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We show that the Word Problem in finitely generated subgroups of $\textsf{GL}_d(\mathbb{Z})$ can be solved in linear average-case complexity. This is done under the bit-complexity model, which accounts for the fact that large integers are…
Scheduling problems are fundamental in combinatorial optimization. Much work has been done on approximation algorithms for NP-hard cases, but relatively little is known about exact solutions when some part of the input is a fixed parameter.…
Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the one-relator Baumslag…
Binary quadratic programming problems have attracted much attention in the last few decades due to their potential applications. This type of problems are NP-hard in general, and still considered a challenge in the design of efficient…
We study the parameterized complexity of the following fundamental geometric problems with respect to the dimension $d$: i) Given $n$ points in $\Rd$, compute their minimum enclosing cylinder. ii) Given two $n$-point sets in $\Rd$, decide…
In the article ''On the (Non) NP-Hardness of Computing Circuit Complexity'', Murray and Williams imply the PARTITION decision problem is not known to be NP-hard via $2^{n^{o(1)}}$-size AC0 reductions. In this note, we show PARTITION is…
Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase…
We prove that approximating the size of stopping and trapping sets in Tanner graphs of linear block codes, and more restrictively, the class of low-density parity-check (LDPC) codes, is NP-hard. The ramifications of our findings are that…
Under a standard assumption in complexity theory (NP not in P/poly), we demonstrate a gap between the minimax prediction risk for sparse linear regression that can be achieved by polynomial-time algorithms, and that achieved by optimal…
The decision problems on matrices were intensively studied for many decades as matrix products play an essential role in the representation of various computational processes. However, many computational problems for matrix semigroups are…
Given a batch of human computation tasks, a commonly ignored aspect is how the price (i.e., the reward paid to human workers) of these tasks must be set or varied in order to meet latency or cost constraints. Often, the price is set…
This paper introduces a concept of a derivative of the optimal value function in linear programming (LP). Basically, it is the the worst case optimal value of an interval LP problem when the nominal data the data are inflated to intervals…
A common practice in large language model (LLM) usage for complex analytical tasks such as code generation, is to sample a solution for the entire task within the model's context window. Previous works have shown that subtask decomposition…
We consider set covering problems where the underlying set system satisfies a particular replacement property w.r.t. a given partial order on the elements: Whenever a set is in the set system then a set stemming from it via the replacement…
This paper studies the lattice agreement problem and proposes a stronger form, $\varepsilon$-bounded lattice agreement, that enforces an additional tightness constraint on the outputs. To formalize the concept, we define a quasi-metric on…
In this paper we investigate computational properties of the Diophantine problem for spherical equations in some classes of finite groups. We classify the complexity of different variations of the problem, e.g., when $G$ is fixed and when…
Seeking tighter relaxations of combinatorial optimization problems, semidefinite programming is a generalization of linear programming that offers better bounds and is still polynomially solvable. Yet, in practice, a semidefinite program is…
We leverage recent advances in high-dimensional statistics to derive new L2 estimation upper bounds for Lasso and Group Lasso in high-dimensions. For Lasso, our bounds scale as $(k^*/n) \log(p/k^*)$---$n\times p$ is the size of the design…
We investigate unitary and state $t$-designs from a computational complexity perspective. First, we address the problems of computing frame potentials that characterize (approximate) $t$-designs. We present a quantum algorithm for computing…
We consider a model of computation motivated by possible limitations on quantum computers. We have a linear array of n wires, and we may perform operations only on pairs of adjacent wires. Our goal is to build a circuits that perform…