The dichromatic number $\vec{\chi}(D)$ of a digraph $D$ is the least integer $k$ for which $D$ has a coloring with $k$ colors such that there is no monochromatic directed cycle in $D$. The digraphs considered here are finite and may have antiparallel arcs, but no parallel arcs. A digraph $D$ is called $k$-critical if each proper subdigraph $D'$ of $D$ satisfies $\vec{\chi}(D')<\vec{\chi}(D)=k$. For integers $k$ and $n$, let $\overrightarrow{\mathrm{ext}}(k,n)$ denote the minimum number of arcs possible in a $k$-critical digraph of order $n$. It is easy to show that $\overrightarrow{\mathrm{ext}}(2,n)=n$ for all $n\geq 2$, and $\overrightarrow{\mathrm{ext}}(3,n)\geq 2n$ for all possible $n$, where equality holds if and only if $n$ is odd and $n\geq 3$. As a main result we prove that if $n, k$ and $p$ are integers with $n=k+p$ and $2\leq p \leq k-1$, then $\overrightarrow{\mathrm{ext}}(k,n)=2({\binom{n}{2}} - (p^2+1))$, and we give an exact characterisation of $k$-critical digraphs for which equality holds. This generalizes a result about critical graphs obtained in 1963 by Tibor Gallai.
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In the average-case $k$-SUM problem, given $r$ integers chosen uniformly at random from $\{0,\dots,M-1\}$, the objective is to find a ``solution'' set of $k$ numbers that sum to $0$ modulo $M$. In the dense regime of $M \leq r^k$, where solutions exist with high probability, the complexity of these problems is well understood. Much less is known in the sparse regime of $M\gg r^k$, where solutions are unlikely to exist. In this work, we initiate the study of the sparse regime for $k$-SUM and its variant $k$-XOR, especially their planted versions, where a random solution is planted in a randomly generated instance and has to be recovered. We provide evidence for the hardness of these problems and suggest new applications to cryptography. Complexity. First we study the complexity of these problems in the sparse regime and show: - Conditional Lower Bounds. Assuming established conjectures about the hardness of average-case (non-planted) $k$-SUM/$k$-XOR when $M = r^k$, we provide non-trivial lower bounds on the running time of algorithms for planted $k$-SUM when $r^k\leq M\leq r^{2k}$. - Hardness Amplification. We show that for any $M \geq r^k$, if an algorithm running in time $T$ solves planted $k$-SUM/$k$-XOR with success probability $\Omega(1/\text{polylog}(r))$, then there is an algorithm running in time $\tilde{O}(T)$ that solves it with probability $(1-o(1))$. - New Reductions and Algorithms. We provide reductions for $k$-SUM/$k$-XOR from search to decision, as well as worst-case and average-case reductions to the Subset Sum problem from $k$-SUM, as well as a new algorithm for average-case $k$-XOR at low densities. Cryptography. We show that by additionally assuming mild hardness of $k$-XOR, we can construct Public Key Encryption (PKE) from a weaker variant of the Learning Parity with Noise (LPN) problem than was known before.
We introduce a new Langevin dynamics based algorithm, called e-TH$\varepsilon$O POULA, to solve optimization problems with discontinuous stochastic gradients which naturally appear in real-world applications such as quantile estimation, vector quantization, CVaR minimization, and regularized optimization problems involving ReLU neural networks. We demonstrate both theoretically and numerically the applicability of the e-TH$\varepsilon$O POULA algorithm. More precisely, under the conditions that the stochastic gradient is locally Lipschitz in average and satisfies a certain convexity at infinity condition, we establish non-asymptotic error bounds for e-TH$\varepsilon$O POULA in Wasserstein distances and provide a non-asymptotic estimate for the expected excess risk, which can be controlled to be arbitrarily small. Three key applications in finance and insurance are provided, namely, multi-period portfolio optimization, transfer learning in multi-period portfolio optimization, and insurance claim prediction, which involve neural networks with (Leaky)-ReLU activation functions. Numerical experiments conducted using real-world datasets illustrate the superior empirical performance of e-TH$\varepsilon$O POULA compared to SGLD, TUSLA, ADAM, and AMSGrad in terms of model accuracy.
