We study two fundamental problems in communication, Document Exchange (DE) and Error Correcting Code (ECC). In the first problem, two parties hold two strings, and one party tries to learn the other party's string through communication. In the second problem, one party tries to send a message to another party through a noisy channel, by adding some redundant information to protect the message. Two

A well-known first-order method for sampling from log-concave probability distributions is the Unadjusted Langevin Algorithm (ULA). This work proposes a new annealing step-size schedule for ULA, which allows to prove new convergence guarantees for sampling from a smooth log-concave distribution, which are not covered by existing state-of-the-art convergence guarantees. To establish this result, we

We present a novel feasibility criteria for the intersection of convex sets given by inequalities. This criteria allows us to easily assert the feasibility by analyzing the unconstrained minimum of a certain convex function, that we form with the given sets. Next an algorithm is presented which extends the idea to a particular non-convex case: assert the inclusion of the intersection of a set of balls

The standard circuit model for quantum computation presumes the ability to directly perform gates between arbitrary pairs of qubits, which is unlikely to be practical for large-scale experiments. Power-law interactions with strength decaying as $1/r^\alpha$ in the distance $r$ provide an experimentally realizable resource for information processing, whilst still retaining long-range connectivity. We

Let $\tilde{f}(X)\in\mathbb{Z}[X]$ be a degree-$n$ polynomial such that $f(X):=\tilde{f}(X)\bmod p$ factorizes into $n$ distinct linear factors over $\mathbb{F}_p$. We study the problem of deterministically factoring $f(X)$ over $\mathbb{F}_p$ given $\tilde{f}(X)$. Under the generalized Riemann hypothesis (GRH), we give an improved deterministic algorithm that computes the complete factorization of

Bayesian adaptive inference is widely used in psychophysics to estimate psychometric parameters. Most applications used myopic one-step ahead strategy which only optimizes the immediate utility. The widely held expectation is that global optimization strategies that explicitly optimize over some horizon can largely improve the performance of the myopic strategy. With limited studies that compared myopic

In this paper, we consider mixtures of multinomial logistic models (MNL), which are known to $\epsilon$-approximate any random utility model. Despite its long history and broad use, rigorous results are only available for learning a uniform mixture of two MNLs. Continuing this line of research, we study the problem of learning an arbitrary mixture of two MNLs. We show that the identifiability of the

Best match graphs (BMGs) are vertex-colored directed graphs that were introduced to model the relationships of genes (vertices) from different species (colors) given an underlying evolutionary tree that is assumed to be unknown. In real-life applications, BMGs are estimated from sequence similarity data. Measurement noise and approximation errors therefore result in empirically determined graphs that

We study the problem of computing the parity of the number of homomorphisms from an input graph $G$ to a fixed graph $H$. Faben and Jerrum [ToC'15] introduced an explicit criterion on the graph $H$ and conjectured that, if satisfied, the problem is solvable in polynomial time and, otherwise, the problem is complete for the complexity class $\oplus\mathrm{P}$ of parity problems. We verify their conjecture

The Ideal Membership Problem (IMP) tests if an input polynomial $f\in \mathbb{F}[x_1,\dots,x_n]$ with coefficients from a field $\mathbb{F}$ belongs to a given ideal $I \subseteq \mathbb{F}[x_1,\dots,x_n]$. It is a well-known fundamental problem with many important applications, though notoriously intractable in the general case. In this paper we consider the IMP for polynomial ideals encoding combinatorial

Multi-access edge computing (MEC) has already shown the potential in enabling mobile devices to bear the computation-intensive applications by offloading some tasks to a nearby access point (AP) integrated with a MEC server (MES). However, due to the varying network conditions and limited computation resources of the MES, the offloading decisions taken by a mobile device and the computational resources

In this article, we study a generalized version of the maximum independent set and minimum dominating set problems, namely, the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem on unit disk graphs for a positive integer $d>0$. We first show that the maximum $d$-distance independent set problem and the minimum $d$-distance dominating set problem belongs

