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We consider the setup of a constrained optimization problem with two agents $E_1$ and $E_2$ who jointly wish to learn the optimal solution set while keeping their feasible sets $\mathcal{P}_1$ and $\mathcal{P}_2$ private from each other. The objective function $f$ is globally known and each feasible set is a collection of points from a global alphabet. We adopt a sequential symmetric private information retrieval (SPIR) framework where one of the agents (say $E_1$) privately checks in $\mathcal{P}_2$, the presence of candidate solutions of the problem constrained to $\mathcal{P}_1$ only, while learning no further information on $\mathcal{P}_2$ than the solution alone. Further, we extract an information theoretically private threshold PSI (ThPSI) protocol from our scheme and characterize its download cost. We show that, compared to privately acquiring the feasible set $\mathcal{P}_1\cap \mathcal{P}_2$ using an SPIR-based private set intersection (PSI) protocol, and finding the optimum, our scheme is better as it incurs less information leakage and less download cost than the former. Over all possible uniform mappings of $f$ to a fixed range of values, our scheme outperforms the former with a high probability.

We consider the k-diameter clustering problem, where the goal is to partition a set of points in a metric space into $k$ clusters, minimizing the maximum distance between any two points in the same cluster. In general metrics, k-diameter is known to be NP-hard, while it has a $2$-approximation algorithm (Gonzalez'85). Complementing this algorithm, it is known that k-diameter is NP-hard to approximate within a factor better than $2$ in the $\ell_1$ and $\ell_\infty$ metrics, and within a factor of $1.969$ in the $\ell_2$ metric (Feder-Greene'88). When $k\geq 3$ is fixed, k-diameter remains NP-hard to approximate within a factor better than $2$ in the $\ell_\infty$ metric (Megiddo'90). However, its approximability in this setting has not previously been studied in the $\ell_1$ and $\ell_2$ metrics, though a $1.415$-approximation algorithm in the $\ell_2$ metric follows from a known result (Badoiu et al.'02). In this paper, we address the remaining gap by showing new hardness of approximation results that hold even when $k=3$. Specifically, we prove that 3-diameter is NP-hard to approximate within a factor better than $1.5$ in the $\ell_1$ metric, and within a factor of $1.304$ in the $\ell_2$ metric.

Let $P$ be a set of at most $n$ points and let $R$ be a set of at most $n$ geometric ranges, such as for example disks or rectangles, where each $p \in P$ has an associated supply $s_{p} > 0$, and each $r \in R$ has an associated demand $d_{r} > 0$. A (many-to-many) matching is a set $\mathcal{A}$ of ordered triples $(p,r,a_{pr}) \in P \times R \times \mathbb{R}_{>0}$ such that $p \in r$ and the $a_{pr}$'s satisfy the constraints given by the supplies and demands. We show how to compute a maximum matching, that is, a matching maximizing $\sum_{(p,r,a_{pr}) \in \mathcal{A}} a_{pr}$. Using our techniques, we can also solve minimum bottleneck problems, such as computing a perfect matching between a set of $n$ red points $P$ and a set of $n$ blue points $Q$ that minimizes the length of the longest edge. For the $L_\infty$-metric, we can do this in time $O(n^{1+\varepsilon})$ in any fixed dimension, for the $L_2$-metric in the plane in time $O(n^{4/3 + \varepsilon})$, for any $\varepsilon > 0$.

We prove that isomorphism of tournaments of twin width at most $k$ can be decided in time $k^{O(\log k)}n^{O(1)}$. This implies that the isomorphism problem for classes of tournaments of bounded or moderately growing twin width is in polynomial time. By comparison, there are classes of undirected graphs of bounded twin width that are isomorphism complete, that is, the isomorphism problem for the classes is as hard as the general graph isomorphism problem. Twin width is a graph parameter that has been introduced only recently (Bonnet et al., FOCS 2020), but has received a lot of attention in structural graph theory since then. On directed graphs, it is functionally smaller than clique width. We prove that on tournaments (but not on general directed graphs) it is also functionally smaller than directed tree width (and thus, the same also holds for cut width and directed path width). Hence, our result implies that tournament isomorphism testing is also fixed-parameter tractable when parameterized by any of these parameters. Our isomorphism algorithm heavily employs group-theoretic techniques. This seems to be necessary: as a second main result, we show that the combinatorial Weisfeiler-Leman algorithm does not decide isomorphism of tournaments of twin width at most 35 if its dimension is $o(\sqrt n)$. (Throughout this abstract, $n$ is the order of the input graphs.)

Consider that there are $k\le n$ agents in a simple, connected, and undirected graph $G=(V,E)$ with $n$ nodes and $m$ edges. The goal of the dispersion problem is to move these $k$ agents to distinct nodes. Agents can communicate only when they are at the same node, and no other means of communication such as whiteboards are available. We assume that the agents operate synchronously. We consider two scenarios: when all agents are initially located at any single node (rooted setting) and when they are initially distributed over any one or more nodes (general setting). Kshemkalyani and Sharma presented a dispersion algorithm for the general setting, which uses $O(m_k)$ time and $\log(k+\delta)$ bits of memory per agent [OPODIS 2021]. Here, $m_k$ is the maximum number of edges in any induced subgraph of $G$ with $k$ nodes, and $\delta$ is the maximum degree of $G$. This algorithm is the fastest in the literature, as no algorithm with $o(m_k)$ time has been discovered even for the rooted setting. In this paper, we present faster algorithms for both the rooted and general settings. First, we present an algorithm for the rooted setting that solves the dispersion problem in $O(k\log \min(k,\delta))=O(k\log k)$ time using $O(\log \delta)$ bits of memory per agent. Next, we propose an algorithm for the general setting that achieves dispersion in $O(k (\log k)\cdot (\log \min(k,\delta))=O(k \log^2 k)$ time using $O(\log (k+\delta))$ bits. Finally, for the rooted setting, we give a time-optimal, i.e.,$O(k)$-time, algorithm with $O(\delta)$ bits of space per agent.

