For a fixed integer $r \geq 1$, a distance-$r$ dominating set of a graph $G = (V, E)$ is a vertex subset $D \subseteq V$ such that every vertex in $V$ is within distance $r$ from some member of $D$. Given two distance-$r$ dominating sets $D_s, D_t$ of $G$, the Distance-$r$ Dominating Set Reconfiguration (D$r$DSR) problem asks if there is a sequence of distance-$r$ dominating sets that transforms $D_s$ into $D_t$ (or vice versa) such that each intermediate member is obtained from its predecessor by applying a given reconfiguration rule exactly once. The problem for $r = 1$ has been well-studied in the literature. We consider D$r$DSR for $r \geq 2$ under two well-known reconfiguration rules: Token Jumping ($\mathsf{TJ}$, which involves replacing a member of the current D$r$DS by a non-member) and Token Sliding ($\mathsf{TS}$, which involves replacing a member of the current D$r$DS by an adjacent non-member). We show that D$r$DSR ($r \geq 2$) is $\mathtt{PSPACE}$-complete under both $\mathsf{TJ}$ and $\mathsf{TS}$ on bipartite graphs, planar graphs of maximum degree six and bounded bandwidth, and chordal graphs. On the positive side, we show that D$r$DSR ($r \geq 2$) can be solved in polynomial time on split graphs and cographs under both $\mathsf{TS}$ and $\mathsf{TJ}$ and on trees and interval graphs under $\mathsf{TJ}$. Along the way, we observe some properties of a shortest reconfiguration sequence in split graphs when $r = 2$, which may be of independent interest.
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Given a matrix $\mathbf{A} \in \mathbb{R}^{k \times n}$, a partitioning of $[k]$ into groups $S_1,\dots,S_m$, an outer norm $p$, and a collection of inner norms such that either $p \ge 1$ and $p_1,\dots,p_m \ge 2$ or $p_1=\dots=p_m=p \ge 1/\log n$, we prove that there is a sparse weight vector $\mathbf{\beta} \in \mathbb{R}^{m}$ such that $\sum_{i=1}^m \beta_i \cdot \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p \approx_{1\pm\varepsilon} \sum_{i=1}^m \|\mathbf{A}_{S_i}\mathbf{x}\|_{p_i}^p$, where the number of nonzero entries of $\mathbf{\beta}$ is at most $O_{p,p_i}(\varepsilon^{-2}n^{\max(1,p/2)}(\log n)^2(\log(n/\varepsilon)))$. When $p_1\dots,p_m \ge 2$, this weight vector arises from an importance sampling procedure based on the block Lewis weights, a recently proposed generalization of Lewis weights. Additionally, we prove that there exist efficient algorithms to find the sparse weight vector $\mathbf{\beta}$ in several important regimes of $p$ and $p_1,\dots,p_m$. Our main technical contribution is a substantial generalization of the change-of-measure method that Bourgain, Lindenstrauss, and Milman used to obtain the analogous result when every group has size $1$. Our generalization allows one to analyze change of measures beyond those implied by D. Lewis's original construction, including the measure implied by the block Lewis weights and natural approximations of this measure.
We present algorithms that compute the terminal configurations for sandpile instances in $O(n \log n)$ time on trees and $O(n)$ time on paths, where $n$ is the number of vertices. The Abelian Sandpile model is a well-known model used in exploring self-organized criticality. Despite a large amount of work on other aspects of sandpiles, there have been limited results in efficiently computing the terminal state, known as the sandpile prediction problem. Our algorithm improves the previous best runtime of $O(n \log^5 n)$ on trees [Ramachandran-Schild SODA '17] and $O(n \log n)$ on paths [Moore-Nilsson '99]. To do so, we move beyond the simulation of individual events by directly computing the number of firings for each vertex. The computation is accelerated using splittable binary search trees. In addition, we give algorithms in $O(n)$ time on cliques and $O(n \log^2 n)$ time on pseudotrees. Towards solving on general graphs, we provide a reduction that transforms the prediction problem on an arbitrary graph into problems on its subgraphs separated by any vertex set $P$. The reduction gives a time complexity of $O(\log^{|P|} n \cdot T)$ where $T$ denotes the total time for solving on each subgraph. We also give algorithms that works well with this reduction scheme.
