Beating Brute Force for Compression Problems
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3618260.3649778.pdf
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Author(s)
Hirahara, Shuichi
Ilango, Rahul
Williams, R. Ryan
Date Issued
June 10, 2024
Publisher
ACM|Proceedings of the 56th Annual ACM Symposium on Theory of Computing
Citation
Hirahara, Shuichi, Ilango, Rahul and Williams, R. Ryan. 2024. "Beating Brute Force for Compression Problems."
Version
Final published version
Abstract
A compression problem is de ned with respect to an e cient encoding function ; given a string , our task is to nd the shortest
such that () = . The obvious brute-force algorithm for solving
this compression task on -bit strings runs in time (2
ℓ
· ()),
where ℓ is the length of the shortest description and () is the
time complexity of when it prints -bit output.
We prove that every compression problem has a Boolean circuit
family which nds short descriptions more e ciently than brute
force. In particular, our circuits have size 2
4ℓ/5
·poly(()), which is
signi cantly more e cient for all ℓ ≫ log(()). Our construction
builds on Fiat-Naor’s data structure for function inversion [SICOMP
1999]: we show how to carefully modify their data structure so that
it can be nontrivially implemented using Boolean circuits, and
we show how to utilize hashing so that the circuit size is only
exponential in the description length.
As a consequence, the Minimum Circuit Size Problem for generic
fan-in two circuits of size () on truth tables of size 2
can be solved
by circuits of size 2
4
5
·+ ()
·poly(2
), where = () log2
(() +
). This improves over the brute-force approach of trying all possible size-() circuits for all () ≥ . Similarly, the task of computing a short description of a string when its K
-complexity is
at most ℓ, has circuits of size 2
4
5
ℓ
· poly(). We also give nontrivial
circuits for computing Kt complexity on average, and for solving
NP relations with “compressible” instance-witness pairs.
Description
STOC ’24, June 24–28, 2024, Vancouver, BC, Canada
MIT Department
Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science
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DOI of Published Version
https://doi.org/10.1145/3618260.3649778
