Beating Exhaustive Search in Boolean
Games and Implications for
Complexity Theory
Evgeny Dantsin1 (joint presentation with
Sergei Soloviev2 and Alexander Wolpert1 )
1 Roosevelt
University, Chicago
2 IRIT,
Toulouse
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Disclaimer
The main results discussed in the presentation are
obtained in the following papers:
Chris Calabro, Russell Impagliazzo, and Ramamohan Paturi.
On the exact complexity of evaluating quantified k-CNF.
Algorithmica, 65(4):817–827, 2013.
Rahul Santhanam and Ryan Williams.
Beating exhaustive search for quantified Boolean formulas and connections
to circuit complexity.
In Proceedings of the 26th Annual ACM-SIAM Symposium on Discrete
Algorithms, SODA 2015, pages 231–241, 2015.
We show how these results can be applied to Boolean
games.
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Our Goal
A slight improvement on an algorithm can imply a lower
bound for a complexity class. Working on algorithms for
Boolean games, you can prove a lower bound like
NEXP 6⊆ NC1 /poly,
even if you don’t know what this notation means.
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Outline
1
Two Recent Algorithms: Beating Exhaustive Search
2
Improvements on Algorithms Imply Lower Bounds
3
Further Improvements for QBFs?
Strong Exponential Time Hypothesis
Lower Bound for NEXP
4
Open Questions
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Outline
1
Two Recent Algorithms: Beating Exhaustive Search
2
Improvements on Algorithms Imply Lower Bounds
3
Further Improvements for QBFs?
Strong Exponential Time Hypothesis
Lower Bound for NEXP
4
Open Questions
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Parameters of QBFs
By a QBF we mean a closed QBF in prenex form:
Q1 X1 Q2 X2 . . . Q` X` φ(X1 , X2 , . . . , X` )
where the prefix is written as ` alternating quantifier
blocks.
Parameters:
n is the number of variables in φ;
m is the size of φ;
` is the number of quantifier blocks in the prefix.
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QBF Modeling in Boolean Games
The existence of a winning strategy for a player in
two-player zero-sum Boolean games. Modeling with
QBFs where ` = 2.
The existence of a winning strategy for a player in
certain extensive-form Boolean games. Modeling
with QBFs where ` ≥ 2.
The existence of a dominant strategy for a player
and the existence of a pure Nash equilibrium.
Modeling with QBFs where ` = 2 and n is the
doubled number of variables in the game.
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QBF Evaluation Using Exhaustive Search
Simple recursive algorithm R for evaluating QBFs
Basis step: R(Φ) is 0 or 1, where Φ is a Boolean
expression without variables.
Recursive steps:
R(∀xΦ) = min{R(Φx=0 ), R(Φx=1 )}
R(∃xΦ) = max{R(Φx=0 ), R(Φx=1 )}
Upper bound:
poly (m) 2n
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Improvements for Special Cases
Can we “beat” exhaustive search? Can we evaluate
QBFs exponentially faster than in 2n time? In the
general case, no faster algorithm is known.
Improvement. However, the 2n upper bound can be
improved for QBFs where
the number of quantifier blocks is “small”;
the matrix is a CNF formula or a DNF formula.
Further improvement. The improved bound for the
special case above can be further improved for
QBFs where
the number of quantifier blocks is 2;
the matrix is a 3-CNF formula or a 3-DNF formula.
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Two Quantifier Blocks and 3-CNF
Theorem (Santhanam and Williams, SODA 2015)
Consider the class of QBFs where
the number of quantifier blocks is 2;
the matrix is a 3-CNF formula or a 3-DNF formula.
There is a deterministic algorithm that evaluates QBFs
from this class in time
√
poly (n) 2n−Ω(
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n)
.
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“Small” Number of Quantifier Blocks
Theorem (Santhanam and Williams, SODA 2015)
Consider the class of QBFs where
the number of quantifier blocks is a function ` of n
such that `(n) ≥ 6 and `(n) = o(log n/ log log n);
the matrix is a CNF formula or a DNF formula of
size poly (n).
There is a probabilistic zero-error algorithm that
evaluates QBFs from this class in expected time
poly (n) 2n−Ω(n
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1/(`+1)
)
Boolean Games and Complexity Theory
.
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Outline
1
Two Recent Algorithms: Beating Exhaustive Search
2
Improvements on Algorithms Imply Lower Bounds
3
Further Improvements for QBFs?
Strong Exponential Time Hypothesis
Lower Bound for NEXP
4
Open Questions
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Complexity Theory vs. Algorithm Design
Popular point of view (before 2010):
1 Proving nontrivial lower bounds is extremely
difficult; most such bounds are far beyond our
current techniques.
2 Slight improvements over exhaustive search (and,
more generally, slight improvements over existing
algorithms) are easy; simple tricks should suffice.
Ryan Williams (2010): One cannot
simultaneously believe in both 1 and 2.
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Example (One of Results by R. Williams)
Informally: Any nontrivial improvement over exhaustive
search for circuit sat would already imply
superpolynomial circuit lower bounds for NEXP.
Theorem (Williams, 2010)
If there exists an algorithm that tests satisfiability of
polynomial-size circuits with n inputs in
poly (n) 2n
s(n)
time, where s(n) is a superpolynomial function, then
NEXP is not in P/poly.
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Optimistic and Pessimistic Views
What such “conditional” results show: The existence of
an algorithm for a problem can be used to prove that
other certain algorithms do not exist.
