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Electronic Colloquium on Computational Complexity

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REPORTS > 2023:
All reports in year 2023:
TR23-008 | 2nd February 2023
Ond?ej Ježil

Limits of structures and Total NP Search Problems

For a class of finite graphs, we define a limit object relative to some computationally restricted class of functions. The properties of the limit object then reflect how a computationally restricted viewer "sees" a generic instance from the class. The construction uses Krají?ek's forcing with random variables [7]. We prove ... more >>>

TR23-007 | 3rd February 2023
Jan Krajicek

Extended Nullstellensatz proof systems

For a finite set $\cal F$ of polynomials over a fixed finite prime field of size $p$ containing all polynomials $x^2 - x$, a Nullstellensatz proof of the unsolvability of the system
f = 0\ ,\ \mbox{ all } f \in {\cal F}
is a linear combination ... more >>>

TR23-006 | 20th January 2023
Nader Bshouty

Superpolynomial Lower Bounds for Learning Monotone Classes

Revisions: 1

Koch, Strassle, and Tan [SODA 2023], show that, under the randomized exponential time hypothesis, there is no distribution-free PAC-learning algorithm that runs in time $n^{\tilde O(\log\log s)}$ for the classes of $n$-variable size-$s$ DNF, size-$s$ Decision Tree, and $\log s$-Junta by DNF (that returns a DNF hypothesis). Assuming a natural ... more >>>

TR23-005 | 13th January 2023
Paul Beame, Niels Kornerup

Cumulative Memory Lower Bounds for Randomized and Quantum Computation

Cumulative memory---the sum of space used over the steps of a computation---is a fine-grained measure of time-space complexity that is a more accurate measure of cost for algorithms with infrequent spikes in memory usage in the context of technologies such as cloud computing that allow dynamic allocation and de-allocation of ... more >>>

TR23-004 | 13th January 2023
Yinan Li, Youming Qiao, Avi Wigderson, Yuval Wigderson, Chuanqi Zhang

On linear-algebraic notions of expansion

A fundamental fact about bounded-degree graph expanders is that three notions of expansion---vertex expansion, edge expansion, and spectral expansion---are all equivalent. In this paper, we study to what extent such a statement is true for linear-algebraic notions of expansion.

There are two well-studied notions of linear-algebraic expansion, namely dimension expansion ... more >>>

TR23-003 | 11th January 2023
Shachar Lovett, Jiapeng Zhang

Streaming Lower Bounds and Asymmetric Set-Disjointness

Frequency estimation in data streams is one of the classical problems in streaming algorithms. Following much research, there are now almost matching upper and lower bounds for the trade-off needed between the number of samples and the space complexity of the algorithm, when the data streams are adversarial. However, in ... more >>>

TR23-002 | 5th January 2023
Noga Alon, Olivier Bousquet, Kasper Green Larsen, Shay Moran, Shlomo Moran

Diagonalization Games

Revisions: 2

We study several variants of a combinatorial game which is based on Cantor's diagonal argument. The game is between two players called Kronecker and Cantor. The names of the players are motivated by the known fact that Leopold Kronecker did not appreciate Georg Cantor's arguments about the infinite, and even ... more >>>

TR23-001 | 5th January 2023
Prerona Chatterjee, Pavel Hrubes

New Lower Bounds against Homogeneous Non-Commutative Circuits

We give several new lower bounds on size of homogeneous non-commutative circuits. We present an explicit homogeneous bivariate polynomial of degree $d$ which requires homogeneous non-commutative circuit of size $\Omega(d/\log d)$. For an $n$-variate polynomial with $n>1$, the result can be improved to $\Omega(nd)$, if $d\leq n$, or $\Omega(nd \frac{\log ... more >>>

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