On ϵ-sensitive monotone computations
We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial f ∈ R [ x 1 , ⋯ , x n ] of degree d has an arithmetic circuit of size s then (...
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| Published in: | Computational complexity Vol. 29; no. 2 |
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| Language: | English |
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01.12.2020
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| Abstract | We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial
f
∈
R
[
x
1
,
⋯
,
x
n
]
of degree
d
has an arithmetic circuit of size
s
then
(
x
1
+
⋯
+
x
n
+
1
)
d
+
ϵ
f
has a monotone arithmetic circuit of size
O
(
s
d
2
+
n
log
n
)
, for some
ϵ
>
0
. Second, if
f
:
{
0
,
1
}
n
→
{
0
,
1
}
is a Boolean function, we associate with
f
an explicit exponential-size matrix
M
(
f
) such that the Boolean circuit size of
f
is at least
Ω
(
min
ϵ
>
0
(
rk
+
(
M
(
f
)
-
ϵ
J
)
)
-
2
n
)
, where
J
is the all-ones matrix and
rk
+
denotes the nonnegative rank of a matrix. In fact, the quantity
min
ϵ
>
0
(
rk
+
(
M
(
f
)
-
ϵ
J
)
)
characterizes how hard is it to distinguish rejecting and accepting inputs of
f
by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar
ϵ
-sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system. |
|---|---|
| AbstractList | We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial
f
∈
R
[
x
1
,
⋯
,
x
n
]
of degree
d
has an arithmetic circuit of size
s
then
(
x
1
+
⋯
+
x
n
+
1
)
d
+
ϵ
f
has a monotone arithmetic circuit of size
O
(
s
d
2
+
n
log
n
)
, for some
ϵ
>
0
. Second, if
f
:
{
0
,
1
}
n
→
{
0
,
1
}
is a Boolean function, we associate with
f
an explicit exponential-size matrix
M
(
f
) such that the Boolean circuit size of
f
is at least
Ω
(
min
ϵ
>
0
(
rk
+
(
M
(
f
)
-
ϵ
J
)
)
-
2
n
)
, where
J
is the all-ones matrix and
rk
+
denotes the nonnegative rank of a matrix. In fact, the quantity
min
ϵ
>
0
(
rk
+
(
M
(
f
)
-
ϵ
J
)
)
characterizes how hard is it to distinguish rejecting and accepting inputs of
f
by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar
ϵ
-sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system. We show that strong-enough lower bounds on monotone arithmetic circuits or the nonnegative rank of a matrix imply unconditional lower bounds in arithmetic or Boolean circuit complexity. First, we show that if a polynomial f∈R[x1,⋯,xn] of degree d has an arithmetic circuit of size s then (x1+⋯+xn+1)d+ϵf has a monotone arithmetic circuit of size O(sd2+nlogn), for some ϵ>0. Second, if f:{0,1}n→{0,1} is a Boolean function, we associate with f an explicit exponential-size matrix M(f) such that the Boolean circuit size of f is at least Ω(minϵ>0(rk+(M(f)-ϵJ))-2n), where J is the all-ones matrix and rk+ denotes the nonnegative rank of a matrix. In fact, the quantity minϵ>0(rk+(M(f)-ϵJ)) characterizes how hard is it to distinguish rejecting and accepting inputs of f by means of a linear program. Finally, we introduce a proof system resembling the monotone calculus of Atserias et al. (J Comput Syst Sci 65:626–638, 2002) and show that similar ϵ-sensitive lower bounds on monotone arithmetic circuits imply lower bounds on proof-size in the system. |
| ArticleNumber | 6 |
| Author | Hrubeš, Pavel |
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| Keywords | Nonnegative rank Extension complexity 69Q09 Arithmetic circuit complexity 68Q17 |
