Related papers: Sorting sums of binary decision summands
We prove lower bounds for higher-order methods in smooth non-convex finite-sum optimization. Our contribution is threefold: We first show that a deterministic algorithm cannot profit from the finite-sum structure of the objective, and that…
Given a multiset $S$ of $n$ positive integers and a target integer $t$, the Subset Sum problem asks to determine whether there exists a subset of $S$ that sums up to $t$. The current best deterministic algorithm, by Koiliaris and Xu…
Let A be a zero-sum free subset of Z_n with |A|=k. We compute for k\le 7 the least possible size of the set of all subset-sums of A.
In the field of algorithmic analysis, one of the more well-known exercises is the subset sum problem. That is, given a set of integers, determine whether one or more integers in the set can sum to a target value. Aside from the brute-force…
We consider the problem of selecting a subset of alternatives given noisy evaluations of the relative strength of different alternatives. We wish to select a k-subset (for a given k) that provides a maximum likelihood estimate for one of…
In many settings people must give numerical scores to entities from a small discrete set. For instance, rating physical attractiveness from 1--5 on dating sites, or papers from 1--10 for conference reviewing. We study the problem of…
In bi-criteria optimization problems, the goal is typically to compute the set of Pareto-optimal solutions. Many algorithms for these types of problems rely on efficient merging or combining of partial solutions and filtering of dominated…
A $k$-sum of a set $A\subseteq \mathbb{Z}$ is an integer that may be expressed as a sum of $k$ distinct elements of $A$. How large can the ratio of the number of $(k+1)$-sums to the number of $k$-sums be? Writing $k\wedge A$ for the set of…
Algorithms are proposed for the computation of set-valued quantiles and the values of the lower cone distribution function for bivariate data sets. These new objects make data analysis possible involving an order relation for the data…
A Direct Sum Theorem holds in a model of computation, when solving some k input instances together is k times as expensive as solving one. We show that Direct Sum Theorems hold in the models of deterministic and randomized decision trees…
Clustering is a powerful machine learning technique that groups "similar" data points based on their characteristics. Many clustering algorithms work by approximating the minimization of an objective function, namely the sum of…
This paper presents a deterministic algorithmic approach of exploring the solution space of the Subset Sum Problem. The algorithm presented is input-robust and structurally adaptive. Exploration is guided and narrows into areas in the…
A natural number is a binary $k$'th power if its binary representation consists of $k$ consecutive identical blocks. We prove an analogue of Waring's theorem for sums of binary $k$'th powers. More precisely, we show that for each integer $k…
A simple method to produce a random order type is to take the order type of a random point set. We conjecture that many probability distributions on order types defined in this way are heavily concentrated and therefore sample inefficiently…
Scaling clustering algorithms to massive data sets is a challenging task. Recently, several successful approaches based on data summarization methods, such as coresets and sketches, were proposed. While these techniques provide provably…
Let $\mathcal{A}$ be a sequence of $rk$ terms which is made up of $k$ distinct integers each appearing exactly $r$ times in $\mathcal{A}$. The sum of all terms of a subsequence of $\mathcal{A}$ is called a subsequence sum of $\mathcal{A}$.…
The problem of ranking can be described as follows. We have a set of combinatorial objects $S$, such as, say, the k-subsets of n things, and we can imagine that they have been arranged in some list, say lexicographically, and we want to…
The histogram is an analysis tool in widespread use within many sciences, with high energy physics as a prime example. However, there exists an inherent bias in the choice of binning for the histogram, with different choices potentially…
In sorting situations where the final destination of each item is known, it is natural to repeatedly choose items and place them where they belong, allowing the intervening items to shift by one to make room. (In fact, a special case of…
Ranking objects is a simple and natural procedure for organizing data. It is often performed by assigning a quality score to each object according to its relevance to the problem at hand. Ranking is widely used for object selection, when…