Neural networks can be used in video coding to improve chroma intra-prediction. In particular, usage of fully-connected networks has enabled better cross-component prediction with respect to traditional linear models. Nonetheless, state-of-the-art architectures tend to disregard the location of individual reference samples in the prediction process. This paper proposes a new neural network architecture

The classical communication complexity of testing closeness of discrete distributions has recently been studied by Andoni, Malkin and Nosatzki (ICALP'19). In this problem, two players each receive $t$ samples from one distribution over $[n]$, and the goal is to decide whether their two distributions are equal, or are $\epsilon$-far apart in the $l_1$-distance. In the present paper we show that the

We study the computational complexity of approximating the partition function of the ferromagnetic Ising model in the Lee-Yang circle of zeros given by $|\lambda|=1$, where $\lambda$ is the external field of the model. Complex-valued parameters for the Ising model are relevant for quantum circuit computations and phase transitions in statistical physics, but have also been key in the recent deterministic

Consider an information diffusion process on a graph $G$ that starts with $k>0$ burnt vertices, and at each subsequent step, burns the neighbors of the currently burnt vertices, as well as $k$ other unburnt vertices. The \emph{$k$-burning number} of $G$ is the minimum number of steps $b_k(G)$ such that all the vertices can be burned within $b_k(G)$ steps. Note that the last step may have smaller than

We study training of Convolutional Neural Networks (CNNs) with ReLU activations and introduce exact convex optimization formulations with a polynomial complexity with respect to the number of data samples, the number of neurons and data dimension. Particularly, we develop a convex analytic framework utilizing semi-infinite duality to obtain equivalent convex optimization problems for several CNN architectures

We consider the general problem of learning about a matrix through vector-matrix-vector queries. These queries provide the value of $\boldsymbol{u}^{\mathrm{T}}\boldsymbol{M}\boldsymbol{v}$ over a fixed field $\mathbb{F}$ for a specified pair of vectors $\boldsymbol{u},\boldsymbol{v} \in \mathbb{F}^n$. To motivate these queries, we observe that they generalize many previously studied models, such as

The disjointness problem - where Alice and Bob are given two subsets of $\{1, \dots, n\}$ and they have to check if their sets intersect - is a central problem in the world of communication complexity. While both deterministic and randomized communication complexities for this problem are known to be $\Theta(n)$, it is also known that if the sets are assumed to be drawn from some restricted set systems

The problem of finding longest common subsequence (LCS) is one of the fundamental problems in computer science, which finds application in fields such as computational biology, text processing, information retrieval, data compression etc. It is well known that (decision version of) the problem of finding the length of a LCS of an arbitrary number of input sequences (which we refer to as Multi-LCS problem)

We reduce the problem of proving a "Boolean Unique Games Conjecture" (with gap 1-delta vs. 1-C*delta, for any C> 1, and sufficiently small delta>0) to the problem of proving a PCP Theorem for a certain non-unique game. In a previous work, Khot and Moshkovitz suggested an inefficient candidate reduction (i.e., without a proof of soundness). The current work is the first to provide an efficient reduction

Aaronson and Ambainis (2009) and Chailloux (2018) showed that fully symmetric (partial) functions do not admit exponential quantum query speedups. This raises a natural question: how symmetric must a function be before it cannot exhibit a large quantum speedup? In this work, we prove that hypergraph symmetries in the adjacency matrix model allow at most a polynomial separation between randomized and

We study the capabilities of probabilistic finite-state machines that act as verifiers for certificates of language membership for input strings, in the regime where the verifiers are restricted to toss some fixed nonzero number of coins regardless of the input size. Say and Yakary{\i}lmaz showed that the class of languages that could be verified by these machines within an error bound strictly less

Although quantum computing is expected to outperform universal classical computing, an unconditional proof of this assertion seems to be hard because an unconditional separation between ${\sf BQP}$ and ${\sf BPP}$ implies ${\sf P}\neq{\sf PSPACE}$. Because of this, the quantum-computational-supremacy approach has been actively studied; it shows that if the output probability distributions from a family

