arXiv subjects

Samuel Kutin

Publications and source records attributed to Samuel Kutin.

6 recordsLinked to original sources

arXiv2026

Permutation Wordle

We introduce a guessing game, ``Permutation Wordle,'' in which a guesser attempts to recover a setter's hidden permutation of the set $\{1, \ldots, n\}$. In each round, the guesser submits a word over the alphabet $\{1, \ldots, n\}$, and, as in the game Wordle, learns which entries are correct. We describe a natural strategy and prove that it is optimal in a strong sense: for every $r$, it solves at least as many secrets within $r$ rounds as any possible strategy. The number of permutations it solves in exactly $k+1$ rounds is the Eulerian number $A(n,k)$.

math.CO↗

arXiv2013

The Promise Polynomial Hierarchy

The polynomial hierarchy is a grading of problems by difficulty, including P, NP and coNP as the best known classes. The promise polynomial hierarchy is similar, but extended to include promise problems. It turns out that the promise polynomial hierarchy is considerably simpler to work with, and many open questions about the polynomial hierarchy can be resolved in the promise polynomial hierarchy. Our main theorem is that, in the world of promise problems, if phi has a weak (Turing, Cook) reduction to SAT then phi has a strong (Karp, many-one) reduction to UVAL2, where UVAL2(f) is the promise problem of finding the unique x such that f(x,y)=1 for all y. We also give a complete promise problem for the promise problem equivalent of UP intersect coUP.

cs.CC↗

arXiv2012

Almost-everywhere algorithmic stability and generalization error

We explore in some detail the notion of algorithmic stability as a viable framework for analyzing the generalization error of learning algorithms. We introduce the new notion of training stability of a learning algorithm and show that, in a general setting, it is sufficient for good bounds on generalization error. In the PAC setting, training stability is both necessary and sufficient for learnability.\ The approach based on training stability makes no reference to VC dimension or VC entropy. There is no need to prove uniform convergence, and generalization error is bounded directly via an extended McDiarmid inequality. As a result it potentially allows us to deal with a broader class of learning algorithms than Empirical Risk Minimization. \ We also explore the relationships among VC dimension, generalization error, and various notions of stability. Several examples of learning algorithms are considered.

cs.LG↗

arXiv2012

Efficient Distributed Quantum Computing

We provide algorithms for efficiently addressing quantum memory in parallel. These imply that the standard circuit model can be simulated with low overhead by the more realistic model of a distributed quantum computer. As a result, the circuit model can be used by algorithm designers without worrying whether the underlying architecture supports the connectivity of the circuit. In addition, we apply our results to existing memory intensive quantum algorithms. We present a parallel quantum search algorithm and improve the time-space trade-off for the Element Distinctness and Collision problems.

quant-ph↗

arXiv2003

A quantum lower bound for the collision problem

We extend Shi's 2002 quantum lower bound for collision in $r$-to-one functions with $n$ inputs. Shi's bound of $Ω((n/r)^{1/3})$ is tight, but his proof applies only in the case where the range has size at least $3n/2$. We give a modified version of Shi's argument which removes this restriction.

quant-ph↗

arXiv2002

On the Quantum Black-Box Complexity of Majority

We describe a quantum black-box network computing the majority of N bits with zero-sided error eps using only 2N/3 + O(sqrt{N (log log N + log 1/eps)}) queries: the algorithm returns the correct answer with probability at least 1 - eps, and "I don't know" otherwise. Our algorithm is given as a randomized "XOR decision tree" for which the number of queries on any input is strongly concentrated around a value of at most 2N/3. We provide a nearly matching lower bound of 2N/3 - O(sqrt(N)) on the expected number of queries on a worst-case input in the randomized XOR decision tree model with zero-sided error o(1). Any classical randomized decision tree computing the majority on N bits with zero-sided error 1/2 has cost N.

quant-ph↗