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Electronic Colloquium on Computational Complexity

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Reports tagged with representation theory:
TR97-048 | 13th October 1997
Soren Riis, Meera Sitharam

Non-constant Degree Lower Bounds imply linear Degree Lower Bounds

The semantics of decision problems are always essentially independent of the
underlying representation. Thus the space of input data (under appropriate
indexing) is closed
under action of the symmetrical group $S_n$ (for a specific data-size)
and the input-output relation is closed under the action of $S_n$.
more >>>

TR06-050 | 18th April 2006
Alexander Razborov, Sergey Yekhanin

An Omega(n^{1/3}) Lower Bound for Bilinear Group Based Private Information Retrieval

A two server private information retrieval (PIR) scheme
allows a user U to retrieve the i-th bit of an
n-bit string x replicated between two servers while each
server individually learns no information about i. The main
parameter of interest in a PIR scheme is its communication
complexity, namely the ... more >>>

TR12-068 | 25th May 2012
Manuel Arora, Gábor Ivanyos, Marek Karpinski, Nitin Saxena

Deterministic Polynomial Factoring and Association Schemes

The problem of finding a nontrivial factor of a polynomial $f(x)$ over a finite field $\mathbb{F}_q$ has many known efficient, but randomized, algorithms. The deterministic complexity of this problem is a famous open question even assuming the generalized Riemann hypothesis (GRH). In this work we improve the state of the ... more >>>

TR16-162 | 18th October 2016
Joshua Grochow

NP-hard sets are not sparse unless P=NP: An exposition of a simple proof of Mahaney's Theorem, with applications

Mahaney's Theorem states that, assuming P $\neq$ NP, no NP-hard set can have a polynomially bounded number of yes-instances at each input length. We give an exposition of a very simple unpublished proof of Manindra Agrawal whose ideas appear in Agrawal-Arvind ("Geometric sets of low information content," Theoret. Comp. Sci., ... more >>>

TR17-033 | 19th February 2017
Daniel Kane, Shachar Lovett, Sankeerth Rao

Labeling the complete bipartite graph with no zero cycles

Revisions: 2

Assume that the edges of the complete bipartite graph $K_{n,n}$ are labeled with elements of $\mathbb{F}_2^d$, such that the sum over
any simple cycle is nonzero. What is the smallest possible value of $d$? This problem was raised by Gopalan et al. [SODA 2017] as it characterizes the alphabet size ... more >>>

TR17-131 | 1st September 2017
Joshua Grochow, Cris Moore

Designing Strassen's algorithm

In 1969, Strassen shocked the world by showing that two n x n matrices could be multiplied in time asymptotically less than $O(n^3)$. While the recursive construction in his algorithm is very clear, the key gain was made by showing that 2 x 2 matrix multiplication could be performed with ... more >>>

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