We revisit a classical crossword filling puzzle which already appeared in Garey\&Jonhson's book. We are given a grid with $n$ vertical and horizontal slots and a dictionary with $m$ words and are asked to place words from the dictionary in the slots so that shared cells are consistent. We attempt to pinpoint the source of intractability of this problem by taking into account the structure of the grid graph, which contains a vertex for each slot and an edge if two slots intersect. Our main approach is to consider the case where this graph has a tree-like structure. Unfortunately, if we impose the common rule that words cannot be reused, we show that the problem remains NP-hard even under very severe structural restrictions. The problem becomes slightly more tractable if word reuse is allowed, as we obtain an $m^{tw}$ algorithm in this case, where $tw$ is the treewidth of the grid graph. However, even in this case, we show that our algorithm cannot be improved. More strongly, we show that under the ETH the problem cannot be solved in time $m^{o(k)}$, where $k$ is the number of horizontal slots of the instance. Motivated by these mostly negative results, we consider the much more restricted case where the problem is parameterized by the number of slots $n$. Here, we show that the problem becomes FPT, but the parameter dependence is exponential in $n^2$. We show that this dependence is also justified: the existence of an algorithm with running time $2^{o(n^2)}$ would contradict the randomized ETH. Finally, we consider an optimization version of the problem, where we seek to place as many words on the grid as possible. Here it is easy to obtain a $\frac{1}{2}$-approximation, even on weighted instances. We show that this algorithm is also likely to be optimal, as obtaining a better approximation ratio in polynomial time would contradict the Unique Games Conjecture.
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In this paper we have to demonstrate that if we claim to have an algorithm that solves CSAT in polynomial time with a DTM (Deterministic Turing Machine), then we have to admit that: there is a counterexample that invalidates the correctness of the algorithm. This is because if we suppose that it can prove that an elenkhos formula (a formula that lists the negated codes of all models) is a contradiction, and if we change exactly a specific boolean variable of that formula, then we have proven that: in this case the algorithm will always fail.
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Faster-than-Nyquist (FTN) signaling is a non-orthogonal transmission technique, which has the potential to provide significant spectral efficiency improvement. This paper studies the capacity of FTN signaling for point-to-point single-input single-output (SISO) and multiple-input multiple-output (MIMO) channels. Although the capacity of SISO FTN was studied previously, due to some incomplete prerequisites, the exact capacity expression was not found. In this paper we resolve these issues for SISO FTN and generalize the result to MIMO FTN. We show that joint waterfilling in time and space is capacity achieving for MIMO FTN.
Given a graph whose nodes may be coloured red, the parity of the number of red nodes can easily be maintained with first-order update rules in the dynamic complexity framework DynFO of Patnaik and Immerman. Can this be generalised to other or even all queries that are definable in first-order logic extended by parity quantifiers? We consider the query that asks whether the number of nodes that have an edge to a red node is odd. Already this simple query of quantifier structure parity-exists is a major roadblock for dynamically capturing extensions of first-order logic. We show that this query cannot be maintained with quantifier-free first-order update rules, and that variants induce a hierarchy for such update rules with respect to the arity of the maintained auxiliary relations. Towards maintaining the query with full first-order update rules, it is shown that degree-restricted variants can be maintained.
In this work we study the orbit recovery problem over $SO(3)$, where the goal is to recover a band-limited function on the sphere from noisy measurements of randomly rotated copies of it. This is a natural abstraction for the problem of recovering the three-dimensional structure of a molecule through cryo-electron tomography. Symmetries play an important role: Recovering the function up to rotation is equivalent to solving a system of polynomial equations that comes from the invariant ring associated with the group action. Prior work investigated this system through computational algebra tools up to a certain size. However many statistical and algorithmic questions remain: How many moments suffice for recovery, or equivalently at what degree do the invariant polynomials generate the full invariant ring? And is it possible to algorithmically solve this system of polynomial equations? We revisit these problems from the perspective of smoothed analysis whereby we perturb the coefficients of the function in the basis of spherical harmonics. Our main result is a quasi-polynomial time algorithm for orbit recovery over $SO(3)$ in this model. We analyze a popular heuristic called frequency marching that exploits the layered structure of the system of polynomial equations by setting up a system of {\em linear} equations to solve for the higher-order frequencies assuming the lower-order ones have already been found. The main questions are: Do these systems have a unique solution? And how fast can the errors compound? Our main technical contribution is in bounding the condition number of these algebraically-structured linear systems. Thus smoothed analysis provides a compelling model in which we can expand the types of group actions we can handle in orbit recovery, beyond the finite and/or abelian case.
The finite models of a universal sentence $\Phi$ in a finite relational signature are the age of a structure if and only if $\Phi$ has the joint embedding property. We prove that the computational problem whether a given universal sentence $\Phi$ has the joint embedding property is undecidable, even if $\Phi$ is additionally Horn and the signature of $\Phi$ only contains relation symbols of arity at most two.