We give a poly-time algorithm for the $k$-edge-connected spanning subgraph ($k$-ECSS) problem that returns a solution of cost no greater than the cheapest $(k+10)$-ECSS on the same graph. Our approach enhances the iterative relaxation framework with a new ingredient, which we call ghost values, that allows for high sparsity in intermediate problems. Our guarantees improve upon the best-known approximation factor of $2$ for $k$-ECSS whenever the optimal value of $(k+10)$-ECSS is close to that of $k$-ECSS. This is a property that holds for the closely related problem $k$-edge-connected spanning multi-subgraph ($k$-ECSM), which is identical to $k$-ECSS except edges can be selected multiple times at the same cost. As a consequence, we obtain a $\left(1+O\left(\frac{1}{k}\right)\right)$-approximation algorithm for $k$-ECSM, which resolves a conjecture of Pritchard and improves upon a recent $\left(1+O\left(\frac{1}{\sqrt{k}}\right)\right)$-approximation algorithm of Karlin, Klein, Oveis Gharan, and Zhang. Moreover, we present a matching lower bound for $k$-ECSM, showing that our approximation ratio is tight up to the constant factor in $O\left(\frac{1}{k}\right)$, unless $P=NP$.
For an odd prime $p$, we say $f(X) \in {\mathbb F}_p[X]$ computes square roots in $\mathbb F_p$ if, for all nonzero perfect squares $a \in \mathbb F_p$, we have $f(a)^2 = a$. When $p \equiv 3 \mod 4$, it is well known that $f(X) = X^{(p+1)/4}$ computes square roots. This degree is surprisingly low (and in fact lowest possible), since we have specified $(p-1)/2$ evaluations (up to sign) of the polynomial $f(X)$. On the other hand, for $p \equiv 1 \mod 4$ there was previously no nontrivial bound known on the lowest degree of a polynomial computing square roots in $\mathbb F_p$; it could have been anywhere between $\frac{p}{4}$ and $\frac{p}{2}$. We show that for all $p \equiv 1 \mod 4$, the degree of a polynomial computing square roots has degree at least $p/3$. Our main new ingredient is a general lemma which may be of independent interest: powers of a low degree polynomial cannot have too many consecutive zero coefficients. The proof method also yields a robust version: any polynomial that computes square roots for 99\% of the squares also has degree almost $p/3$. In the other direction, we also show that for infinitely many $p \equiv 1 \mod 4$, the degree of a polynomial computing square roots can be $(\frac{1}{2} - \Omega(1))p$.
We prove and collect numerous explicit and computable results for the fractional Laplacian $(-\Delta)^s f(x)$ with $s>0$ as well as its whole space inverse, the Riesz potential, $(-\Delta)^{-s}f(x)$ with $s\in\left(0,\frac{1}{2}\right)$. Choices of $f(x)$ include weighted classical orthogonal polynomials such as the Legendre, Chebyshev, Jacobi, Laguerre and Hermite polynomials, or first and second kind Bessel functions with or without sinusoid weights. Some higher dimensional fractional Laplacians and Riesz potentials of generalized Zernike polynomials on the unit ball and its complement as well as whole space generalized Laguerre polynomials are also discussed. The aim of this paper is to aid in the continued development of numerical methods for problems involving the fractional Laplacian or the Riesz potential in bounded and unbounded domains -- both directly by providing useful basis or frame functions for spectral method approaches and indirectly by providing accessible ways to construct computable toy problems on which to test new numerical methods.
We use a labelled deduction system ( LND$_{ED-}$TRS ) based on the concept of computational paths (sequences of rewrites) as equalities between two terms of the same type, which allowed us to carry out in homotopic theory an approach using the concept of computational paths. From this, we show that the computational paths can be used to perform the proofs of the $LND_{EQ}-TRS_{2}$ rewriting system.
A subsequence of a word $w$ is a word $u$ such that $u = w[i_1] w[i_2] \dots w[i_{k}]$, for some set of indices $1 \leq i_1 < i_2 < \dots < i_k \leq \lvert w\rvert$. A word $w$ is $k$-subsequence universal over an alphabet $\Sigma$ if every word in $\Sigma^k$ appears in $w$ as a subsequence. In this paper, we study the intersection between the set of $k$-subsequence universal words over some alphabet $\Sigma$ and regular languages over $\Sigma$. We call a regular language $L$ \emph{$k$-$\exists$-subsequence universal} if there exists a $k$-subsequence universal word in $L$, and \emph{$k$-$\forall$-subsequence universal} if every word of $L$ is $k$-subsequence universal. We give algorithms solving the problems of deciding if a given regular language, represented by a finite automaton recognising it, is \emph{$k$-$\exists$-subsequence universal} and, respectively, if it is \emph{$k$-$\forall$-subsequence universal}, for a given $k$. The algorithms are FPT w.r.t.~the size of the input alphabet, and their run-time does not depend on $k$; they run in polynomial time in the number $n$ of states of the input automaton when the size of the input alphabet is $O(\log n)$. Moreover, we show that the problem of deciding if a given regular language is \emph{$k$-$\exists$-subsequence universal} is NP-complete, when the language is over a large alphabet. Further, we provide algorithms for counting the number of $k$-subsequence universal words (paths) accepted by a given deterministic (respectively, nondeterministic) finite automaton, and ranking an input word (path) within the set of $k$-subsequence universal words accepted by a given finite automaton.