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| References | Pavel Hrubeš & Amir YehudayoffHomogeneous formulas and symmetric polynomialsComputational Complexity2011203559578282480910.1007/s00037-011-0007-3 S. A. Vavasis (2008). On the complexity of nonnegative matrix factorization. SIAM Journal on Optimization20(3), 1364-1377 HrubešPOn the nonnegative rank of distance matricesInformation Processing Letters201211211457461290514810.1016/j.ipl.2012.02.009 D. deCaen, D. Gregory & N. Pullman (1981). The Boolean rank of zero one matrices. In Proceedings of the Third Caribean Conference on Combinatorics and Computing, 169–173 G. Braun, S. Fiorini, S. Pokutta & D. Steuer (2012). Approximation Limits of Linear Programs (Beyond Hierarchies). In FOCS, 480–489 GoosMJainRWatsonTExtension complexity of independent set polytopesSIAM J. Comput.2018471241269376592410.1137/16M109884X BeasleyLLaffeyTReal rank versus nonnegative rankLinear Algebra and its Applications20094311223302335256302510.1016/j.laa.2009.02.034 A. Yehudayoff (2019). Separating monotone VP and VNP. In STOC Thomas Rothvoß (2011). Some 0/1 polytopes need exponential size extended formulations. CoRRarXiv:1105.0036 Amir Shpilka & Avi Wigderson (1999). Depth-3 arithmetic formulae over fields of characteristic zero. In In CCC, 87. IEEE Computer Society S. Basu, R. Pollack & M.F. Roy (2006). Algorithms in real algebraic geometry. Springer-Verlag. P. Pudlák & M. de Oliveira Oliveira (2017). Representations of monotone Boolean functions by linear programs. In Proceedings of the 32nd Computational Complexity Conference ValiantLGNegation is powerless for Boolean slice functionsSIAM J. Comput.198615253153583760110.1137/0215037 P. Bürgisser, M. Clausen & M. A. Shokrollahi (1997). Algebraic complexity theory, volume 315 of A series of comprehensive studies in mathematics. Springer ValiantLGNegation can be exponentially powerfulTheoretical Computer Science19801230331458931110.1016/0304-3975(80)90060-2 YannakakisMihalisExpressing combinatorial optimization problems by linear programsJournal of Computer and System Sciences1991433441466113547210.1016/0022-0000(91)90024-Y Amir Shpilka & Amir YehudayoffArithmetic Circuits: A survey of recent results and open questionsFoundations and Trends in Theoretical Computer Science20105320738827561661205.68175 L. G. Valiant (1979). Completeness Classes in Algebra. In STOC, 249–261 P. Hrubeš (2019). On the complexity of computing a random Boolean function over the reals. ECCC ValiantLGReducibility by algebraic projectionsEnseign. Math.1982282532686842360493.68045 A. Atserias, A. Dawar & J. Ochremiak (2019). On the power of symmetric linear programs. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) A. Razborov (1992). On submodular complexity measures. In Boolean functions complexity, 76–83. Cambridge University Press KrajíčekJBounded arithmetic, propositional logic, and complexity theory1995USACambridge University Press10.1017/CBO9780511529948 N. Nisan (1991). Lower bounds for non-commutative computation. In Proceeding of the 23th STOC, 410–418 Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary & Ronald de Wolf (2011). Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds. CoRRarXiv:1111.0837 ShamirESnirMOn the depth complexity of formulasJournal Theory of Computing Systems19791313013225910100445.68031 R. Raz (2004). Multi-linear formulas for Permanent and Determinant are of super-polynomial size. In Proceeding of the 36th STOC, 633–641 AtseriasAGalesiNPudlákPMonotone simulations of non-monotone proofsJ. of Computer and System Sciences200265626638196464610.1016/S0022-0000(02)00020-X T. Rothvoss (2017). The matching polytope has exponential extension complexity. J. of the ACM 64(6). I. Wegener (1987). The complexity of Boolean functions 196_CR30 196_CR8 L Beasley (196_CR4) 2009; 431 LG Valiant (196_CR24) 1980; 12 Pavel Hrubeš & Amir Yehudayoff (196_CR12) 2011; 20 196_CR3 196_CR1 J Krajíček (196_CR13) 1995 196_CR6 196_CR7 196_CR5 P Hrubeš (196_CR10) 2012; 112 196_CR27 196_CR28 196_CR23 Amir Shpilka & Amir Yehudayoff (196_CR22) 2010; 5 196_CR21 LG Valiant (196_CR26) 1986; 15 LG Valiant (196_CR25) 1982; 28 E Shamir (196_CR20) 1979; 13 196_CR19 196_CR15 196_CR16 196_CR17 196_CR18 A Atserias (196_CR2) 2002; 65 M Goos (196_CR9) 2018; 47 196_CR11 Mihalis Yannakakis (196_CR29) 1991; 43 196_CR14 |
