search (or function)
Problems

• SAT
• Input boolean CNF-formula f
• Output yes or no
• FSAT
• Input boolean CNF-formula f
• Output satisfying assignment or no if none
  exist

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Are search problems harder?

• They are definitely not easier
• a poly-time algorithm for FSAT can be easily
  tweaked to give a poly-time algorithm for SAT
• and vice versa, FSAT reduces to SAT
• we can figure out a satisfying assignment by
  running poly-time algorithm for SAT n-times

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The Classes FP and FNP

• L NP iff there exists poly-time computable
  RL(x,y) s.t.
• X L ? y y p(x) RL(x,y)
• Note how RL defines the problem-language L
• The corresponding search problem ?R(L) FNP is
• given an x find any y s.t. RL(x,y) and reply
  no if none exist
• FSAT FNP what about FTSP?
• Are all FNP problems self-reducible like FSAT?
  open?
• FP is the subclass of FNP where we only consider
  problems for which a poly-time algorithm is known

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Reductions and completeness

• A function problem ?R reduces to a function
  problem ?S if there exist log-space computable
  string functions f and g, s.t.
• R(x,g(y)) ? S(f(x),y)
• intuitively f reduces problem ?R to ?S
• and g transforms a solution of ?S to one of ?R
• Standard notion of completeness works fine

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FP lt?gt FNP

• A proof a-la-Cook shows that FSAT is FNP-complete
• Hence, if FSAT FP then FNP FP
• But we showed self-reducibility for SAT, so the
  theorem follows
• Theorem FP FNP iff PNP
• So, why care for function problems anyway??

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Outline

• Generally on Search Problems
• The Class TFNP
• Subclasses of TFNP part I PPA, PPAD
• Problems in PPA, PPAD
• Completeness in PPAD
• Subclasses of TFNP part II PPP, PLS
• PPAD-completeness of NASH the complexity of
  computing equilibria in congestion games

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On total functions the class TFNP

• What happens if the relation R is total?
• i.e., for each x there is at least one y s.t.
  R(x,y)
• Define TFNP to be the subclass of FNP where the
  relation R is total
• TFNP contains problems that always have a
  solution, e.g. factoring, fix-point theorems,
  graph-theoretic problems,
• How do we know a solution exists?
• By an inefficient proof of existence, i.e.
  non-(efficiently)-constructive proof
• The idea is to categorize the problems in TFNP
  based on the type of inefficient argument that
  guarantees their solution

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Basic stuff about TFNP

• FP TFNP FNP
• TFNP F(NP coNP)
• NP problems with yes certificate y s.t.
  R1(x,y)
• coNP problems with no certificate z s.t.
  R2(x,y)
• for TFNP F(NP coNP) take R R1 U R2
• for F(NP coNP) TFNP take R1 R and R2
  ø
• There is an FNP-complete problem in TFNP iff NP
  coNP
• ? If NP coNP then trivially FNP TFNP
• ? If the FNP-complete problem ?R is in TFNP
  thenFSAT reduces to ?R via f and g, hence any
  unsatisfiable formula f has a no certificate y,
  s.t. R(f(f),y) (y exists since ?R is in TFNP) and
  g(y)no
• TFNP is a semantic complexity class ? no complete
  problems!
• note how telling whether a relation is total is
  undecidable (and not even RE!!)