CSCI 3210: Computational Game Theory Computational Complexity Ref: - - PDF document
10/26/20 CSCI 3210: Computational Game Theory Computational Complexity Ref: Algorithm Design (Ch 8) on Blackboard Mohammad T . Irfan Email: mirfan@bowdoin.edu Web: www.bowdoin.edu/~mirfan Famous complexity classes u NP (non-deterministic
u NP (non-deterministic
u Class of decision problems
for which we can verify a solution (certificate/witness) in polynomial time u P (deterministic
u Class of decision problems
that can be solved in polynomial time
u NP-Hard
u "As hard as NP" u Every NP problem reduces
to problems in this class in polynomial time
u May not be in NP (e.g.,
may not have polynomial- time verifier)
u NP-Complete
u Decision problems that
are:
1.
In NP
2.
NP-Hard (i.e., every NP problem reduces to these problem!)
Implication Fact? Alternative Fact? Unresolved!
u EXPTIME
u Evaluating a position in chess/checkers/Go
u Undecidable
u Halting problem
u A ton of other classes...
u Want to prove problem B is NP-complete u 2 main steps:
u Show that we can verify an answer to B in
polynomial time
u Pick a known NP-complete problem A u Reduce A to B in a polynomial-time (i.e., A is not harder than B è B is at least as hard as the already-known hard problem A)
Caution: not the
u Boolean satisfiability problem (SAT) is NP-
u P = NP if and only if SAT is in P
Stephen Cook
u Operators
u AND ∧ u OR ∨ u NOT ¬ (also, over-score)
u Given a boolean formula, is there a satisfying
(x1 v x2 v x3) ∧ ¬x1 ∧ ¬x3 ? (¬x1 v ¬x2) ∧ (¬x1 v x2) ∧ (x1 v ¬x2) ∧ (x1 v x2) ?
u Stylized version of SAT u (... V ... V ...) ∧ (... V ... V ...) ∧ (... V ... V …) ∧ ...
u Each parenthetic expression is a clause u Each green box is a literal (e.g., x1, ¬x2, etc.)
u 3-SAT is NP-complete
u Clique: A subset of nodes of a graph such that
there is an edge between every pair of nodes in that subset.
u CLIQUE problem: Given a graph and a number k,
Is there a clique of size >= k?
A clique of size 5
Example: Costas Busch- LSU
u We can verify a solution (k-clique) in
u
Check whether there is an edge between every pair of nodes in the given set of k nodes
u Take our favorite NP-complete problem 3-
u Reduce 3-SAT to CLIQUE
u
Given k, take any instance of 3-SAT with k clauses
u
Convert it to an instance of CLIQUE in polynomial time
u
Show that the answer to CLIQUE is yes if and only if the answer to 3-SAT is yes
Again, caution: not the other way!
Costas Busch - LSU
) ( ) ( ) ( ) (
4 3 2 3 2 1 4 2 1 4 2 1
x x x x x x x x x x x x Ú Ú Ù Ú Ú Ù Ú Ú Ù Ú Ú
1
x
2
x
1
x
2
x
4
x 1
x
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4
x
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3
x
Node: every literal of 3-SAT instance (k = 4 here)
Costas Busch - LSU
1
x
2
x
1
x
2
x
4
x 1
x
2
x
2
x
4
x
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3
Edge: Connect each node/literal with every literal of a different clause, except two literals that are negations of each other.
) ( ) ( ) ( ) (
4 3 2 3 2 1 4 2 1 4 2 1
x x x x x x x x x x x x Ú Ú Ù Ú Ú Ù Ú Ú Ù Ú Ú
Do it for every node!
Costas Busch - LSU
1
2
x
1
2
4
x 1
2
x
3
2
x
4
x
4
x
3
x
) ( ) ( ) ( ) (
4 3 2 3 2 1 4 2 1 4 2 1
x x x x x x x x x x x x Ú Ú Ù Ú Ú Ù Ú Ú Ù Ú Ú
Costas Busch - LSU
1 1
4 3 2 1
= = = = x x x x
4 3 2 3 2 1 4 2 1 4 2 1
1
2
x
1
2
4
x 1
2
3
2
x
4
x
4
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x
This proves CLIQUE is NP-Hard (3-SAT is not harder than CLIQUE!)
