Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits

Hrubeš, Pavel; Ramamoorthy, Sivaramakrishnan Natarajan; Rao, Anup; Yehudayoff, Amir

doi:10.4230/lipics.icalp.2019.72

Cited by 3 publications

(6 citation statements)

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“…This lemma was rst proved in [23] using Gröbner basis techniques. An elementary proof of this was recently given by the author and independently by Alon (see [25]) using the Combinatorial Nullstellensatz.…”

Section: Introductionmentioning

confidence: 91%

“…Hegedűs used the above lemma to give an alternate proof of a lower bound of 𝑛 in the case that 𝑡 is an odd prime. His proof was subsequently strengthened to a linear lower bound for all 𝑡 by Alon et al [5] and more recently to a near-tight lower bound of (𝑛/2) − 𝑜(𝑛) for all 𝑡 by Hrubeš et al [25]. Both these results used the lemma above.…”

Section: Introductionmentioning

confidence: 93%

“…The lemma is usually stated [23,5,25] for a more restricted choice of parameters. However, the known proofs extend to yield the stronger statement given here.…”

Section: Introductionmentioning

confidence: 99%

“…Hrubeš et al [25] also used Hegedűs's lemma to answer the following question of Kulikov and Podolskii [30] on depth-2 threshold circuits. What is the smallest 𝑘 = 𝑘(𝑛)…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

A Robust Version of Heged\H{u}s's Lemma, with Applications

Srinivasan¹

2023

TheoretiCS

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Heged\H{u}s's lemma is the following combinatorial statement regarding polynomials over finite fields. Over a field $\mathbb{F}$ of characteristic $p > 0$ and for $q$ a power of $p$, the lemma says that any multilinear polynomial $P\in \mathbb{F}[x_1,\ldots,x_n]$ of degree less than $q$ that vanishes at all points in $\{0,1\}^n$ of some fixed Hamming weight $k\in [q,n-q]$ must also vanish at all points in $\{0,1\}^n$ of weight $k + q$. This lemma was used by Heged\H{u}s (2009) to give a solution to \emph{Galvin's problem}, an extremal problem about set systems; by Alon, Kumar and Volk (2018) to improve the best-known multilinear circuit lower bounds; and by Hrube\v{s}, Ramamoorthy, Rao and Yehudayoff (2019) to prove optimal lower bounds against depth-$2$ threshold circuits for computing some symmetric functions. In this paper, we formulate a robust version of Heged\H{u}s's lemma. Informally, this version says that if a polynomial of degree $o(q)$ vanishes at most points of weight $k$, then it vanishes at many points of weight $k+q$. We prove this lemma and give three different applications.

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Section: Introductionmentioning

confidence: 91%

Section: Introductionmentioning

confidence: 93%

“…The lemma is usually stated [23,5,25] for a more restricted choice of parameters. However, the known proofs extend to yield the stronger statement given here.…”

Section: Introductionmentioning

confidence: 99%

“…Hrubeš et al [25] also used Hegedűs's lemma to answer the following question of Kulikov and Podolskii [30] on depth-2 threshold circuits. What is the smallest 𝑘 = 𝑘(𝑛)…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

A Robust Version of Heged\H{u}s's Lemma, with Applications

Srinivasan¹

2023

TheoretiCS

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“…• The following lemma was proven by Hrǔbes, Ramamoorthy, Rao and Yehudayoff [HRRY19] to solve a different version of the balancing problem. They used this lemma to exploit a connection between balancing set systems and depth-2 threshold circuits, which are an important class of Boolean circuits studied in the theory of computation.…”

Section: Motivationmentioning

confidence: 97%

On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic

Srinivasan¹,

Venkitesh²

2021

Preprint

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The finite-degree Zariski (Z-) closure is a classical algebraic object, that has found a key place in several applications of the polynomial method in combinatorics. In this work, we characterize the finite-degree Z-closures of a subclass of symmetric sets (subsets that are invariant under permutations of coordinates) of the Boolean cube, in positive characteristic.Our results subsume multiple statements on finite-degree Z-closures that have found applications in extremal combinatorial problems, for instance, pertaining to set systems (Hegedűs, Stud. Sci. Math. Hung. 2010; Hegedűs, arXiv 2021), and Boolean circuits (Hrǔbes et al., ICALP 2019). Our characterization also establishes that for the subclasses of symmetric sets that we consider, the finite-degree Z-closures have low computational complexity.A key ingredient in our characterization is a new variant of finite-degree Z-closures, defined using vanishing conditions on only symmetric polynomials satisfying a degree bound.

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CNF Encodings of Symmetric Functions

Emdin,

Kulikov,

Mihajlin

et al. 2024

Theory Comput Syst

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Many Boolean functions that need to be encoded as CNF in practice, have only exponential size CNF representations. To avoid this effect, one usually introduces nondeterministic variables. For example, whereas the minimum number of clauses in a CNF computing the parity functionIn this paper, we prove tradeoffs between various parameters (the number of clauses, the width of clauses, and the number of nondeterministic variables) of CNF encodings of various symmetric functions. In particular, we show that a folklore way of encoding parity as CNF is provably optimal. We do this by using a tight connection between CNF encodings and depth-3 circuits. This connection shows that CNF encodings is an interesting computational model for Boolean functions: on the one hand, it is routinely used in practice when translating a practical computational problem to a format acceptable by a SAT solver, on the other hand, lower bounds on the size of CNF encodings imply depth-3 circuit lower bounds.

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Lower Bounds on Balancing Sets and Depth-2 Threshold Circuits

Cited by 3 publications

References 0 publications

A Robust Version of Heged\H{u}s's Lemma, with Applications

A Robust Version of Heged\H{u}s's Lemma, with Applications

On Vanishing Properties of Polynomials on Symmetric Sets of the Boolean Cube, in Positive Characteristic

CNF Encodings of Symmetric Functions

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