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A subset $S$ of vertices in a graph $G=(V, E)$ is a Dominating Set if each vertex in $V(G)\setminus S$ is adjacent to at least one vertex in $S$. Chellali et al. in 2013, by restricting the number of neighbors in $S$ of a vertex outside $S$, introduced the concept of $[1,j]$-dominating set. A set $D \subseteq V$ of a graph $G = (V, E)$ is called a $[1,j]$-Dominating Set of $G$ if every vertex not in $D$ has at least one neighbor and at most $j$ neighbors in $D$. The Minimum $[1,j]$-Domination problem is the problem of finding the minimum $[1,j]$-dominating set $D$. Given a positive integer $k$ and a graph $G = (V, E)$, the $[1,j]$-Domination Decision problem is to decide whether $G$ has a $[1,j]$-dominating set of cardinality at most $k$. A polynomial-time algorithm was obtained in split graphs for a constant $j$ in contrast to the Dominating Set problem which is NP-hard for split graphs. This result motivates us to investigate the effect of restriction $j$ on the complexity of $[1,j]$-domination problem on various classes of graphs. Although for $j\geq 3$, it has been proved that the minimum of classical domination is equal to minimum $[1,j]$-domination in interval graphs, the complexity of finding the minimum $[1,2]$-domination in interval graphs is still outstanding. In this paper, we propose a polynomial-time algorithm for computing a minimum $[1,2]$-dominating set on interval graphs by a dynamic programming technique. Next, on the negative side, we show that the minimum $[1,2]$-dominating set problem on circle graphs is $NP$-complete.

We provide a perfect sampling algorithm for the hard-sphere model on subsets of $\mathbb{R}^d$ with expected running time linear in the volume under the assumption of strong spatial mixing. A large number of perfect and approximate sampling algorithms have been devised to sample from the hard-sphere model, and our perfect sampling algorithm is efficient for a range of parameters for which only efficient approximate samplers were previously known and is faster than these known approximate approaches. Our methods also extend to the more general setting of Gibbs point processes interacting via finite-range, repulsive potentials.

We revisit the sample and computational complexity of completing a rank-1 tensor in $\otimes_{i=1}^{N} \mathbb{R}^{d}$, given a uniformly sampled subset of its entries. We present a characterization of the problem (i.e. nonzero entries) which admits an algorithm amounting to Gauss-Jordan on a pair of random linear systems. For example, when $N = \Theta(1)$, we prove it uses no more than $m = O(d^2 \log d)$ samples and runs in $O(md^2)$ time. Moreover, we show any algorithm requires $\Omega(d\log d)$ samples. By contrast, existing upper bounds on the sample complexity are at least as large as $d^{1.5} \mu^{\Omega(1)} \log^{\Omega(1)} d$, where $\mu$ can be $\Theta(d)$ in the worst case. Prior work obtained these looser guarantees in higher rank versions of our problem, and tend to involve more complicated algorithms.

We introduce the novel class $(E_\alpha)_{\alpha \in [-\infty,1)}$ of reverse map projection embeddings, each one defining a unique new method of encoding classical data into quantum states. Inspired by well-known map projections from the unit sphere onto its tangent planes, used in practice in cartography, these embeddings address the common drawback of the amplitude embedding method, wherein scalar multiples of data points are identified and information about the norm of data is lost. We show how reverse map projections can be utilised as equivariant embeddings for quantum machine learning. Using these methods, we can leverage symmetries in classical datasets to significantly strengthen performance on quantum machine learning tasks. Finally, we select four values of $\alpha$ with which to perform a simple classification task, taking $E_\alpha$ as the embedding and experimenting with both equivariant and non-equivariant setups. We compare their results alongside those of standard amplitude embedding.

Suppose Alice has a distribution $P$ and Bob has a distribution $Q$. Alice wants to generate a sample $a\sim P$ and Bob a sample $b \sim Q$ such that $a = b$ with has as high of probability as possible. It is well-known that, by sampling from an optimal coupling between the distributions, Alice and Bob can achieve $Pr[a = b] = 1 - D_{TV}(P,Q)$, where $D_{TV}(P,Q)$ is the total variation distance. What if Alice and Bob must solve this same problem without communicating at all? Perhaps surprisingly, with access to public randomness, they can still achieve $Pr[a=b] \geq \frac{1-D_{TV}(P,Q)}{1+D_{TV}(P,Q)} \geq 1-2D_{TV}(P,Q)$. In fact, this bound can be obtained using a simple protocol based on the Weighted MinHash algorithm. In this work, we explore the communication-free coupling problem in greater depth. First, we show that an equally simple protocol based on Gumbel sampling matches the worst-case guarantees of the Weighted MinHash approach, but tends to perform better in practice. Conversely, we prove that both approaches are actually sharp: no communication-free protocol can achieve $Pr[a=b]>\frac{1-D_{TV}(P,Q)}{1+D_{TV}(P,Q)}$ in the worst-case. Finally, we prove that, for distributions over $n$ items, there exists a scheme that uses just $O(\log(n/\epsilon))$ bits of communication to achieve $Pr[a = b] = 1 - D_{TV}(P,Q) - \epsilon$, i.e. to essentially match optimal coupling. Beyond our theoretical results, we demonstrate an application of communication-free coupling to speculative decoding, a recent method for accelerating autoregressive large language models [Leviathan, Kalman, Matias, ICML 2023]. We show that communication-free protocols yield a variant of speculative decoding that we call Drafter-Invariant Speculative Decoding, which has the desirable property that the output of the method is fixed given a fixed random seed, regardless of what drafter is used for speculation.

