On the cut-query complexity of approximating max-cut

We consider the problem of query-efficient global max-cut on a weighted undirected graph in the value oracle model examined by [RSW18]. This model arises as a natural special case of submodular function maximization: on query $S \subseteq V$, the oracle returns the total weight of the cut between $S$ and $V \backslash S$. For most constants $c \in (0,1]$, we nail down the query complexity of achieving a $c$-approximation, for both deterministic and randomized algorithms (up to logarithmic factors). Analogously to general submodular function maximization in the same model, we observe a phase transition at $c = 1/2$: we design a deterministic algorithm for global $c$-approximate max-cut in $O(\log n)$ queries for any $c < 1/2$, and show that any randomized algorithm requires $\tilde{\Omega}(n)$ queries to find a $c$-approximate max-cut for any $c > 1/2$. Additionally, we show that any deterministic algorithm requires $\Omega(n^2)$ queries to find an exact max-cut (enough to learn the entire graph), and develop a $\tilde{O}(n)$-query randomized $c$-approximation for any $c < 1$. Our approach provides two technical contributions that may be of independent interest. One is a query-efficient sparsifier for undirected weighted graphs (prior work of [RSW18] holds only for unweighted graphs). Another is an extension of the cut dimension to rule out approximation (prior work of [GPRW20] introducing the cut dimension only rules out exact solutions)

A Proof of The Triangular Ashbaugh-Benguria-Payne-P\'{o}lya-Weinberger Inequality

In this paper, we show that for all triangles in the plane, the equilateral triangle maximizes the ratio of the first two Dirichlet-Laplacian eigenvalues. This is an extension of work by Siudeja, who proved the inequality in the case of acute triangles. The proof utilizes inequalities due to Siudeja and Freitas, together with improved variational bounds.Comment: 16 pages, 4 figure

The Query Complexity of Approximating Max-Cut in the Value Oracle Model

This thesis considers the problem of global max-cut on a weighted undirected graph in the \emph{value oracle model}. In this model, the algorithm does not have direct access to the graph and instead only has access to an oracle that can answer queries about the value of any cut. The algorithm's cost is simply the number of queries made. This model arises as a natural special case of submodular function maximisation with a value oracle. We consider both deterministic and randomised algorithms, and investigate the query complexity of achieving a $c$-approximation for any positive $c \leq 1$. We make substantial progress towards identifying the query complexity for all values of $c$ and both the deterministic and randomised settings up to logarithmic factors. We observe a ``phase transition" analogous to general submodular function maximisation at $c = 1/2$, where the complexity of solving the problem for $c > 1/2$ is much harder than solving it for $c 1/2$. We also observe what could be an additional jump at $c = 1$, showing that deterministic algorithms require at least $\Omega(n^2)$ queries to find the exact value of the max cut

Optimizing Space in Regev\u27s Factoring Algorithm

We improve the space efficiency of Regev\u27s quantum factoring algorithm [Reg23] while keeping the circuit size the same. Our main result constructs a quantum factoring circuit using $O(n \log n)$ qubits and $O(n^{3/2} \log n)$ gates. In contrast, Regev\u27s circuit requires $O(n^{3/2})$ qubits, while Shor\u27s circuit requires $O(n^2)$ gates. As with Regev, to factor an $n$-bit integer $N$, one runs this circuit independently $\approx \sqrt{n}$ times and apply Regev\u27s classical post-processing procedure. Our optimization is achieved by implementing efficient and reversible exponentiation with Fibonacci numbers in the exponent, rather than the usual powers of 2