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The Voronoi diagrams technique was introduced by Cabello to compute the diameter of planar graphs in subquadratic time. We present novel applications of this technique in static, fault-tolerant, and partially-dynamic undirected unweighted planar graphs, as well as some new limitations. 1. In the static case, we give $n^{3+o(1)}/D^2$ and $\tilde{O}(n\cdot D^2)$ time algorithms for computing the diameter of a planar graph $G$ with diameter $D$. These are faster than the state of the art $\tilde{O}(n^{5/3})$ when $D<n^{1/3}$ or $D>n^{2/3}$. 2. In the fault-tolerant setting, we give an $n^{7/3+o(1)}$ time algorithm for computing the diameter of $G\setminus \{e\}$ for every edge $e$ in $G$ the replacement diameter problem. Compared to the naive $\tilde{O}(n^{8/3})$ time algorithm that runs the static algorithm for every edge. 3. In the incremental setting, where we wish to maintain the diameter while while adding edges, we present an algorithm with total running time $n^{7/3+o(1)}$. Compared to the naive $\tilde{O}(n^{8/3})$ time algorithm that runs the static algorithm after every update. 4. We give a lower bound (conditioned on the SETH) ruling out an amortized $O(n^{1-\varepsilon})$ update time for maintaining the diameter in *weighted* planar graph. The lower bound holds even for incremental or decremental updates. Our upper bounds are obtained by novel uses and manipulations of Voronoi diagrams. These include maintaining the Voronoi diagram when edges of the graph are deleted, allowing the sites of the Voronoi diagram to lie on a BFS tree level (rather than on boundaries of $r$-division), and a new reduction from incremental diameter to incremental distance oracles that could be of interest beyond planar graphs. Our lower bound is the first lower bound for a dynamic planar graph problem that is conditioned on the SETH.

Motivated by an application from geodesy, we introduce a novel clustering problem which is a $k$-center (or k-diameter) problem with a side constraint. For the side constraint, we are given an undirected connectivity graph $G$ on the input points, and a clustering is now only feasible if every cluster induces a connected subgraph in $G$. We call the resulting problems the connected $k$-center problem and the connected $k$-diameter problem. We prove several results on the complexity and approximability of these problems. Our main result is an $O(\log^2{k})$-approximation algorithm for the connected $k$-center and the connected $k$-diameter problem. For Euclidean metrics and metrics with constant doubling dimension, the approximation factor of this algorithm improves to $O(1)$. We also consider the special cases that the connectivity graph is a line or a tree. For the line we give optimal polynomial-time algorithms and for the case that the connectivity graph is a tree, we either give an optimal polynomial-time algorithm or a $2$-approximation algorithm for all variants of our model. We complement our upper bounds by several lower bounds.

Given a matroid $M=(E,{\cal I})$, and a total ordering over the elements $E$, a broken circuit is a circuit where the smallest element is removed and an NBC independent set is an independent set in ${\cal I}$ with no broken circuit. The set of NBC independent sets of any matroid $M$ define a simplicial complex called the broken circuit complex which has been the subject of intense study in combinatorics. Recently, Adiprasito, Huh and Katz showed that the face of numbers of any broken circuit complex form a log-concave sequence, proving a long-standing conjecture of Rota. We study counting and optimization problems on NBC bases of a generic matroid. We find several fundamental differences with the independent set complex: for example, we show that it is NP-hard to find the max-weight NBC base of a matroid or that the convex hull of NBC bases of a matroid has edges of arbitrary large length. We also give evidence that the natural down-up walk on the space of NBC bases of a matroid may not mix rapidly by showing that for some family of matroids it is NP-hard to count the number of NBC bases after certain conditionings.