The orthogonality dimension of a graph over $\mathbb{R}$ is the smallest integer $d$ for which one can assign to every vertex a nonzero vector in $\mathbb{R}^d$ such that every two adjacent vertices receive orthogonal vectors. For an integer $d$, the $d$-Ortho-Dim$_\mathbb{R}$ problem asks to decide whether the orthogonality dimension of a given graph over $\mathbb{R}$ is at most $d$. We prove that for every integer $d \geq 3$, the $d$-Ortho-Dim$_\mathbb{R}$ problem parameterized by the vertex cover number $k$ admits a kernel with $O(k^{d-1})$ vertices and bit-size $O(k^{d-1} \cdot \log k)$. We complement this result by a nearly matching lower bound, showing that for any $\varepsilon > 0$, the problem admits no kernel of bit-size $O(k^{d-1-\varepsilon})$ unless $\mathsf{NP} \subseteq \mathsf{coNP/poly}$. We further study the kernelizability of orthogonality dimension problems in additional settings, including over general fields and under various structural parameterizations.

We consider the problem of linearizing a pseudo-Boolean function $f : \{0,1\}^n \to \mathbb{R}$ by means of $k$ Boolean functions. Such a linearization yields an integer linear programming formulation with only $k$ auxiliary variables. This motivates the definition of the linarization complexity of $f$ as the minimum such $k$. Our theoretical contributions are the proof that random polynomials almost surely have a high linearization complexity and characterizations of its value in case we do or do not restrict the set of admissible Boolean functions. The practical relevance is shown by devising and evaluating integer linear programming models of two such linearizations for the low auto-correlation binary sequences problem. Still, many problems around this new concept remain open.

We study the problem of searching for a target at some unknown location in $\mathbb{R}^d$ when additional information regarding the position of the target is available in the form of predictions. In our setting, predictions come as approximate distances to the target: for each point $p\in \mathbb{R}^d$ that the searcher visits, we obtain a value $\lambda(p)$ such that $|p\mathbf{t}|\le \lambda(p) \le c\cdot |p\mathbf{t}|$, where $c\ge 1$ is a fixed constant, $\mathbf{t}$ is the position of the target, and $|p\mathbf{t}|$ is the Euclidean distance of $p$ to $\mathbf{t}$. The cost of the search is the length of the path followed by the searcher. Our main positive result is a strategy that achieves $(12c)^{d+1}$-competitive ratio, even when the constant $c$ is unknown. We also give a lower bound of roughly $(c/16)^{d-1}$ on the competitive ratio of any search strategy in $\mathbb{R}^d$.

Given a graph $G=(V,E)$ and a set $T=\{ (s_i, t_i) : 1\leq i\leq k \}\subseteq V\times V$ of $k$ pairs, the $k$-vertex-disjoint-paths (resp. $k$-edge-disjoint-paths) problem asks to determine whether there exist~$k$ pairwise vertex-disjoint (resp. edge-disjoint) paths $P_1, P_2, ..., P_k$ in $G$ such that, for each $1\leq i\leq k$, $P_i$ connects $s_i$ to $t_i$. Both the edge-disjoint and vertex-disjoint versions in undirected graphs are famously known to be FPT (parameterized by $k$) due to the Graph Minor Theory of Robertson and Seymour. Eilam-Tzoreff [DAM `98] introduced a variant, known as the $k$-disjoint-shortest-paths problem, where each individual path is further required to be a shortest path connecting its pair. They showed that the $k$-disjoint-shortest-paths problem is NP-complete on both directed and undirected graphs; this holds even if the graphs are planar and have unit edge lengths. We focus on four versions of the problem, corresponding to considering edge/vertex disjointness, and to considering directed/undirected graphs. Building on the reduction of Chitnis [SIDMA `23] for $k$-edge-disjoint-paths on planar DAGs, we obtain the following inapproximability lower bound for each of the four versions of $k$-disjoint-shortest-paths on $n$-vertex graphs: - Under Gap-ETH, there exists a constant $\delta>0$ such that for any constant $0<\epsilon\leq \frac{1}{2}$ and any computable function $f$, there is no $(\frac{1}{2}+\epsilon)$-approx in $f(k)\cdot n^{\delta\cdot k}$ time. We further strengthen our results as follows: Directed: Inapprox lower bound for edge-disjoint (resp. vertex-disjoint) paths holds even if the input graph is a planar (resp. 1-planar) DAG with max in-degree and max out-degree at most $2$. Undirected: Inapprox lower bound for edge-disjoint (resp. vertex-disjoint) paths hold even if the input graph is planar (resp. 1-planar) and has max degree $4$.

Hausdorff $\Phi$-dimension is a notion of Hausdorff dimension developed using a restricted class of coverings of a set. We introduce a constructive analogue of $\Phi$-dimension using the notion of constructive $\Phi$-$s$-supergales. We prove a Point-to-Set Principle for $\Phi$-dimension, through which we get Point-to-Set Principles for Hausdorff dimension, continued-fraction dimension and dimension of Cantor coverings as special cases. We also provide a Kolmogorov complexity characterization of constructive $\Phi$-dimension. A class of covering sets $\Phi$ is said to be "faithful" to Hausdorff dimension if the $\Phi$-dimension and Hausdorff dimension coincide for every set. Similarly, $\Phi$ is said to be "faithful" to constructive dimension if the constructive $\Phi$-dimension and constructive dimension coincide for every set. Using the Point-to-Set Principle for Cantor coverings and a new technique for the construction of sequences satisfying a certain Kolmogorov complexity condition, we show that the notions of ``faithfulness'' of Cantor coverings at the Hausdorff and constructive levels are equivalent. We adapt the result by Albeverio, Ivanenko, Lebid, and Torbin to derive the necessary and sufficient conditions for the constructive dimension faithfulness of the coverings generated by the Cantor series expansion, based on the terms of the expansion.