A word~$w$ has a border $u$ if $u$ is a non-empty proper prefix and suffix of $u$. A word~$w$ is said to be \emph{closed} if $w$ is of length at most $1$ or if $w$ has a border that occurs exactly twice in $w$. A word~$w$ is said to be \emph{privileged} if $w$ is of length at most $1$ or if $w$ has a privileged border that occurs exactly twice in $w$. Let $C_k(n)$ (resp.~$P_k(n)$) be the number of length-$n$ closed (resp. privileged) words over a $k$-letter alphabet. In this paper, we improve existing upper and lower bounds on $C_k(n)$ and $P_k(n)$. We completely resolve the asymptotic behaviour of $C_k(n)$. We also nearly completely resolve the asymptotic behaviour of $P_k(n)$ by giving a family of upper and lower bounds that are separated by a factor that grows arbitrarily slowly.
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The article concerns low-rank approximation of matrices generated by sampling a smooth function of two $m$-dimensional variables. We refute an argument made in the literature that, for a specific class of analytic functions, such matrices admit accurate entrywise approximation of rank that is independent of $m$. We provide a theoretical explanation of the numerical results presented in support of this argument, describing three narrower classes of functions for which $n \times n$ function-generated matrices can be approximated within an entrywise error of order $\varepsilon$ with rank $\mathcal{O}(\log(n) \varepsilon^{-2} \mathrm{polylog}(\varepsilon^{-1}))$ that is independent of the dimension $m$: (i) functions of the inner product of the two variables, (ii) functions of the squared Euclidean distance between the variables, and (iii) shift-invariant positive-definite kernels. We extend our argument to low-rank tensor-train approximation of tensors generated with functions of the multi-linear product of their $m$-dimensional variables. We discuss our results in the context of low-rank approximation of attention in transformer neural networks.
We obtain the smallest unsatisfiable formulas in subclasses of $k$-CNF (exactly $k$ distinct literals per clause) with bounded variable or literal occurrences. Smaller unsatisfiable formulas of this type translate into stronger inapproximability results for MaxSAT in the considered formula class. Our results cover subclasses of 3-CNF and 4-CNF; in all subclasses of 3-CNF we considered we were able to determine the smallest size of an unsatisfiable formula; in the case of 4-CNF with at most 5 occurrences per variable we decreased the size of the smallest known unsatisfiable formula. Our methods combine theoretical arguments and symmetry-breaking exhaustive search based on SAT Modulo Symmetries (SMS), a recent framework for isomorph-free SAT-based graph generation. To this end, and as a standalone result of independent interest, we show how to encode formulas as graphs efficiently for SMS.
In this paper, various properties of core-EP matrices are investigated. We introduce the MPDMP matrix associated with $A$ and by means of it, some properties and equivalent conditions of core-EP matrices can be obtained. Also, properties of MPD, DMP, and CMP inverses are studied and we prove that in the class of core-EP matrices, DMP, MPD, and Drazin inverses are the same. Moreover, DMP and MPD binary relation orders are introduced and the relationship between these orders and other binary relation orders are considered.
We present fully abstract encodings of the call-by-name and call-by-value $\lambda$-calculus into HOcore, a minimal higher-order process calculus with no name restriction. We consider several equivalences on the $\lambda$-calculus side -- normal-form bisimilarity, applicative bisimilarity, and contextual equivalence -- that we internalize into abstract machines in order to prove full abstraction of the encodings. We also demonstrate that this technique scales to the $\lambda\mu$-calculus, i.e., a standard extension of the $\lambda$-calculus with control operators.
We prove that the Weihrauch degree of the problem of finding a bad sequence in a non-well quasi order ($\mathsf{BS}$) is strictly above that of finding a descending sequence in an ill-founded linear order ($\mathsf{DS}$). This corrects our mistaken claim in arXiv:2010.03840, which stated that they are Weihrauch equivalent. We prove that K\"onig's lemma $\mathsf{KL}$ and the problem $\mathsf{wList}_{2^{\mathbb{N}},\leq\omega}$ of enumerating a given non-empty countable closed subset of $2^\mathbb{N}$ are not Weihrauch reducible to $\mathsf{DS}$ either, resolving two main open questions raised in arXiv:2010.03840.
A separating system of a graph $G$ is a family $\mathcal{S}$ of subgraphs of $G$ for which the following holds: for all distinct edges $e$ and $f$ of $G$, there exists an element in $\mathcal{S}$ that contains $e$ but not $f$. Recently, it has been shown that every graph of order $n$ admits a separating system consisting of $19n$ paths [Bonamy, Botler, Dross, Naia, Skokan, Separating the Edges of a Graph by a Linear Number of Paths, Adv. Comb., October 2023], improving the previous almost linear bound of $\mathrm{O}(n\log^\star n)$ [S. Letzter, Separating paths systems of almost linear size, Trans. Amer. Math. Soc., to appear], and settling conjectures posed by Balogh, Csaba, Martin, and Pluh\'ar and by Falgas-Ravry, Kittipassorn, Kor\'andi, Letzter, and Narayanan. We investigate a natural generalization of these results to subdivisions of cliques, showing that every graph admits both a separating system consisting of $41n$ edges and cycles, and a separating system consisting of $82 n$ edges and subdivisions of $K_4$.