We study the problem of efficiently answering strong connectivity queries under two vertex failures. Given a directed graph $G$ with $n$ vertices, we provide a data structure with $O(nh)$ space and $O(h)$ query time, where $h$ is the height of a decomposition tree of $G$ into strongly connected subgraphs. This immediately implies data structures with $O(n \log{n})$ space and $O(\log{n})$ query time for graphs of constant treewidth, and $O(n^{3/2})$ space and $O(\sqrt{n})$ query time for planar graphs. For general directed graphs, we give a refined version of our data structure that achieves $O(n\sqrt{m})$ space and $O(\sqrt{m})$ query time, where $m$ is the number of edges of the graph. We also provide some simple BFS-based heuristics that seem to work remarkably well in practice. In the experimental part, we first evaluate various methods to construct a decomposition tree with small height $h$ in practice. Then we provide efficient implementations of our data structures, and evaluate their empirical performance by conducting an extensive experimental study on graphs taken from real-world applications.

The classical way of extending an $[n, k, d]$ linear code $\C$ is to add an overall parity-check coordinate to each codeword of the linear code $\C$. This extended code, denoted by $\overline{\C}(-\bone)$ and called the standardly extended code of $\C$, is a linear code with parameters $[n+1, k, \bar{d}]$, where $\bar{d}=d$ or $\bar{d}=d+1$. This is one of the two extending techniques for linear codes in the literature. The standardly extended codes of some families of binary linear codes have been studied to some extent. However, not much is known about the standardly extended codes of nonbinary codes. For example, the minimum distances of the standardly extended codes of the nonbinary Hamming codes remain open for over 70 years. The first objective of this paper is to introduce the nonstandardly extended codes of a linear code and develop some general theory for this type of extended linear codes. The second objective is to study this type of extended codes of a number of families of linear codes, including cyclic codes and nonbinary Hamming codes. Four families of distance-optimal or dimension-optimal linear codes are obtained with this extending technique. The parameters of certain extended codes of many families of linear codes are settled in this paper.

The Djokovi\'{c}-Winkler relation $\Theta$ is a binary relation defined on the edge set of a given graph that is based on the distances of certain vertices and which plays a prominent role in graph theory. In this paper, we explore the relatively uncharted ``reflexive complement'' $\overline\Theta$ of $\Theta$, where $(e,f)\in \overline\Theta$ if and only if $e=f$ or $(e,f)\notin \Theta$ for edges $e$ and $f$. We establish the relationship between $\overline\Theta$ and the set $\Delta_{ef}$, comprising the distances between the vertices of $e$ and $f$ and shed some light on the intricacies of its transitive closure $\overline\Theta^*$. Notably, we demonstrate that $\overline\Theta^*$ exhibits multiple equivalence classes only within a restricted subclass of complete multipartite graphs. In addition, we characterize non-trivial relations $R$ that coincide with $\overline\Theta$ as those where the graph representation is disconnected, with each connected component being the (join of) Cartesian product of complete graphs. The latter results imply, somewhat surprisingly, that knowledge about the distances between vertices is not required to determine $\overline\Theta^*$. Moreover, $\overline\Theta^*$ has either exactly one or three equivalence classes.

We revisit the well-known Curve Shortening Flow for immersed curves in the $d$-dimensional Euclidean space. We exploit a fundamental structure of the problem to derive a new global construction of a solution, that is, a construction that is valid for all times and is insensitive to singularities. The construction is characterized by discretization in time and the approximant, while still exhibiting the possibile formation of finitely many singularities at a finite set of singular times, exists globally and is well behaved and simpler to analyze than a solution of the CSF. A solution of the latter is obtained in the limit. Estimates for a natural (geometric) norm involving length and total absolute curvature allow passage to the limit. Many classical qualitative results about the flow can be recovered by exploiting the simplicity of the approximant and new ones can be proved. The construction also suggests a numerical procedure for the computation of the flow which proves very effective as demonstrated by a series of numerical experiments scattered throughout the paper.

Let $G$ be a graph with $n$ vertices and $m$ edges. One of several hierarchies towards the stability number of $G$ is the exact subgraph hierarchy (ESH). On the first level it computes the Lov\'{a}sz theta function $\vartheta(G)$ as semidefinite program (SDP) with a matrix variable of order $n+1$ and $n+m+1$ constraints. On the $k$-th level it adds all exact subgraph constraints (ESC) for subgraphs of order $k$ to the SDP. An ESC ensures that the submatrix of the matrix variable corresponding to the subgraph is in the correct polytope. By including only some ESCs into the SDP the ESH can be exploited computationally. In this paper we introduce a variant of the ESH that computes $\vartheta(G)$ through an SDP with a matrix variable of order $n$ and $m+1$ constraints. We show that it makes sense to include the ESCs into this SDP and introduce the compressed ESH (CESH) analogously to the ESH. Computationally the CESH seems favorable as the SDP is smaller. However, we prove that the bounds based on the ESH are always at least as good as those of the CESH. In computational experiments sometimes they are significantly better. We also introduce scaled ESCs (SESCs), which are a more natural way to include exactness constraints into the smaller SDP and we prove that including an SESC is equivalent to including an ESC for every subgraph.