One of the most studied extensions of the famous Traveling Salesperson Problem (TSP) is the {\sc Multiple TSP}: a set of $m\geq 1$ salespersons collectively traverses a set of $n$ cities by $m$ non-trivial tours, to minimize the total length of their tours. This problem can also be considered to be a variant of {\sc Uncapacitated Vehicle Routing} where the objective function is the sum of all tour lengths. When all $m$ tours start from a single common \emph{depot} $v_0$, then the metric {\sc Multiple TSP} can be approximated equally well as the standard metric TSP, as shown by Frieze (1983). The {\sc Multiple TSP} becomes significantly harder to approximate when there is a \emph{set} $D$ of $d \geq 1$ depots that form the starting and end points of the $m$ tours. For this case only a $(2-1/d)$-approximation in polynomial time is known, as well as a $3/2$-approximation for \emph{constant} $d$ which requires a prohibitive run time of $n^{\Theta(d)}$ (Xu and Rodrigues, \emph{INFORMS J. Comput.}, 2015). A recent work of Traub, Vygen and Zenklusen (STOC 2020) gives another approximation algorithm for {\sc Multiple TSP} running in time $n^{\Theta(d)}$ and reducing the problem to approximating TSP. In this paper we overcome the $n^{\Theta(d)}$ time barrier: we give the first efficient approximation algorithm for {\sc Multiple TSP} with a \emph{variable} number $d$ of depots that yields a better-than-2 approximation. Our algorithm runs in time $(1/\varepsilon)^{\mathcal O(d\log d)}\cdot n^{\mathcal O(1)}$, and produces a $(3/2+\varepsilon)$-approximation with constant probability. For the graphic case, we obtain a deterministic $3/2$-approximation in time $2^d\cdot n^{\mathcal O(1)}$.ithm for metric {\sc Multiple TSP} with run time $n^{\Theta(d)}$, which reduces the problem to approximating metric TSP.
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We study $L_2$-approximation problems $\text{APP}_d$ in the worst case setting in the weighted Korobov spaces $H_{d,\a,{\bm \ga}}$ with parameter sequences ${\bm \ga}=\{\ga_j\}$ and $\a=\{\az_j\}$ of positive real numbers $1\ge \ga_1\ge \ga_2\ge \cdots\ge 0$ and $\frac1 2<\az_1\le \az_2\le \cdots$. We consider the minimal worst case error $e(n,\text{APP}_d)$ of algorithms that use $n$ arbitrary continuous linear functionals with $d$ variables. We study polynomial convergence of the minimal worst case error, which means that $e(n,\text{APP}_d)$ converges to zero polynomially fast with increasing $n$. We recall the notions of polynomial, strongly polynomial, weak and $(t_1,t_2)$-weak tractability. In particular, polynomial tractability means that we need a polynomial number of arbitrary continuous linear functionals in $d$ and $\va^{-1}$ with the accuracy $\va$ of the approximation. We obtain that the matching necessary and sufficient condition on the sequences ${\bm \ga}$ and $\a$ for strongly polynomial tractability or polynomial tractability is $$\dz:=\liminf_{j\to\infty}\frac{\ln \ga_j^{-1}}{\ln j}>0,$$ and the exponent of strongly polynomial tractability is $$p^{\text{str}}=2\max\big\{\frac 1 \dz, \frac 1 {2\az_1}\big\}.$$
A pair of linear codes whose intersection is of dimension $\ell$, where $\ell$ is a non-negetive integer, is called an $\ell$-intersection pair of codes. This paper focuses on studying $\ell$-intersection pairs of $\lambda_i$-constacyclic, $i=1,2,$ and conjucyclic codes. We first characterize an $\ell$-intersection pair of $\lambda_i$-constacyclic codes. A formula for $\ell$ has been established in terms of the degrees of the generator polynomials of $\lambda_i$-constacyclic codes. This allows obtaining a condition for $\ell$-linear complementary pairs (LPC) of constacyclic codes. Later, we introduce and characterize the $\ell$-intersection pair of conjucyclic codes over $\mathbb{F}_{q^2}$. The first observation in the process is that there are no non-trivial linear conjucyclic codes over finite fields. So focus on the characterization of additive conjucyclic (ACC) codes. We show that the largest $\mathbb{F}_q$-subcode of an ACC code over $\mathbb{F}_{q^2}$ is cyclic and obtain its generating polynomial. This enables us to find the size of an ACC code. Furthermore, we discuss the trace code of an ACC code and show that it is cyclic. Finally, we determine $\ell$-intersection pairs of trace codes of ACC codes over $\mathbb{F}_4$.