This paper explores the Ziv-Zakai bound (ZZB), which is a well-known Bayesian lower bound on the Minimum Mean Squared Error (MMSE). First, it is shown that the ZZB holds without any assumption on the distribution of the estimand, that is, the estimand does not necessarily need to have a probability density function. The ZZB is then further analyzed in the high-noise and low-noise regimes and shown to always tensorize. Finally, the tightness of the ZZB is investigated under several aspects, such as the number of hypotheses and the usefulness of the valley-filling function. In particular, a sufficient and necessary condition for the tightness of the bound with continuous inputs is provided, and it is shown that the bound is never tight for discrete input distributions with a support set that does not have an accumulation point at zero.

We consider the problem of computing compact routing tables for a (weighted) planar graph $G:= (V, E,w)$ in the PRAM, CONGEST, and the novel HYBRID communication model. We present algorithms with polylogarithmic work and communication that are almost optimal in all relevant parameters, i.e., computation time, table sizes, and stretch. All algorithms are heavily randomized, and all our bounds hold w.h.p. For a given parameter $\epsilon>0$, our scheme computes labels of size $\widetilde{O}(\epsilon^{-1})$ and is computed in $\widetilde{O}(\epsilon^{-2})$ time and $\widetilde{O}(n)$ work in the PRAM and a HYBRID model and $\widetilde{O}(\epsilon^{-2} \cdot HD)$ (Here, $HD$ denotes the network's hop-diameter) time in CONGEST. The stretch of the resulting routing scheme is $1+\epsilon$. To achieve these results, we extend the divide-and-conquer framework of Li and Parter [STOC '19] and combine it with state-of-the-art distributed distance approximation algorithms [STOC '22]. Furthermore, we provide a distributed decomposition scheme, which may be of independent interest.

We prove that every $n$-vertex planar graph $G$ with no triangle sharing an edge with a 4-cycle has independence ratio $n/\alpha(G) \leq 4 - \varepsilon$ for $\varepsilon = 1/30$. This result implies that the same bound holds for 4-cycle-free planar graphs and planar graphs with no adjacent triangles and no triangle sharing an edge with a 5-cycle. For the latter case we strengthen the bound to $\varepsilon = 2/9$.

An edge $e$ of a graph $G$ is called deletable for some orientation $o$ if the restriction of $o$ to $G-e$ is a strong orientation. In 2021, H\"orsch and Szigeti proposed a new parameter for $3$-edge-connected graphs, called the Frank number, which refines $k$-edge-connectivity. The Frank number is defined as the minimum number of orientations of $G$ for which every edge of $G$ is deletable in at least one of them. They showed that every $3$-edge-connected graph has Frank number at most $7$ and that in case these graphs are also $3$-edge-colourable graphs the parameter is at most $3$. Here we strengthen the latter result by showing that such graphs have Frank number $2$, which also confirms a conjecture by Bar\'at and Bl\'aszik. Furthermore, we prove two sufficient conditions for cubic graphs to have Frank number $2$ and use them in an algorithm to computationally show that the Petersen graph is the only cyclically $4$-edge-connected cubic graph up to $36$ vertices having Frank number greater than $2$.