Chan, Har-Peled, and Jones [SICOMP 2020] developed locality-sensitive orderings (LSO) for Euclidean space. A $(\tau,\rho)$-LSO is a collection $\Sigma$ of orderings such that for every $x,y\in\mathbb{R}^d$ there is an ordering $\sigma\in\Sigma$, where all the points between $x$ and $y$ w.r.t. $\sigma$ are in the $\rho$-neighborhood of either $x$ or $y$. In essence, LSO allow one to reduce problems to the $1$-dimensional line. Later, Filtser and Le [STOC 2022] developed LSO's for doubling metrics, general metric spaces, and minor free graphs. For Euclidean and doubling spaces, the number of orderings in the LSO is exponential in the dimension, which made them mainly useful for the low dimensional regime. In this paper, we develop new LSO's for Euclidean, $\ell_p$, and doubling spaces that allow us to trade larger stretch for a much smaller number of orderings. We then use our new LSO's (as well as the previous ones) to construct path reporting low hop spanners, fault tolerant spanners, reliable spanners, and light spanners for different metric spaces. While many nearest neighbor search (NNS) data structures were constructed for metric spaces with implicit distance representations (where the distance between two metric points can be computed using their names, e.g. Euclidean space), for other spaces almost nothing is known. In this paper we initiate the study of the labeled NNS problem, where one is allowed to artificially assign labels (short names) to metric points. We use LSO's to construct efficient labeled NNS data structures in this model.
We show that the problem of whether a query is equivalent to a query of tree-width $k$ is decidable, for the class of Unions of Conjunctive Regular Path Queries with two-way navigation (UC2RPQs). A previous result by Barcel\'o, Romero, and Vardi [SIAM Journal on Computing, 2016] has shown decidability for the case $k=1$, and here we extend this result showing that decidability in fact holds for any arbitrary $k\geq 1$. The algorithm is in 2ExpSpace, but for the restricted but practically relevant case where all regular expressions of the query are of the form $a^*$ or $(a_1 + \dotsb + a_n)$ we show that the complexity of the problem drops to $\Pi^P_2$. We also investigate the related problem of approximating a UC2RPQ by queries of small tree-width. We exhibit an algorithm which, for any fixed number $k$, builds the maximal under-approximation of tree-width $k$ of a UC2RPQ. The maximal under-approximation of tree-width $k$ of a query $q$ is a query $q'$ of tree-width $k$ which is contained in $q$ in a maximal and unique way, that is, such that for every query $q''$ of tree-width $k$, if $q''$ is contained in $q$ then $q''$ is also contained in $q'$. Our approach is shown to be robust, in the sense that it allows also to test equivalence with queries of a given path-width, it also covers the previously known result for $k=1$, and it allows to test for equivalence of whether a (one-way) UCRPQ is equivalent to a UCRPQ of a given tree-width (or path-width).
The orthogonality dimension of a graph $G$ over $\mathbb{R}$ is the smallest integer $k$ for which one can assign a nonzero $k$-dimensional real vector to each vertex of $G$, such that every two adjacent vertices receive orthogonal vectors. We prove that for every sufficiently large integer $k$, it is $\mathsf{NP}$-hard to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $k$ or at least $2^{(1-o(1)) \cdot k/2}$. We further prove such hardness results for the orthogonality dimension over finite fields as well as for the closely related minrank parameter, which is motivated by the index coding problem in information theory. This in particular implies that it is $\mathsf{NP}$-hard to approximate these graph quantities to within any constant factor. Previously, the hardness of approximation was known to hold either assuming certain variants of the Unique Games Conjecture or for approximation factors smaller than $3/2$. The proofs involve the concept of line digraphs and bounds on their orthogonality dimension and on the minrank of their complement.
We find a succinct expression for computing the sequence $x_t = a_t x_{t-1} + b_t$ in parallel with two prefix sums, given $t = (1, 2, \dots, n)$, $a_t \in \mathbb{R}^n$, $b_t \in \mathbb{R}^n$, and initial value $x_0 \in \mathbb{R}$. On $n$ parallel processors, the computation of $n$ elements incurs $\mathcal{O}(\log n)$ time and $\mathcal{O}(n)$ space. Sequences of this form are ubiquitous in science and engineering, making efficient parallelization useful for a vast number of applications. We implement our expression in software, test it on parallel hardware, and verify that it executes faster than sequential computation by a factor of $\frac{n}{\log n}$.