In this paper, we propose a rate-splitting design and characterize the sum-degrees-of-freedom (DoF) for the $K$-user multiple-input-single-output (MISO) broadcast channel with mixed channel state information at the transmitter (CSIT) and order-$(K-1)$ messages, where mixed CSIT refers to the delayed and imperfect-current CSIT, and order-$(K-1)$ message refers to the message desired by $K-1$ users simultaneously. In particular, for the sum-DoF lower bound, we propose a rate-splitting scheme embedding with retrospective interference alignment. In addition, we propose a matching sum-DoF upper bound via genie signalings and extremal inequality. Opposed to existing works for $K=2$, our results show that the sum-DoF is saturated with CSIT quality when CSIT quality thresholds are satisfied for $K>2$.
Given an image $u_0$, the aim of minimising the Mumford-Shah functional is to find a decomposition of the image domain into sub-domains and a piecewise smooth approximation $u$ of $u_0$ such that $u$ varies smoothly within each sub-domain. Since the Mumford-Shah functional is highly non-smooth, regularizations such as the Ambrosio-Tortorelli approximation can be considered which is one of the most computationally efficient approximations of the Mumford-Shah functional for image segmentation. While very impressive numerical results have been achieved in a large range of applications when minimising the functional, no analytical results are currently available for minimizers of the functional in the piecewise smooth setting, and this is the goal of this work. Our main result is the $\Gamma$-convergence of the Ambrosio-Tortorelli approximation of the Mumford-Shah functional for piecewise smooth approximations. This requires the introduction of an appropriate function space. As a consequence of our $\Gamma$-convergence result, we can infer the convergence of minimizers of the respective functionals.
The Longest Common Substring (LCS) and Longest Palindromic Substring (LPS) are classical problems in computer science, representing fundamental challenges in string processing. Both problems can be solved in linear time using a classical model of computation, by means of very similar algorithms, both relying on the use of suffix trees. Very recently, two sublinear algorithms for LCS and LPS in the quantum query model have been presented by Le Gall and Seddighin~\cite{GallS23}, requiring $\tilde{\mathcal{O}}(n^{5/6})$ and $\tilde{\mathcal{O}}(\sqrt{n})$ queries, respectively. However, while the query model is fascinating from a theoretical standpoint, its practical applicability becomes limited when it comes to crafting algorithms meant for actual execution on real hardware. In this paper we present, for the first time, a $\tilde{\mathcal{O}}(\sqrt{n})$ quantum algorithm for both LCS and LPS working in the circuit model of computation. Our solutions are simpler than previous ones and can be easily translated into quantum procedures. We also present actual implementations of the two algorithms as quantum circuits working in $\mathcal{O}(\sqrt{n}\log^5(n))$ and $\mathcal{O}(\sqrt{n}\log^4(n))$ time, respectively.
This paper presents a Multiple Kernel Learning (abbreviated as MKL) framework for the Support Vector Machine (SVM) with the $(0, 1)$ loss function. Some KKT-like first-order optimality conditions are provided and then exploited to develop a fast ADMM algorithm to solve the nonsmooth nonconvex optimization problem. Numerical experiments on real data sets show that the performance of our MKL-$L_{0/1}$-SVM is comparable with the one of the leading approaches called SimpleMKL developed by Rakotomamonjy, Bach, Canu, and Grandvalet [Journal of Machine Learning Research, vol. 9, pp. 2491-2521, 2008].
We study the time complexity of computing the $(\min,+)$ matrix product of two $n\times n$ integer matrices in terms of $n$ and the number of monotone subsequences the rows of the first matrix and the columns of the second matrix can be decomposed into. In particular, we show that if each row of the first matrix can be decomposed into at most $m_1$ monotone subsequences and each column of the second matrix can be decomposed into at most $m_2$ monotone subsequences such that all the subsequences are non-decreasing or all of them are non-increasing then the $(\min,+)$ product of the matrices can be computed in $O(m_1m_2n^{2.569})$ time. On the other hand, we observe that if all the rows of the first matrix are non-decreasing and all columns of the second matrix are non-increasing or {\em vice versa} then this case is as hard as the general one. Similarly, we also study the time complexity of computing the $(\min,+)$ convolution of two $n$-dimensional integer vectors in terms of $n$ and the number of monotone subsequences the two vectors can be decomposed into. We show that if the first vector can be decomposed into at most $m_1$ monotone subsequences and the second vector can be decomposed into at most $m_2$ subsequences such that all the subsequences of the first vector are non-decreasing and all the subsequences of the second vector are non-increasing or {\em vice versa} then their $(\min,+)$ convolution can be computed in $\tilde{O}(m_1m_2n^{1.5})$ time. On the other, the case when both vectors are non-decreasing or both of them are non-increasing is as hard as the general case.