Recent advances in the field of adiabatic quantum computing and the closely related field of quantum annealers has centered around using more advanced and novel Hamiltonian representations to solve optimization problems. One of these advances has centered around the development of driver Hamiltonians that commute with the constraints of an optimization problem - allowing for another avenue to satisfying

A homomorphism from a graph $G$ to a graph $H$ is an edge-preserving mapping from $V(G)$ to $V(H)$. Let $H$ be a fixed graph with possible loops. In the list homomorphism problem, denoted by LHom($H$), we are given a graph $G$, whose every vertex $v$ is assigned with a list $L(v)$ of vertices of $H$. We ask whether there exists a homomorphism $h$ from $G$ to $H$, which respects lists $L$, i.e., for

Suppose we have randomized decision trees for an outer function $f$ and an inner function $g$. The natural approach for obtaining a randomized decision tree for the composed function $(f\circ g^n)(x^1,\ldots,x^n)=f(g(x^1),\ldots,g(x^n))$ involves amplifying the success probability of the decision tree for $g$, so that a union bound can be used to bound the error probability over all the coordinates

Forgetting variables from a propositional formula may increase its size. Introducing new variables is a way to shorten it. Both operations can be expressed in terms of common equivalence, a weakened version of equivalence. In turn, common equivalence can be expressed in terms of forgetting. An algorithm for forgetting and checking common equivalence in polynomial space is given for the Horn case; it

Detecting and eliminating logic hazards in Boolean circuits is a fundamental problem in logic circuit design. We show that there is no $O(3^{(1-\epsilon)n} \text{poly}(s))$ time algorithm, for any $\epsilon > 0$, that detects logic hazards in Boolean circuits of size $s$ on $n$ variables under the assumption that the strong exponential time hypothesis is true. This lower bound holds even when the input

We study the Independent Set (IS) problem in $H$-free graphs, i.e., graphs excluding some fixed graph $H$ as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halld\'orsson [SODA 1995] showed that for every $\delta>0$ IS has a polynomial-time $(\frac{d-1}{2}+\delta)$-approximation in $K_{1,d}$-free graphs. We extend this result by

Political campaigns are full of political ads posted by candidates on social media. Political advertisements constitute a basic form of campaigning, subjected to various social requirements. We present the first publicly open dataset for detecting specific text chunks and categories of political advertising in the Polish language. It contains 1,705 human-annotated tweets tagged with nine categories

We give an algorithm for solving unique games (UG) instances whose constraints correspond to edges of graphs with a sum-of-squares (SoS) small-set-expansion certificate. As corollaries, we obtain the first polynomial-time algorithms for solving UG on the noisy hypercube and the short code graphs. The prior best algorithm for such instances was the eigenvalue enumeration algorithm of Arora, Barak, and

The Artificial Hormone System (AHS) is a self-organizing middleware to allocate tasks in a distributed system. We extended it by so-called negator hormones to enable conditional task structures. However, this extension increases the computational complexity of seemingly simple decision problems in the system: In [1] and [2], we defined the problems Negator-Path and Negator-Sat and proved their NP-completeness

Igusa's local zeta function $Z_{f,p}(s)$ is the generating function that counts the number of integral roots, $N_{k}(f)$, of $f(\mathbf x) \bmod p^k$, for all $k$. It is a famous result, in analytic number theory, that $Z_{f,p}$ is a rational function in $\mathbb{Q}(p^s)$. We give an elementary proof of this fact for a univariate polynomial $f$. Our proof is constructive as it gives a closed-form expression

In this paper we introduce a new problem in the field of production planning which we call the Production Leveling Problem. The task is to assign orders to production periods such that the load in each period and on each production resource is balanced, capacity limits are not exceeded and the orders' priorities are taken into account. Production Leveling is an important intermediate step between long-term

We investigate the relationship between algorithmic fractal dimensions and the classical local fractal dimensions of outer measures in Euclidean spaces. We introduce global and local optimality conditions for lower semicomputable outer measures. We prove that globally optimal outer measures exist. Our main theorem states that the classical local fractal dimensions of any locally optimal outer measure