We consider the following general model of a sorting procedure: we fix a hereditary permutation class $\mathcal{C}$, which corresponds to the operations that the procedure is allowed to perform in a single step. The input of sorting is a permutation $\pi$ of the set $[n]=\{1,2,\dotsc,n\}$, i.e., a sequence where each element of $[n]$ appears once. In every step, the sorting procedure picks a permutation $\sigma$ of length $n$ from $\mathcal{C}$, and rearranges the current permutation of numbers by composing it with $\sigma$. The goal is to transform the input $\pi$ into the sorted sequence $1,2,\dotsc,n$ in as few steps as possible. This model of sorting captures not only classical sorting algorithms, like insertion sort or bubble sort, but also sorting by series of devices, like stacks or parallel queues, as well as sorting by block operations commonly considered, e.g., in the context of genome rearrangement. Our goal is to describe the possible asymptotic behavior of the worst-case number of steps needed when sorting with a hereditary permutation class. As the main result, we show that any hereditary permutation class $\mathcal{C}$ falls into one of five distinct categories. Disregarding the classes that cannot sort all permutations, the number of steps needed to sort any permutation of $[n]$ with $\mathcal{C}$ is either $\Theta(n^2)$, a function between $O(n)$ and $\Omega(\sqrt{n})$, a function betwee $O(\log^2 n)$ and $\Omega(\log n), or $1$, and for each of these cases we provide a structural characterization of the corresponding hereditary classes.
We investigate the linear chromatic number $\chi_{\text{lin}}(G(n,p))$ of the binomial random graph $G(n,p)$ on $n$ vertices in which each edge appears independently with probability $p=p(n)$. For dense random graphs ($np \to \infty$ as $n \to \infty$), we show that asymptotically almost surely $\chi_{\text{lin}}(G(n,p)) \ge n (1 - O( (np)^{-1/2} ) ) = n(1-o(1))$. Understanding the order of the linear chromatic number for subcritical random graphs ($np < 1$) and critical ones ($np=1$) is relatively easy. However, supercritical sparse random graphs ($np = c$ for some constant $c > 1$) remain to be investigated.
Given a graph, a $k$-plex is a set of vertices in which each vertex is not adjacent to at most $k-1$ other vertices in the set. The maximum $k$-plex problem, which asks for the largest $k$-plex from the given graph, is an important but computationally challenging problem in applications such as graph mining and community detection. So far, there are many practical algorithms, but without providing theoretical explanations on their efficiency. We define a novel parameter of the input instance, $g_k(G)$, the gap between the degeneracy bound and the size of the maximum $k$-plex in the given graph, and present an exact algorithm parameterized by this $g_k(G)$, which has a worst-case running time polynomial in the size of the input graph and exponential in $g_k(G)$. In real-world inputs, $g_k(G)$ is very small, usually bounded by $O(\log{(|V|)})$, indicating that the algorithm runs in polynomial time. We further extend our discussion to an even smaller parameter $cg_k(G)$, the gap between the community-degeneracy bound and the size of the maximum $k$-plex, and show that without much modification, our algorithm can also be parameterized by $cg_k(G)$. To verify the empirical performance of these algorithms, we carry out extensive experiments to show that these algorithms are competitive with the state-of-the-art algorithms. In particular, for large $k$ values such as $15$ and $20$, our algorithms dominate the existing algorithms. Finally, empirical analysis is performed to illustrate the effectiveness of the parameters and other key components in the implementation.
A matrix $\Phi \in \mathbb{R}^{Q \times N}$ satisfies the restricted isometry property if $\|\Phi x\|_2^2$ is approximately equal to $\|x\|_2^2$ for all $k$-sparse vectors $x$. We give a construction of RIP matrices with the optimal $Q = O(k \log(N/k))$ rows using $O(k\log(N/k)\log(k))$ bits of randomness. The main technical ingredient is an extension of the Hanson-Wright inequality to $\epsilon$-biased distributions.
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