The $k$-center problem is a classical clustering problem in which one is asked to find a partitioning of a point set $P$ into $k$ clusters such that the maximum radius of any cluster is minimized. It is well-studied. But what if we add up the radii of the clusters instead of only considering the cluster with maximum radius? This natural variant is called the $k$-min-sum-radii problem. It has become the subject of more and more interest in recent years, inspiring the development of approximation algorithms for the $k$-min-sum-radii problem in its plain version as well as in constrained settings. We study the problem for Euclidean spaces $\mathbb{R}^d$ of arbitrary dimension but assume the number $k$ of clusters to be constant. In this case, a PTAS for the problem is known (see Bandyapadhyay, Lochet and Saurabh, SoCG, 2023). Our aim is to extend the knowledge base for $k$-min-sum-radii to the domain of fair clustering. We study several group fairness constraints, such as the one introduced by Chierichetti et al. (NeurIPS, 2017). In this model, input points have an additional attribute (e.g., colors such as red and blue), and clusters have to preserve the ratio between different attribute values (e.g., have the same fraction of red and blue points as the ground set). Different variants of this general idea have been studied in the literature. To the best of our knowledge, no approximative results for the fair $k$-min-sum-radii problem are known, despite the immense amount of work on the related fair $k$-center problem. We propose a PTAS for the fair $k$-min-sum-radii problem in Euclidean spaces of arbitrary dimension for the case of constant $k$. To the best of our knowledge, this is the first PTAS for the problem. It works for different notions of group fairness.
We derive an intuitionistic version of G\"odel-L\"ob modal logic ($\sf{GL}$) in the style of Simpson, via proof theoretic techniques. We recover a labelled system, $\sf{\ell IGL}$, by restricting a non-wellfounded labelled system for $\sf{GL}$ to have only one formula on the right. The latter is obtained using techniques from cyclic proof theory, sidestepping the barrier that $\sf{GL}$'s usual frame condition (converse well-foundedness) is not first-order definable. While existing intuitionistic versions of $\sf{GL}$ are typically defined over only the box (and not the diamond), our presentation includes both modalities. Our main result is that $\sf{\ell IGL}$ coincides with a corresponding semantic condition in birelational semantics: the composition of the modal relation and the intuitionistic relation is conversely well-founded. We call the resulting logic $\sf{IGL}$. While the soundness direction is proved using standard ideas, the completeness direction is more complex and necessitates a detour through several intermediate characterisations of $\sf{IGL}$.
Querying cohesive subgraphs on temporal graphs (e.g., social network, finance network, etc.) with various conditions has attracted intensive research interests recently. In this paper, we study a novel Temporal $(k,\mathcal{X})$-Core Query (TXCQ) that extends a fundamental Temporal $k$-Core Query (TCQ) proposed in our conference paper by optimizing or constraining an arbitrary metric $\mathcal{X}$ of $k$-core, such as size, engagement, interaction frequency, time span, burstiness, periodicity, etc. Our objective is to address specific TXCQ instances with conditions on different $\mathcal{X}$ in a unified algorithm framework that guarantees scalability. For that, this journal paper proposes a taxonomy of measurement $\mathcal{X}(\cdot)$ and achieve our objective using a two-phase framework while $\mathcal{X}(\cdot)$ is time-insensitive or time-monotonic. Specifically, Phase 1 still leverages the query processing algorithm of TCQ to induce all distinct $k$-cores during a given time range, and meanwhile locates the "time zones" in which the cores emerge. Then, Phase 2 conducts fast local search and $\mathcal{X}$ evaluation in each time zone with respect to the time insensitivity or monotonicity of $\mathcal{X}(\cdot)$. By revealing two insightful concepts named tightest time interval and loosest time interval that bound time zones, the redundant core induction and unnecessary $\mathcal{X}$ evaluation in a zone can be reduced dramatically. Our experimental results demonstrate that TXCQ can be addressed as efficiently as TCQ, which achieves the latest state-of-the-art performance, by using a general algorithm framework that leaves $\mathcal{X}(\cdot)$ as a user-defined function.
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