Equivalence testing for a polynomial family {g_m} over a field F is the following problem: Given black-box access to an n-variate polynomial f(x), where n is the number of variables in g_m, check if there exists an A in GL(n,F) such that f(x) = g_m(Ax). If yes, then output such an A. The complexity of equivalence testing has been studied for a number of important polynomial families, including the

In neuroscience, an important aspect of understanding the function of a neural circuit is to determine which, if any, of the neurons in the circuit are vital for the biological behavior governed by the neural circuit. A similar problem is to determine whether a given small set of neurons may be enough for the behavior to be displayed, even if all other neurons in the circuit are deactivated. Such a

In this work we resolve conjectures of Beecken, Mitmann and Saxena [BMS13] and Gupta [Gup14], by proving an analog of a theorem of Edelstein and Kelly for quadratic polynomials. As immediate corollary we obtain the first deterministic polynomial time black-box algorithm for testing zeroness of $\Sigma^{[3]}\Pi\Sigma\Pi^{[2]}$ circuits.

We study the problem of testing forbidden patterns. The patterns that are of significant interest include monotone pattern and $(1,3,2)$-pattern. For the problem of testing monotone patterns, \cite{newman2019testing} propose a non-adaptive algorithm with query complexity $(\log n)^{O(k^2)}$. \cite{ben2019finding} then improve the query complexity of non-adaptive algorithm to $\Omega((\log n)^{\lfloor\log

We study the notion of boundedness in the context of positive existential rules, that is, whether there exists an upper bound to the depth of the chase procedure, that is independent from the initial instance. By focussing our attention on the oblivious and the semi-oblivious chase variants, we give a characterization of boundedness in terms of FO-rewritability and chase termination. We show that it

We show that the problem of deciding whether in a multi-player perfect information recursive game (i.e. a stochastic game with terminal rewards) there exists a stationary Nash equilibrium ensuring each player a certain payoff is Existential Theory of the Reals complete. Our result holds for acyclic games, where a Nash equilibrium may be computed efficiently by backward induction, and even for deterministic

We prove that the finite-difference based derivative-free descent (FD-DFD) methods have a capability to find the global minima for a class of multiple minima problems. Our main result shows that, for a class of multiple minima objectives that is extended from strongly convex functions with Lipschitz-continuous gradients, the iterates of FD-DFD converge to the global minimizer $x_*$ with the linear

Radical membership testing, and the special case of Hilbert's Nullstellensatz (HN), is a fundamental computational algebra problem. It is NP-hard; and has a famous PSPACE algorithm due to effective Nullstellensatz bounds. We identify a useful case of these problems where practical algorithms, and improved bounds, could be given, when the transcendence degree $r$ of the input polynomials is smaller

The Skolem Problem asks, given a linear recurrence sequence $(u_n)$, whether there exists $n\in\mathbb{N}$ such that $u_n=0$. In this paper we consider the following specialisation of the problem: given in addition $c\in\mathbb{N}$, determine whether there exists $n\in\mathbb{N}$ of the form $n=lp^k$, with $k,l\leq c$ and $p$ any prime number, such that $u_n=0$.

We study the applicability of quantum algorithms in computational game theory and generalize some results related to Subtraction games, which are sometimes referred to as one-heap Nim games. In quantum game theory, a subset of Subtraction games became the first explicitly defined class of zero-sum combinatorial games with provable separation between quantum and classical complexity of solving them

Deep learning has shown great potential in image and video compression tasks. However, it brings bit savings at the cost of significant increases in coding complexity, which limits its potential for implementation within practical applications. In this paper, a novel neural network-based tool is presented which improves the interpolation of reference samples needed for fractional precision motion compensation

In function inversion, we are given a function $f: [N] \mapsto [N]$, and want to prepare some advice of size $S$, such that we can efficiently invert any image in time $T$. This is a well studied problem with profound connections to cryptography, data structures, communication complexity, and circuit lower bounds. Investigation of this problem in the quantum setting was initiated by Nayebi, Aaronson