| References_xml | – reference: N. Nisan (1991). Lower bounds for non-commutative computation. In Proceeding of the 23th STOC, 410–418 – reference: A. Razborov (1992). On submodular complexity measures. In Boolean functions complexity, 76–83. Cambridge University Press – reference: L. G. Valiant (1979). Completeness Classes in Algebra. In STOC, 249–261 – reference: P. Pudlák & M. de Oliveira Oliveira (2017). Representations of monotone Boolean functions by linear programs. In Proceedings of the 32nd Computational Complexity Conference – reference: Pavel Hrubeš & Amir YehudayoffHomogeneous formulas and symmetric polynomialsComputational Complexity2011203559578282480910.1007/s00037-011-0007-3 – reference: GoosMJainRWatsonTExtension complexity of independent set polytopesSIAM J. Comput.2018471241269376592410.1137/16M109884X – reference: T. Rothvoss (2017). The matching polytope has exponential extension complexity. J. of the ACM 64(6). – reference: ValiantLGNegation can be exponentially powerfulTheoretical Computer Science19801230331458931110.1016/0304-3975(80)90060-2 – reference: Amir Shpilka & Amir YehudayoffArithmetic Circuits: A survey of recent results and open questionsFoundations and Trends in Theoretical Computer Science20105320738827561661205.68175 – reference: AtseriasAGalesiNPudlákPMonotone simulations of non-monotone proofsJ. of Computer and System Sciences200265626638196464610.1016/S0022-0000(02)00020-X – reference: S. A. Vavasis (2008). On the complexity of nonnegative matrix factorization. SIAM Journal on Optimization20(3), 1364-1377 – reference: S. Basu, R. Pollack & M.F. Roy (2006). Algorithms in real algebraic geometry. Springer-Verlag. – reference: I. Wegener (1987). The complexity of Boolean functions – reference: YannakakisMihalisExpressing combinatorial optimization problems by linear programsJournal of Computer and System Sciences1991433441466113547210.1016/0022-0000(91)90024-Y – reference: Thomas Rothvoß (2011). Some 0/1 polytopes need exponential size extended formulations. CoRRarXiv:1105.0036 – reference: ShamirESnirMOn the depth complexity of formulasJournal Theory of Computing Systems19791313013225910100445.68031 – reference: ValiantLGNegation is powerless for Boolean slice functionsSIAM J. Comput.198615253153583760110.1137/0215037 – reference: P. Hrubeš (2019). On the complexity of computing a random Boolean function over the reals. ECCC – reference: R. Raz (2004). Multi-linear formulas for Permanent and Determinant are of super-polynomial size. In Proceeding of the 36th STOC, 633–641 – reference: D. deCaen, D. Gregory & N. Pullman (1981). The Boolean rank of zero one matrices. In Proceedings of the Third Caribean Conference on Combinatorics and Computing, 169–173 – reference: HrubešPOn the nonnegative rank of distance matricesInformation Processing Letters201211211457461290514810.1016/j.ipl.2012.02.009 – reference: G. Braun, S. Fiorini, S. Pokutta & D. Steuer (2012). Approximation Limits of Linear Programs (Beyond Hierarchies). In FOCS, 480–489 – reference: P. Bürgisser, M. Clausen & M. A. Shokrollahi (1997). Algebraic complexity theory, volume 315 of A series of comprehensive studies in mathematics. Springer – reference: A. Atserias, A. Dawar & J. Ochremiak (2019). On the power of symmetric linear programs. In 34th Annual ACM/IEEE Symposium on Logic in Computer Science (LICS) – reference: KrajíčekJBounded arithmetic, propositional logic, and complexity theory1995USACambridge University Press10.1017/CBO9780511529948 – reference: ValiantLGReducibility by algebraic projectionsEnseign. Math.1982282532686842360493.68045 – reference: A. Yehudayoff (2019). Separating monotone VP and VNP. In STOC – reference: Amir Shpilka & Avi Wigderson (1999). Depth-3 arithmetic formulae over fields of characteristic zero. In In CCC, 87. IEEE Computer Society – reference: BeasleyLLaffeyTReal rank versus nonnegative rankLinear Algebra and its Applications20094311223302335256302510.1016/j.laa.2009.02.034 – reference: Samuel Fiorini, Serge Massar, Sebastian Pokutta, Hans Raj Tiwary & Ronald de Wolf (2011). Linear vs. Semidefinite Extended Formulations: Exponential Separation and Strong Lower Bounds. CoRRarXiv:1111.0837 – volume: 65 start-page: 626 year: 2002 ident: 196_CR2 publication-title: J. of Computer and System Sciences doi: 10.1016/S0022-0000(02)00020-X – volume: 47 start-page: 241 issue: 1 year: 2018 ident: 196_CR9 publication-title: SIAM J. Comput. doi: 10.1137/16M109884X – ident: 196_CR30 doi: 10.1145/3313276.3316311 – volume: 12 start-page: 303 year: 1980 ident: 196_CR24 publication-title: Theoretical Computer Science doi: 10.1016/0304-3975(80)90060-2 – ident: 196_CR3 doi: 10.1007/3-540-33099-2 – ident: 196_CR7 – volume: 5 start-page: 207 issue: 3 year: 2010 ident: 196_CR22 publication-title: Foundations and Trends in Theoretical Computer Science – ident: 196_CR8 doi: 10.1145/2213977.2213988 – ident: 196_CR19 doi: 10.1007/s10107-012-0574-3 – ident: 196_CR28 doi: 10.1007/3-540-18170-9_185 – volume: 15 start-page: 531 issue: 2 year: 1986 ident: 196_CR26 publication-title: SIAM J. Comput. doi: 10.1137/0215037 – volume: 28 start-page: 253 year: 1982 ident: 196_CR25 publication-title: Enseign. Math. – ident: 196_CR16 doi: 10.1145/1007352.1007353 – ident: 196_CR1 doi: 10.1109/LICS.2019.8785792 – volume: 13 start-page: 301 issue: 1 year: 1979 ident: 196_CR20 publication-title: Journal Theory of Computing Systems – ident: 196_CR17 doi: 10.1017/CBO9780511526633.007 – ident: 196_CR14 doi: 10.1145/103418.103462 – ident: 196_CR18 doi: 10.1145/3127497 – volume: 43 start-page: 441 issue: 3 year: 1991 ident: 196_CR29 publication-title: Journal of Computer and System Sciences doi: 10.1016/0022-0000(91)90024-Y – ident: 196_CR15 – ident: 196_CR11 – ident: 196_CR23 doi: 10.1145/800135.804419 – ident: 196_CR5 doi: 10.1109/FOCS.2012.10 – volume-title: Bounded arithmetic, propositional logic, and complexity theory year: 1995 ident: 196_CR13 doi: 10.1017/CBO9780511529948 – volume: 431 start-page: 2330 issue: 12 year: 2009 ident: 196_CR4 publication-title: Linear Algebra and its Applications doi: 10.1016/j.laa.2009.02.034 – ident: 196_CR27 doi: 10.1137/070709967 – ident: 196_CR6 doi: 10.1007/978-3-662-03338-8 – volume: 112 start-page: 457 issue: 11 year: 2012 ident: 196_CR10 publication-title: Information Processing Letters doi: 10.1016/j.ipl.2012.02.009 – volume: 20 start-page: 559 issue: 3 year: 2011 ident: 196_CR12 publication-title: Computational Complexity doi: 10.1007/s00037-011-0007-3 – ident: 196_CR21 |
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| SubjectTerms | Algorithm Analysis and Problem Complexity Arithmetic Boolean Boolean algebra Boolean functions Circuits Computational Mathematics and Numerical Analysis Computer Science Lower bounds Mathematical analysis Polynomials |
| Title | On ϵ-sensitive monotone computations |
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