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We initiate the study of combinatorial algorithms for Triangle Detection in $H$-free graphs. The goal is to decide if a graph that forbids a fixed pattern $H$ as a subgraph contains a triangle, using only "combinatorial" methods that…
Gaussian boson sampling (GBS) is a promising protocol for demonstrating quantum computational advantage. One of the key steps for proving classical hardness of GBS is the so-called ``hiding conjecture'', which asserts that one can ``hide''…
Strip packing is a classical packing problem, where the goal is to pack a set of rectangular objects into a strip of a given width, while minimizing the total height of the packing. The problem has multiple applications, e.g. in scheduling…
The success of machine learning solutions for reasoning about discrete structures has brought attention to its adoption within combinatorial optimization algorithms. Such approaches generally rely on supervised learning by leveraging…
This work considers the sample complexity of obtaining an $\varepsilon$-optimal policy in an average reward Markov Decision Process (AMDP), given access to a generative model (simulator). When the ground-truth MDP is weakly communicating,…
The Subset Sum problem, which asks whether a set of $n$ integers has a subset summing to a target $t$, is a fundamental NP-complete problem in cryptography and combinatorial optimization. The classical meet-in-the-middle (MIM) algorithm of…
Estimating high-dimensional precision matrices is a fundamental problem in modern statistics, with the graphical lasso and its $\ell_1$-penalty being a standard approach for recovering sparsity patterns. However, many statistical models,…
We give a short, self-contained, and easily verifiable proof that determining the outerthickness of a general graph is NP-hard. This resolves a long-standing open problem on the computational complexity of outerthickness. Moreover, our…
We introduce a new conjecture on the computational hardness of detecting random lifts of graphs: we claim that there is no polynomial-time algorithm that can distinguish between a large random $d$-regular graph and a large random lift of a…
The random assignment problem asks for the minimum-cost perfect matching in the complete $n\times n$ bipartite graph $\Knn$ with i.i.d. edge weights, say uniform on $[0,1]$. In a remarkable work by Aldous (2001), the optimal cost was shown…
We consider a class of multivariate recurrences frequently arising in the worst case analysis of Davis-Putnam-style exponential time backtracking algorithms for NP-hard problems. We describe a technique for proving asymptotic upper bounds…
Recent years have seen significant progress in the study of dynamic graph algorithms, and most notably, the introduction of strong lower bound techniques for them (e.g., Henzinger, Krinninger, Nanongkai and Saranurak, STOC 2015; Larsen and…
Binary embedding is the problem of mapping points from a high-dimensional space to a Hamming cube in lower dimension while preserving pairwise distances. An efficient way to accomplish this is to make use of fast embedding techniques…
For many fundamental problems in computational topology, such as unknot recognition and $3$-sphere recognition, the existence of a polynomial-time solution remains unknown. A major algorithmic tool behind some of the best known algorithms…
We propose a new identification system based on algorithmic problems related to computing isomorphisms between central simple algebras. We design a statistical zero knowledge protocol which relies on the hardness of computing isomorphisms…
We study the minimum number of constraints needed to formulate random instances of the maximum stable set problem via linear programs (LPs), in two distinct models. In the uniform model, the constraints of the LP are not allowed to depend…
This paper explores a theory of generalization for learning problems on product distributions, complementing the existing learning theories in the sense that it does not rely on any complexity measures of the hypothesis classes. The main…
While modern large-scale datasets often consist of heterogeneous subpopulations -- for example, multiple demographic groups or multiple text corpora -- the standard practice of minimizing average loss fails to guarantee uniformly low losses…
We introduce a new approach for establishing fixed-parameter tractability of problems parameterized above tight lower bounds. To illustrate the approach we consider three problems of this type of unknown complexity that were introduced by…
The basic problem in the PAC model of computational learning theory is to determine which hypothesis classes are efficiently learnable. There is presently a dearth of results showing hardness of learning problems. Moreover, the existing…