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Electronic Colloquium on Computational Complexity

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TR25-084 | 28th June 2025
Somnath Bhattacharjee, Mrinal Kumar, Shanthanu Rai, Varun Ramanathan, Ramprasad Saptharishi, Shubhangi Saraf

Closure under factorization from a result of Furstenberg

We show that algebraic formulas and constant-depth circuits are \emph{closed} under taking factors. In other words, we show that if a multivariate polynomial over a field of characteristic zero has a small constant-depth circuit or formula, then all its factors can be computed by small constant-depth circuits or formulas ... more >>>


TR25-083 | 24th June 2025
C.S. Bhargav, Prateek Dwivedi, Nitin Saxena

A primer on the closure of algebraic complexity classes under factoring

Polynomial factorization is a fundamental problem in computational algebra. Over the past half century, a variety of algorithmic techniques have been developed to tackle different variants of this problem. In parallel, algebraic complexity theory classifies polynomials into complexity classes based on their perceived `hardness'. This raises a natural question: Do ... more >>>


TR25-082 | 23rd June 2025
Per Austrin, Johan Håstad, Björn Martinsson

On the usefulness of Promises

A Boolean predicate $A$ is defined to be promise-useful if $PCSP(A,B)$ is tractable for some non-trivial $B$ and otherwise it is promise-useless. We initiate investigations of this notion and derive sufficient conditions for both promise-usefulness and promise-uselessness (assuming $\text{P} \ne \text{NP}$). While we do not obtain a complete characterization, our ... more >>>



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