Optimistic view. A “conditional” result can be used
to obtain an “unconditional” result. We can prove a
nontrivial lower bound by designing a slightly better
algorithm.
Pessimistic view. A “conditional” result shows that
it may be extremely difficult (if not impossible) to
design a slightly better algorithm.
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Example of “Unconditional” Result
In 2011, Ryan Williams showed that the “optimistic
approach” works. He proved two theorems:
“Conditional” result. A certain slight improvement
over exhaustive search for acc circuit sat
implies that NEXP is not contained in ACC.
Improvement. There exists a required improved
algorithm for acc circuit sat.
Hence, NEXP is not contained in ACC.
Both theorems were proved using many deep results of
complexity theory. Scott Aaronson has called this result
“one of the most spectacular of the decade”.
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What Is ACC?
AC0 circuits: polynomial size, constant depth,
AND, OR, NOT gates where AND and OR have
unbounded fan-in.
ACC circuits: AC0 circuits with the ability to
count. That is, ACC circuits have polynomial size,
constant depth, AND, OR, NOT, and MODm gates
where AND, OR, and MODm have unbounded
fan-in.
ACC: The class of problems that can be solved by
ACC circuits.
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Outline
1
Two Recent Algorithms: Beating Exhaustive Search
2
Improvements on Algorithms Imply Lower Bounds
3
Further Improvements for QBFs?
Strong Exponential Time Hypothesis
Lower Bound for NEXP
4
Open Questions
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Outline
1
Two Recent Algorithms: Beating Exhaustive Search
2
Improvements on Algorithms Imply Lower Bounds
3
Further Improvements for QBFs?
Strong Exponential Time Hypothesis
Lower Bound for NEXP
4
Open Questions
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Boolean Satisfiability: Open Questions
Satisfiability of k-CNF formulas. Known facts about
solving ksat.
Notation. Let {sk }k≥2 be a sequence of real
numbers defined by
sk = inf{δ | ksat can be solved in O(2δn ) time}
Exponential Time Hypothesis (ETH)
s3 > 0
Strong Exponential Time Hypothesis (SETH)
sk → 1 as k → ∞
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ETH and SETH
ETH implies that 3sat cannot be solved in O(2o(n) )
time.
ETH implies that P 6= NP.
ETH implies that {sk } increases infinitely often
(very nontrivial fact!).
SETH implies that sat (satisfiability of CNF
formulas with no restriction on the width of clauses)
cannot be solved in O(2δn ) time where δ < 1.
Researchers’ opinions about whether the hypotheses
are true or false.
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Quantified 3-CNF with Two Blocks
Currently fastest algorithm. Consider the class of
QBFs where ` = 2 and the matrix is a 3-CNF or
3-DNF formula. The algorithm by Santhanam and
Williams evaluates QBFs from this class in time
poly (m) 2n−Ω(
√
n)
.
Is there a faster algorithm? Possibly yes, but if the
faster algorithm runs in time
poly (n) 2δn where δ < 1,
then SETH is false. This follows from the results by
Calabro, Impagliazzo, Paturi (Algorithmica, 2013).
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Outline
1
Two Recent Algorithms: Beating Exhaustive Search
2
Improvements on Algorithms Imply Lower Bounds
3
Further Improvements for QBFs?
Strong Exponential Time Hypothesis
Lower Bound for NEXP
4
Open Questions
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“Small” Number of Blocks
Currently fastest algorithm. Consider the class of
QBFs where `(n) = o(log n/ log log n) and the
matrix is a CNF or DNF formula. The algorithm by
Santhanam and Williams evaluates QBFs from this
class in expected time
poly (n) 2n−Ω(n
1/(`+1)
)
.
Faster algorithm? Can we improve this bound to
poly (n) 2n−Ω(n
f (`)
)
where f is any function of ` such that
f (`) = ω(1/`)?
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Circuit Lower Bound for NEXP
It is proved in [Santhanam and Williams, SODA 15] that
such an improvement would imply
NEXP 6⊆ NC1 /poly.
This means that NEXP contains some languages that
cannot be decided by nonuniform families of Boolean
circuits with n inputs and
size poly (n);
depth O(log n);
fan-in 2.
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Outline
1
Two Recent Algorithms: Beating Exhaustive Search
2
Improvements on Algorithms Imply Lower Bounds
3
Further Improvements for QBFs?
Strong Exponential Time Hypothesis
Lower Bound for NEXP
4
Open Questions
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Question 1
Computational problem: The existence of a winning
strategy for a player in two-player zero-sum Boolean
games with 3-CNF or 3-DNF formulas.
Question: Is it possible to solve this problem in time
O(2δn )
where δ < 1?
Implication for complexity theory: If so, SETH is
false.
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Question 2
Computational problem: The existence of a winning
strategy for a player in Boolean games that can be
modeled with QBFs where ` is “small” and matrices
are polynomial-size CNF or DNF formulas.
Question: Is it possible to solve this problem in time
2n−Ω(n
f (`)
)
where f (`) = ω(1/`)?
Implication for complexity theory: If so, we have
NEXP 6⊆ NC1 /poly.
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Questions About Pure Nash Equilibria
Computational problem: The existence of a pure
Nash equilibrium in Boolean games.
Questions:
What is the exponential complexity of this problem?
In particular, is it possible to solve this problem within
polynomial space in 2n time (up to a factor polynomial in
the size of formulas), where n is the number of variables
in the game?
Is there any connection between algorithms for this
problem and lower bounds in complexity theory?
Remark: Similar questions about the existence of a
dominant strategy for a player in Boolean games.
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