Related papers: Mastermind with a Linear Number of Queries
Recent advancements in large language models (LLMs) have led to remarkable performance across a wide range of language understanding and mathematical tasks. As a result, increasing attention has been given to assessing the true reasoning…
We study the classic set cover problem from the perspective of sub-linear algorithms. Given access to a collection of $m$ sets over $n$ elements in the query model, we show that sub-linear algorithms derived from existing techniques have…
We consider the following two-player game: Maxi and Mini start with the empty graph on $n$ vertices and take turns, always adding one additional edge to the graph such that the chromatic number is at most $k$, where $k \in \mathbb{N}$ is a…
We study a generalization of the classical median finding problem to batched query case: given an array of unsorted $n$ items and $k$ (not necessarily disjoint) intervals in the array, the goal is to determine the median in {\em each} of…
The GRAPH MOTIF problem asks whether a given multiset of colors appears on a connected subgraph of a vertex-colored graph. The fastest known parameterized algorithm for this problem is based on a reduction to the $k$-Multilinear Detection…
We study a geometric facility location problem under imprecision. Given $n$ unit intervals in the real line, each with one of $k$ colors, the goal is to place one point in each interval such that the resulting \emph{minimum color-spanning…
In this note we study the number of quantum queries required to identify an unknown multilinear polynomial of degree d in n variables over a finite field F_q. Any bounded-error classical algorithm for this task requires Omega(n^d) queries…
In the Maker-Breaker vertex colouring game, first publicised by Gardner in 1981, Maker and Breaker alternately colour vertices of a graph using a fixed palette, maintaining a proper colouring at all times. Maker aims to colour the whole…
We prove a tight quantum query lower bound $\Omega(n^{k/(k+1)})$ for the problem of deciding whether there exist $k$ numbers among $n$ that sum up to a prescribed number, provided that the alphabet size is sufficiently large. This is an…
A tournament is a complete directed graph. It is well known that every tournament contains at least one vertex v such that every other vertex is reachable from v by a path of length at most 2. All such vertices v are called *kings* of the…
In collective tree exploration, a team of $k$ mobile agents is tasked to go through all edges of an unknown tree as fast as possible. An edge of the tree is revealed to the team when one agent becomes adjacent to that edge. The agents start…
In this paper, we prove that, given a clique-width $k$-expression of an $n$-vertex graph, \textsc{Hamiltonian Cycle} can be solved in time $n^{\mathcal{O}(k)}$. This improves the naive algorithm that runs in time $n^{\mathcal{O}(k^2)}$ by…
An edge-colouring of a graph $G$ is said to be colour-balanced if there are equally many edges of each available colour. We are interested in finding a colour-balanced perfect matching within a colour-balanced clique $K_{2nk}$ with a…
We present a number of new results about range searching for colored (or "categorical") data: 1. For a set of $n$ colored points in three dimensions, we describe randomized data structures with $O(n\mathop{\rm polylog}n)$ space that can…
Linear arrangements of graphs are a well-known type of graph labeling and are found in many important computational problems, such as the Minimum Linear Arrangement Problem ($\texttt{minLA}$). A linear arrangement is usually defined as a…
Given a graph $G=(V,E)$ where each vertex is assigned a color from the set $C=\{c_1, c_2, .., c_\sigma\}$. In the (approximate) nearest colored node problem, we want to query, given $v \in V$ and $c \in C$, for the (approximate) distance…
We study extensions of the classic \emph{Line Cover} problem, which asks whether a set of $n$ points in the plane can be covered using $k$ lines. Line Cover is known to be NP-hard, and we focus on two natural generalizations. The first is…
A famous result in arithmetic Ramsey theory says that for many linear homogeneous equations $E$ there is a threshold value $R_k(E)$ (the Rado number of $E$) such that for any $k$-coloring of the integers in the interval $[1,n]$, with $n \ge…
Given a set of $m$ points and a set of $n$ lines in the plane, we consider the problem of computing the faces of the arrangement of the lines that contain at least one point. In this paper, we present an $O(m^{2/3}n^{2/3}+(n+m)\log n)$ time…
his paper presents two novel approaches to solving the classic board game mastermind, including a variant of simulated annealing (SA) and a technique we term maximum expected reduction in consistency (MERC). In addition, we compare search…