Related papers: Sum-Of-Squares To Approximate Knapsack
In this work, we consider the Submodular Maximization under Knapsack (SMK) constraint problem over the ground set of size $n$. The problem recently attracted a lot of attention due to its applications in various domains of combination…
Sets represent a fundamental abstraction across many types of data. To handle the unordered nature of set-structured data, models such as DeepSets and PointNet rely on fixed, non-learnable pooling operations (e.g., sum or max) -- a design…
Knapsack problems (KPs) are common in industry, but solving KPs is known to be NP-hard and has been tractable only at a relatively small scale. This paper examines KPs in a slightly generalized form and shows that they can be solved nearly…
In a common formulation of semi-infinite programs, the infinite constraint set is a requirement that a function parametrized by the decision variables is nonnegative over an interval. If this function is sufficiently closely approximable by…
Let $P:\{0,1\}^k \to \{0,1\}$ be a nontrivial $k$-ary predicate. Consider a random instance of the constraint satisfaction problem $\mathrm{CSP}(P)$ on $n$ variables with $\Delta n$ constraints, each being $P$ applied to $k$ randomly chosen…
We investigate a more generalized form of submodular maximization, referred to as $k$-submodular maximization, with applications across social networks and machine learning domains. In this work, we propose the multilinear extension of…
Equiangular tight frames (ETFs) may be used to construct examples of feasible points for semidefinite programs arising in sum-of-squares (SOS) optimization. We show how generalizing the calculations in a recent work of the authors' that…
We propose a general framework using spike-and-slab prior distributions to aid with the development of high-dimensional Bayesian inference. Our framework allows inference with a general quasi-likelihood function. We show that highly…
Learning the embedding space, where semantically similar objects are located close together and dissimilar objects far apart, is a cornerstone of many computer vision applications. Existing approaches usually learn a single metric in the…
We introduce a new framework for unifying and systematizing the performance analysis of first-order black-box optimization algorithms for unconstrained convex minimization. The low-cost iteration complexity enjoyed by first-order algorithms…
We study the Min-Weighted Sum Bin Packing problem, a variant of the classical Bin Packing problem in which items have a weight, and each item induces a cost equal to its weight multiplied by the index of the bin in which it is packed. This…
We study sum of squares (SOS) relaxations to optimize polynomial functions over a set $V\cap R^n$, where $V$ is a complex algebraic variety. We propose a new methodology that, rather than relying on some algebraic description, represents…
The Subset Sum Problem is a fundamental NP-complete problem in cryptography and combinatorial optimization, with many real-world applications. The Random Subset Sum Problem (RSSP) is a more applicable version of subset sum, where numbers…
The set of 2-dimensional packing problems builds an important class of optimization problems and Strip Packing together with 2-dimensional Bin Packing and 2-dimensional Knapsack is one of the most famous of these problems. Given a set of…
The Sum-of-Squares (SoS) hierarchy is a semi-definite programming meta-algorithm that captures state-of-the-art polynomial time guarantees for many optimization problems such as Max-$k$-CSPs and Tensor PCA. On the flip side, a SoS lower…
Clustering is a fundamental technique in data analysis and machine learning, used to group similar data points together. Among various clustering methods, the Minimum Sum-of-Squares Clustering (MSSC) is one of the most widely used. MSSC…
We initiate the algorithmic study of the Quantum Max-$d$-Cut problem, a quantum generalization of the well-known Max-$d$-Cut problem. The Quantum Max-$d$-Cut problem involves finding a quantum state that maximizes the expected energy…
The fast computation of large kernel sums is a challenging task, which arises as a subproblem in any kernel method. We approach the problem by slicing, which relies on random projections to one-dimensional subspaces and fast Fourier…
The $k$-median and $k$-means clustering objectives are classic objectives for modeling clustering in a metric space. Given a set of points in a metric space, the goal of the $k$-median (resp. $k$-means) problem is to find $k$ representative…
A natural and important generalization of submodularity -- $k$-submodularity -- applies to set functions with $k$ arguments and appears in a broad range of applications, such as infrastructure design, machine learning, and healthcare. In…