u Independent set: a subset of nodes of a
u INDEPNDENT SET problem
u Given a graph and a number k, does there exist an
independent set of size >= k?
u Is there an independent set of size ³ 6? Yes u Is there an independent set of size ³ 7? No
Nodes in independent set
Hantao Zhang – U Iowa
u Reduce a known NP-complete problem to
INDEPENDENT SET
u Art: Which NP-complete problem would you like to
pick? How would you do the reduction? REDUCTION: A à B Given any instance of problem A, convert it to an instance of B in polynomial time such that B instance has answer yes if and only if A instance has answer yes
u Vertex cover: a subset of nodes of a graph
u VERTEX COVER problem:
u Given a graph and a number k, is there a vertex
cover of size <= k?
u Is there a vertex cover of size £ 4? Yes. u Is there a vertex cover of size £ 3? No.
Hantao Zhang – U Iowa
vertex cover
u Reduce a known NP-complete problem to VERTEX
COVER
independent set
Hantao Zhang – U Iowa
vertex cover
u Special cases of 3-SAT u SET COVER u HITTING SET u SUBSET-SUM u PARTITION u GRAPH COLORING u HAMILTONIAN PATH (TSP) u ...
SHORTEST-PATH FRACTIONAL KNAPSACK FACTORIZATION/RSA GRAPH ISOMORPHISM TSP CLIQUE INDEPENDENT SET HALTING PROBLEM
any NP-complete problem
u (Gottlob et al., 2005)
u Proof.
u Reduction: 3-SAT' à GG
u
For any 3-SAT' instance (where each variable
u
Each clause and variable is a player
u
Neighbors/friends based on clause-var relations
u
Actions: {t, f, u}
u
Payoffs: next slide
u A clause in 3-SAT’ instance is different from
u A clause in 3-SAT’ can only have a truth value of
true or false.
u A clause player in GG can play any one of the three
actions t, f, and u.
Pay
Case 3 c plays t & all neighboring variable players play t/f making the clause in 3-SAT’ true 2 c plays u & all neighboring variable players play t/f making the clause in 3-SAT’ false 2 c plays f & some neighboring variable player plays u 1 All other cases Payoff of a clause player c Pay
Case 3 v plays t/f & all neighboring clause players play t/f 2 v plays u & some neighboring clause player plays u 1 All other cases Payoff of a variable player v
3-SAT' instance is satisfiable GG instance has a pure NE
Pay
Case 3 c plays t & all neighboring variable players play t/f making the clause in 3-SAT’ true 2 c plays u & all neighboring variable players play t/f making the clause in 3-SAT’ false 2 c plays f & some neighboring variable player plays u 1 All other cases Payoff of a clause player c Pay
Case 3 v plays t/f & all neighboring clause players play t/f 2 v plays u & some neighboring clause player plays u 1 All other cases Payoff of a variable player v
3-SAT' instance is satisfiable è GG instance has a pure NE Easy to prove!
Pay
Case 3 c plays t & all neighboring variable players play t/f making the clause in 3-SAT’ true 2 c plays u & all neighboring variable players play t/f making the clause in 3-SAT’ false 2 c plays f & some neighboring variable player plays u 1 All other cases Payoff of a clause player c Pay
Case 3 v plays t/f & all neighboring clause players play t/f 2 v plays u & some neighboring clause player plays u 1 All other cases Payoff of a variable player v
GG instance has a pure NE è 3-SAT' instance is satisfiable Not that easy! Outline: (1)A var v playing u cannot be a NE. (2)A clause c playing u cannot be a NE. (3)All players playing t/f and some clause c becoming false cannot be a NE: c gets 1, can get 2 by playing u. (4)All players playing t/f and some clause c playing f cannot be a NE: c gets 1 -> can get 3 if true or 2 if false. Details: page 222 of Gottlob et al. paper
u NP-complete, even for very special types of
u Games in "normal form" (with payoff matrices)
u Always exists (so not NP-complete!) u PPAD-complete even for 2-player games
u PPAD: Polynomial Parity Arguments on Directed
graphs (Papadimitriou, 1994)
u PPAD-Complete:
u Mixed-strategy NE computation for a graphical
game with maximum degree 3
u Computationally equivalent to solving 4-player
games in normal form [Goldberg & Papad., 2006]
u Polynomial time algorithms for special
u Heuristics for general instances u Approximation algorithms for general