Spatial modulation (SM) is a promising multiple-input multiple-output system used to increase spectral efficiency. The maximum likelihood (ML) decoder jointly detects the transmitted SM symbol, which is of high complexity. In this paper, a novel reliable sphere decoder (RSD) algorithm based on tree-search is proposed for the SM system. The basic idea of the proposed RSD algorithm is to reduce the size

We give a quantum logspace algorithm for powering contraction matrices, that is, matrices with spectral norm at most~1. The algorithm gets as an input an arbitrary $n\times n$ contraction matrix $A$, and a parameter $T \leq \mathrm{poly}(n)$ and outputs the entries of $A^T$, up to (arbitrary) polynomially small additive error. The algorithm applies only unitary operators, without intermediate measurements

The Quantum Approximate Optimization Algorithm (QAOA) is a hybrid quantum-classical algorithm to solve binary-variable optimization problems. Due to its expected robustness to systematic errors and the short circuit depth, it is one of the promising candidates likely to run on near-term quantum devices. We project the performance of QAOA applied to the Max-Cut problem and compare it with some of the

We present a quantum algorithm to compute the discrete Legendre-Fenchel transform. Given access to a convex function evaluated at $N$ points, the algorithm outputs a quantum-mechanical representation of its corresponding discrete Legendre-Fenchel transform evaluated at $K$ points in the transformed space. For a fixed regular discretizaton of the dual space the expected running time scales as $O(\sqrt{\kappa}\

A Boolean function $f:\{0,1\}^n\to \{0,1\}$ is $k$-linear if it returns the sum (over the binary field $F_2$) of $k$ coordinates of the input. In this paper, we study property testing of the classes $k$-Linear, the class of all $k$-linear functions, and $k$-Linear$^*$, the class $\cup_{j=0}^kj$-Linear. We give a non-adaptive distribution-free two-sided $\epsilon$-tester for $k$-Linear that makes $$O\left(k\log

We prove PSPACE-completeness of all but one problem in a large space of pulling-block problems where the goal is for the agent to reach a target destination. The problems are parameterized by whether pulling is optional, the number of blocks which can be pulled simultaneously, whether there are fixed blocks or thin walls, and whether there is gravity. We show NP-hardness for the remaining problem,

Parameterization and approximation are two popular ways of coping with NP-hard problems. More recently, the two have also been combined to derive many interesting results. We survey developments in the area both from the algorithmic and hardness perspectives, with emphasis on new techniques and potential future research directions.

In this paper, we consider the Delay Constrained Unsplittable Shortest Path Routing problem which arises in the field of traffic engineering for IP networks. This problem consists, given a directed graph and a set of commodities, to compute a set of routing paths and the associated administrative weights such that each commodity is routed along the unique shortest path between its origin and its destination

The Continuous Periodic Joint Replenishment Problem (CPJRP) has been one of the core and most studied problems in supply chain management for the last half a century. Nonetheless, despite the vast effort put into studying the problem, its complexity has eluded researchers for years. Although the CPJRP has one of the tighter constant approximation ratio of 1.02, a polynomial optimal solution to it was

We consider the task of proving integer infeasibility of a bounded convex $K$ in $\mathbb{R}^n$ using a general branching proof system. In a general branching proof, one constructs a branching tree by adding an integer disjunction $\mathbf{a} \mathbf{x} \leq b$ or $\mathbf{a} \mathbf{x} \geq b+1$, $\mathbf{a} \in \mathbb{Z}^n$, $b \in \mathbb{Z}$, at each node, such that the leaves of the tree correspond

Although it has been claimed in two different papers that the maximum cardinality cut problem is polynomial-time solvable for proper interval graphs, both of them turned out to be erroneous. In this paper, we give FPT algorithms for the maximum cardinality cut problem in classes of graphs containing proper interval graphs and mixed unit interval graphs when parameterized by some new parameters that

We study a new version of the Euclidean TSP called VectorTSP (VTSP for short) where a mobile entity is allowed to move according to a set of physical constraints inspired from the pen-and-pencil game Racetrack (also known as Vector Racer ). In contrast to other versions of TSP accounting for physical constraints, such as Dubins TSP, the spirit of this model is that (1) no speed limitations apply, and