The adaptive probability $P_{\text{\tiny{adp}}}$ formalized in Adapt-$P$ is developed based on the remaining number of SNs $\zeta$ and optimal clustering $\kappa_{\text{\tiny{max}}}$, yet $P_{\text{\tiny{adp}}}$ does not implement the probabilistic ratios of energy and distance factors in the network. Furthermore, Adapt-$P$ does not localize cluster-heads in the first round properly because of its reliance on distance computations defined in LEACH, that might result in uneven distribution of cluster-heads in the WSN area and hence might at some rounds yield inefficient consumption of energy. This paper utilizes \nolinebreak{$k$\small{-}means\small{++}} and Adapt-$P$ to propose \nolinebreak{$P_{\text{c}} \kappa_{\text{\tiny{max}}}$\small{-}means\small{++}} clustering algorithm that better manages the distribution of cluster-heads and produces an enhanced performance. The algorithm employs an optimized cluster-head election probability $P_\text{c}$ developed based on energy-based $P_{\eta(j,i)}$ and distance-based $P\!\!\!_{\psi(j,i)}$ quality probabilities along with the adaptive probability $P_{\text{\tiny{adp}}}$, utilizing the energy $\varepsilon$ and distance optimality $d\!_{\text{\tiny{opt}}}$ factors. Furthermore, the algorithm utilizes the optimal clustering $\kappa_{\text{\tiny{max}}}$ derived in Adapt-$P$ to perform adaptive clustering through \nolinebreak{$\kappa_{\text{\tiny{max}}}$\small{-}means\small{++}}. The proposed \nolinebreak{$P_{\text{c}} \kappa_{\text{\tiny{max}}}${\small{-}}means{\small{++}}} is compared with the energy-based algorithm \nolinebreak{$P_\eta \varepsilon \kappa_{\text{\tiny{max}}}${\small{-}}means{\small{++}}} and distance-based \nolinebreak{$P_\psi d_{\text{\tiny{opt}}} \kappa_{\text{\tiny{max}}}${\small{-}}means{\small{++}}} algorithm, and has shown an optimized performance in term of residual energy and stability period of the network.
We present an algorithm which can generate all pairwise non-isomorphic $K_2$-hypohamiltonian graphs, i.e. non-hamiltonian graphs in which the removal of any pair of adjacent vertices yields a hamiltonian graph, of a given order. We introduce new bounding criteria specifically designed for $K_2$-hypohamiltonian graphs, allowing us to improve upon earlier computational results. Specifically, we characterise the orders for which $K_2$-hypohamiltonian graphs exist and improve existing lower bounds on the orders of the smallest planar and the smallest bipartite $K_2$-hypohamiltonian graphs. Furthermore, we describe a new operation for creating $K_2$-hypohamiltonian graphs that preserves planarity under certain conditions and use it to prove the existence of a planar $K_2$-hypohamiltonian graph of order $n$ for every integer $n\geq 134$. Additionally, motivated by a theorem of Thomassen on hypohamiltonian graphs, we show the existence $K_2$-hypohamiltonian graphs with large maximum degree and size.
For a function $F: X \to Y$ between real Banach spaces, we show how continuation methods to solve $F(u) = g$ may improve from basic understanding of the critical set $C$ of $F$. The algorithm aims at special points with a large number of preimages, which in turn may be used as initial conditions for standard continuation methods applied to the solution of the desired equation. A geometric model based on the sets $C$ and $F^{-1}(F(C))$ substantiate our choice of curves $c \in X$ with abundant intersections with $C$. We consider three classes of examples. First we handle functions $F: R^2 \to R^2$, for which the reasoning behind the techniques is visualizable. The second set of examples, between spaces of dimension 15, is obtained by discretizing a nonlinear Sturm-Liouville problem for which special points admit a high number of solutions. Finally, we handle a semilinear elliptic operator, by computing the six solutions of an equation of the form $-\Delta - f(u) = g$ studied by Solimini.
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