We consider a dynamic model of traffic that has received a lot of attention in the past few years. Users control infinitesimal flow particles aiming to travel from a source to destination as quickly as possible. Flow patterns vary over time, and congestion effects are modeled via queues, which form whenever the inflow into a link exceeds its capacity. Despite lots of interest, some very basic questions remain open in this model. We resolve a number of them: - We show uniqueness of journey times in equilibria. - We show continuity of equilibria: small perturbations to the instance or to the traffic situation at some moment cannot lead to wildly different equilibrium evolutions. - We demonstrate that, assuming constant inflow into the network at the source, equilibria always settle down into a "steady state" in which the behavior extends forever in a linear fashion. One of our main conceptual contributions is to show that the answer to the first two questions, on uniqueness and continuity, are intimately connected to the third. Our result also shows very clearly that resolving uniqueness and continuity, despite initial appearances, cannot be resolved by analytic techniques, but are related to very combinatorial aspects of the model. To resolve the third question, we substantially extend the approach of Cominetti et al., who show a steady-state result in the regime where the input flow rate is smaller than the network capacity.
This paper studies the expressive power of artificial neural networks (NNs) with rectified linear units. To study them as a model of real-valued computation, we introduce the concept of Max-Affine Arithmetic Programs and show equivalence between them and NNs concerning natural complexity measures. We then use this result to show that two fundamental combinatorial optimization problems can be solved with polynomial-size NNs, which is equivalent to the existence of very special strongly polynomial time algorithms. First, we show that for any undirected graph with $n$ nodes, there is an NN of size $\mathcal{O}(n^3)$ that takes the edge weights as input and computes the value of a minimum spanning tree of the graph. Second, we show that for any directed graph with $n$ nodes and $m$ arcs, there is an NN of size $\mathcal{O}(m^2n^2)$ that takes the arc capacities as input and computes a maximum flow. These results imply in particular that the solutions of the corresponding parametric optimization problems where all edge weights or arc capacities are free parameters can be encoded in polynomial space and evaluated in polynomial time, and that such an encoding is provided by an NN.
Hierarchical Clustering has been studied and used extensively as a method for analysis of data. More recently, Dasgupta [2016] defined a precise objective function. Given a set of $n$ data points with a weight function $w_{i,j}$ for each two items $i$ and $j$ denoting their similarity/dis-similarity, the goal is to build a recursive (tree like) partitioning of the data points (items) into successively smaller clusters. He defined a cost function for a tree $T$ to be $Cost(T) = \sum_{i,j \in [n]} \big(w_{i,j} \times |T_{i,j}| \big)$ where $T_{i,j}$ is the subtree rooted at the least common ancestor of $i$ and $j$ and presented the first approximation algorithm for such clustering. Then Moseley and Wang [2017] considered the dual of Dasgupta's objective function for similarity-based weights and showed that both random partitioning and average linkage have approximation ratio $1/3$ which has been improved in a series of works to $0.585$ [Alon et al. 2020]. Later Cohen-Addad et al. [2019] considered the same objective function as Dasgupta's but for dissimilarity-based metrics, called $Rev(T)$. It is shown that both random partitioning and average linkage have ratio $2/3$ which has been only slightly improved to $0.667078$ [Charikar et al. SODA2020]. Our first main result is to consider $Rev(T)$ and present a more delicate algorithm and careful analysis that achieves approximation $0.71604$. We also introduce a new objective function for dissimilarity-based clustering. For any tree $T$, let $H_{i,j}$ be the number of $i$ and $j$'s common ancestors. Intuitively, items that are similar are expected to remain within the same cluster as deep as possible. So, for dissimilarity-based metrics, we suggest the cost of each tree $T$, which we want to minimize, to be $Cost_H(T) = \sum_{i,j \in [n]} \big(w_{i,j} \times H_{i,j} \big)$. We present a $1.3977$-approximation for this objective.
Retrieving a signal from the Fourier transform of its third-order statistics or bispectrum arises in a wide range of signal processing problems. Conventional methods do not provide a unique inversion of bispectrum. In this paper, we present a an approach that uniquely recovers signals with finite spectral support (band-limited signals) from at least $3B$ measurements of its bispectrum function (BF), where $B$ is the signal's bandwidth. Our approach also extends to time-limited signals. We propose a two-step trust region algorithm that minimizes a non-convex objective function. First, we approximate the signal by a spectral algorithm. Then, we refine the attained initialization based upon a sequence of gradient iterations. Numerical experiments suggest that our proposed algorithm is able to estimate band/time-limited signals from its BF for both complete and undersampled observations.
While existing work in robust deep learning has focused on small pixel-level $\ell_p$ norm-based perturbations, this may not account for perturbations encountered in several real world settings. In many such cases although test data might not be available, broad specifications about the types of perturbations (such as an unknown degree of rotation) may be known. We consider a setup where robustness is expected over an unseen test domain that is not i.i.d. but deviates from the training domain. While this deviation may not be exactly known, its broad characterization is specified a priori, in terms of attributes. We propose an adversarial training approach which learns to generate new samples so as to maximize exposure of the classifier to the attributes-space, without having access to the data from the test domain. Our adversarial training solves a min-max optimization problem, with the inner maximization generating adversarial perturbations, and the outer minimization finding model parameters by optimizing the loss on adversarial perturbations generated from the inner maximization. We demonstrate the applicability of our approach on three types of naturally occurring perturbations -- object-related shifts, geometric transformations, and common image corruptions. Our approach enables deep neural networks to be robust against a wide range of naturally occurring perturbations. We demonstrate the usefulness of the proposed approach by showing the robustness gains of deep neural networks trained using our adversarial training on MNIST, CIFAR-10, and a new variant of the CLEVR dataset.
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