Our work concerns algorithms for an unweighted variant of Maximum Flow. In the All-Pairs Connectivity (APC) problem, we are given a graph $G$ on $n$ vertices and $m$ edges, and are tasked with computing the maximum number of edge-disjoint paths from $s$ to $t$ (equivalently, the size of a minimum $(s,t)$-cut) in $G$, for all pairs of vertices $(s,t)$. Although over undirected graphs APC can be solved in essentially optimal $n^{2+o(1)}$ time, the true time complexity of APC over directed graphs remains open: this problem can be solved in $\tilde{O}(m^\omega)$ time, where $\omega \in [2, 2.373)$ is the exponent of matrix multiplication, but no matching conditional lower bound is known. We study a variant of APC called the $k$-Bounded All Pairs Connectivity ($k$-APC) problem. In this problem, we are given an integer $k$ and graph $G$, and are tasked with reporting the size of a minimum $(s,t)$-cut only for pairs $(s,t)$ of vertices with a minimum cut size less than $k$ (if the minimum $(s,t)$-cut has size at least $k$, we just report it is "large" instead of computing the exact value). We present an algorithm solving $k$-APC in directed graphs in $\tilde{O}((kn)^\omega)$ time. This runtime is $\tilde O(n^\omega)$ for all $k$ polylogarithmic in $n$, which is essentially optimal under popular conjectures from fine-grained complexity. Previously, this runtime was only known for $k\le 2$ [Georgiadis et al., ICALP 2017]. We also study a variant of $k$-APC, the $k$-Bounded All-Pairs Vertex Connectivity ($k$-APVC) problem, which considers internally vertex-disjoint paths instead of edge-disjoint paths. We present an algorithm solving $k$-APVC in directed graphs in $\tilde{O}(k^2n^\omega)$ time. Previous work solved an easier version of the $k$-APVC problem in $\tilde O((kn)^\omega)$ time [Abboud et al, ICALP 2019].

A framework consists of an undirected graph $G$ and a matroid $M$ whose elements correspond to the vertices of $G$. Recently, Fomin et al. [SODA 2023] and Eiben et al. [ArXiV 2023] developed parameterized algorithms for computing paths of rank $k$ in frameworks. More precisely, for vertices $s$ and $t$ of $G$, and an integer $k$, they gave FPT algorithms parameterized by $k$ deciding whether there is an $(s,t)$-path in $G$ whose vertex set contains a subset of elements of $M$ of rank $k$. These algorithms are based on Schwartz-Zippel lemma for polynomial identity testing and thus are randomized, and therefore the existence of a deterministic FPT algorithm for this problem remains open. We present the first deterministic FPT algorithm that solves the problem in frameworks whose underlying graph $G$ is planar. While the running time of our algorithm is worse than the running times of the recent randomized algorithms, our algorithm works on more general classes of matroids. In particular, this is the first FPT algorithm for the case when matroid $M$ is represented over rationals. Our main technical contribution is the nontrivial adaptation of the classic irrelevant vertex technique to frameworks to reduce the given instance to one of bounded treewidth. This allows us to employ the toolbox of representative sets to design a dynamic programming procedure solving the problem efficiently on instances of bounded treewidth.

Over the years, much research involving mobile computational entities has been performed. From modeling actual microscopic (and smaller) robots, to modeling software processes on a network, many important problems have been studied in this context. Gathering is one such fundamental problem in this area. The problem of gathering $k$ robots, initially arbitrarily placed on the nodes of an $n$-node graph, asks that these robots coordinate and communicate in a local manner, as opposed to global, to move around the graph, find each other, and settle down on a single node as fast as possible. A more difficult problem to solve is gathering with detection, where once the robots gather, they must subsequently realize that gathering has occurred and then terminate. In this paper, we propose a deterministic approach to solve gathering with detection for any arbitrary connected graph that is faster than existing deterministic solutions for even just gathering (without the requirement of detection) for arbitrary graphs. In contrast to earlier work on gathering, it leverages the fact that there are more robots present in the system to achieve gathering with detection faster than those previous papers that focused on just gathering. The state of the art solution for deterministic gathering~[Ta-Shma and Zwick, TALG, 2014] takes $\Tilde{O}$$(n^5 \log \ell)$ rounds, where $\ell$ is the smallest label among robots and $\Tilde{O}$ hides a polylog factor. We design a deterministic algorithm for gathering with detection with the following trade-offs depending on how many robots are present: (i) when $k \geq \lfloor n/2 \rfloor + 1$, the algorithm takes $O(n^3)$ rounds, (ii) when $k \geq \lfloor n/3 \rfloor + 1$, the algorithm takes $O(n^4 \log n)$ rounds, and (iii) otherwise, the algorithm takes $\Tilde{O}$$(n^5)$ rounds. The algorithm is not required to know $k$, but only $n$.