Let $\mathsf{TH}_k$ denote the $k$-out-of-$n$ threshold function: given $n$ input Boolean variables, the output is $1$ if and only if at least $k$ of the inputs are $1$. We consider the problem of computing the $\mathsf{TH}_k$ function using noisy readings of the Boolean variables, where each reading is incorrect with some fixed and known probability $p \in (0,1/2)$. As our main result, we show that it is sufficient to use $(1+o(1)) \frac{n\log \frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ queries in expectation to compute the $\mathsf{TH}_k$ function with a vanishing error probability $\delta = o(1)$, where $m\triangleq \min\{k,n-k\}$ and $D_{\mathsf{KL}}(p \| 1-p)$ denotes the Kullback-Leibler divergence between $\mathsf{Bern}(p)$ and $\mathsf{Bern}(1-p)$ distributions. Conversely, we show that any algorithm achieving an error probability of $\delta = o(1)$ necessitates at least $(1-o(1))\frac{(n-m)\log\frac{m}{\delta}}{D_{\mathsf{KL}}(p \| 1-p)}$ queries in expectation. The upper and lower bounds are tight when $m=o(n)$, and are within a multiplicative factor of $\frac{n}{n-m}$ when $m=\Theta(n)$. In particular, when $k=n/2$, the $\mathsf{TH}_k$ function corresponds to the $\mathsf{MAJORITY}$ function, in which case the upper and lower bounds are tight up to a multiplicative factor of two. Compared to previous work, our result tightens the dependence on $p$ in both the upper and lower bounds.
In the $K_r$-Cover problem, given a graph $G$ and an integer $k$ one has to decide if there exists a set of at most $k$ vertices whose removal destroys all $r$-cliques of $G$. In this paper we give an algorithm for $K_r$-Cover that runs in subexponential FPT time on graph classes satisfying two simple conditions related to cliques and treewidth. As an application we show that our algorithm solves $K_r$-Cover in time * $2^{O_r\left (k^{(r+1)/(r+2)}\log k \right)} \cdot n^{O_r(1)}$ in pseudo-disk graphs and map-graphs; * $2^{O_{t,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $K_{t,t}$-subgraph-free string graphs; and * $2^{O_{H,r}(k^{2/3}\log k)} \cdot n^{O_r(1)}$ in $H$-minor-free graphs.
We consider the problem of enumerating all minimal transversals (also called minimal hitting sets) of a hypergraph $\mathcal{H}$. An equivalent formulation of this problem known as the \emph{transversal hypergraph} problem (or \emph{hypergraph dualization} problem) is to decide, given two hypergraphs, whether one corresponds to the set of minimal transversals of the other. The existence of a polynomial time algorithm to solve this problem is a long standing open question. In \cite{fredman_complexity_1996}, the authors present the first sub-exponential algorithm to solve the transversal hypergraph problem which runs in quasi-polynomial time, making it unlikely that the problem is (co)NP-complete. In this paper, we show that when one of the two hypergraphs is of bounded VC-dimension, the transversal hypergraph problem can be solved in polynomial time, or equivalently that if $\mathcal{H}$ is a hypergraph of bounded VC-dimension, then there exists an incremental polynomial time algorithm to enumerate its minimal transversals. This result generalizes most of the previously known polynomial cases in the literature since they almost all consider classes of hypergraphs of bouded VC-dimension. As a consequence, the hypergraph transversal problem is solvable in polynomial time for any class of hypergraphs closed under partial subhypergraphs. We also show that the proposed algorithm runs in quasi-polynomial time in general hypergraphs and runs in polynomial time if the conformality of the hypergraph is bounded, which is one of the few known polynomial cases where the VC-dimension is unbounded.
A subset of vertices $S$ of a graph $G$ is a dominating set if every vertex in $V \setminus S$ has at least one neighbor in $S$. A domatic partition is a partition of the vertices of a graph $G$ into disjoint dominating sets. The domatic number $d(G)$ is the maximum size of a domatic partition. Suppose that $dp(G,i)$ is the number of distinct domatic partition of $G$ with cardinality $i$. In this paper, we consider the generating function of $dp(G,i)$, i.e., $DP(G,x)=\sum_{i=1}^{d(G)}dp(G,i)x^i$ which we call it the domatic partition polynomial. We explore the domatic polynomial for trees, providing a quadratic time algorithm for its computation based on weak 2-coloring numbers. Our results include specific findings for paths and certain graph products, demonstrating practical applications of our theoretical framework.
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