We study the problem of deterministically exploring an undirected and initially unknown graph with $n$ vertices either by a single agent equipped with a set of pebbles, or by a set of collaborating agents. The vertices of the graph are unlabeled and cannot be distinguished by the agents, but the edges incident to a vertex have locally distinct labels. The graph is explored when all vertices have been visited by at least one agent. In this setting, it is known that for a single agent without pebbles $Θ(\log n)$ bits of memory are necessary and sufficient to explore any graph with at most $n$ vertices. We are interested in how the memory requirement decreases as the agent may mark vertices by dropping and retrieving distinguishable pebbles, or when multiple agents jointly explore the graph. We give tight results for both questions showing that for a single agent with constant memory $Θ(\log \log n)$ pebbles are necessary and sufficient for exploration. We further prove that the same bound holds for the number of collaborating agents needed for exploration.
For the upper bound, we devise an algorithm for a single agent with constant memory that explores any $n$-vertex graph using $\mathcal{O}(\log \log n)$ pebbles, even when $n$ is unknown. The algorithm terminates after polynomial time and returns to the starting vertex. Since an additional agent is at least as powerful as a pebble, this implies that $\mathcal{O}(\log \log n)$ agents with constant memory can explore any $n$-vertex graph. For the lower bound, we show that the number of agents needed for exploring any graph of size $n$ is already $Ω(\log \log n)$ when we allow each agent to have at most $\mathcal{O}( \log n ^{1-\varepsilon})$ bits of memory for any $\varepsilon>0$. This also implies that a single agent with sublogarithmic memory needs $Θ(\log \log n)$ pebbles to explore any $n$-vertex graph.
For an undirected tree with $n$ edges labelled by single letters, we consider its substrings, which are labels of the simple paths between pairs of nodes. We prove that there are $O(n^{1.5})$ different palindromic substrings. This solves an open problem of Brlek, Lafrenière, and Provençal (DLT 2015), who gave a matching lower-bound construction. Hence, we settle the tight bound of $Θ(n^{1.5})$ for the maximum palindromic complexity of trees. For standard strings, i.e., for paths, the palindromic complexity is $n+1$. We also propose $O(n^{1.5} \log{n})$-time algorithm for reporting all distinct…
This work studies the generalized Moran process, as introduced by Lieberman et al. [Nature, 433:312-316, 2005]. We introduce the parameterized notions of selective amplifiers and selective suppressors of evolution, i.e. of networks (graphs) with many "strong starts" and many "weak starts" for the mutant, respectively. We first prove the existence of strong selective amplifiers and of (quite) strong selective suppressors. Furthermore we provide strong upper bounds and almost tight lower bounds (by proving the "Thermal Theorem") for the traditional notion of fixation probability of Lieberman et …
Fine-grained reductions have established equivalences between many core problems with $\tilde{O}(n^3)$-time algorithms on $n$-node weighted graphs, such as Shortest Cycle, All-Pairs Shortest Paths (APSP), Radius, Replacement Paths, Second Shortest Paths, and so on. These problems also have $\tilde{O}(mn)$-time algorithms on $m$-edge $n$-node weighted graphs, and such algorithms have wider applicability. Are these $mn$ bounds optimal when $m \ll n^2$? Starting from the hypothesis that the minimum weight $(2\ell+1)$-Clique problem in edge weighted graphs requires $n^{2\ell+1-o(1)}$ time, we prov…
We study maximal identifiability, a measure recently introduced in Boolean Network Tomography to characterize networks' capability to localize failure nodes in end-to-end path measurements. We prove tight upper and lower bounds on the maximal identifiability of failure nodes for specific classes of network topologies, such as trees and $d$-dimensional grids, in both directed and undirected cases. We prove that directed $d$-dimensional grids with support $n$ have maximal identifiability $d$ using $2d(n-1)+2$ monitors; and in the undirected case we show that $2d$ monitors suffice to get identifi…
Among the most fundamental graph parameters is the Diameter, the largest distance between any pair of vertices. Computing the Diameter of a graph with $m$ edges requires $m^{2-o(1)}$ time under the Strong Exponential Time Hypothesis (SETH), which can be prohibitive for very large graphs, so efficient approximation algorithms for Diameter are desired. There is a folklore algorithm that gives a $2$-approximation for Diameter in $\tilde{O}(m)$ time. Additionally, a line of work concludes with a $3/2$-approximation algorithm for Diameter in weighted directed graphs that runs in $\tilde{O}(m^{3/2})…
Evolutionary graph theory studies the evolutionary dynamics in a population structure given as a connected graph. Each node of the graph represents an individual of the population, and edges determine how offspring are placed. We consider the classical birth-death Moran process where there are two types of individuals, namely, the residents with fitness 1 and mutants with fitness r. The fitness indicates the reproductive strength. The evolutionary dynamics happens as follows: in the initial step, in a population of all resident individuals a mutant is introduced, and then at each step, an indi…
This paper focuses on finding a spanning tree of a graph to maximize the number of its internal vertices. We present an approximation algorithm for this problem which can achieve a performance ratio $\frac{4}{3}$ on undirected simple graphs. This improves upon the best known approximation algorithm with performance ratio $\frac{5}{3}$ before. Our algorithm benefits from a new observation for bounding the number of internal vertices of a spanning tree, which reveals that a spanning tree of an undirected simple graph has less internal vertices than the edges a maximum path-cycle cover of that gr…
We consider the influence maximization problem (selecting $k$ seeds in a network maximizing the expected total influence) on undirected graphs under the linear threshold model. On the one hand, we prove that the greedy algorithm always achieves a $(1 - (1 - 1/k)^k + Ω(1/k^3))$-approximation, showing that the greedy algorithm does slightly better on undirected graphs than the generic $(1- (1 - 1/k)^k)$ bound which also applies to directed graphs. On the other hand, we show that substantial improvement on this bound is impossible by presenting an example where the greedy algorithm can obtain at …
We study the structure of solutions to linear programming formulations for the traveling salesperson problem (TSP). We perform a detailed analysis of the support of the subtour elimination linear programming relaxation, which leads to algorithms that find 2-matchings with few components in polynomial time. The number of components directly leads to integrality gap upper bounds for the TSP with distances one and two, for both undirected and directed graphs. Our main results concern the subtour elimination relaxation with one additional cutting plane inequality: - For undirected instances we obt…
Some of the most fundamental and well-studied graph parameters are the Diameter (the largest shortest paths distance) and Radius (the smallest distance for which a "center" node can reach all other nodes). The natural and important $ST$-variant considers two subsets $S$ and $T$ of the vertex set and lets the $ST$-diameter be the maximum distance between a node in $S$ and a node in $T$, and the $ST$-radius be the minimum distance for a node of $S$ to reach all nodes of $T$. The bichromatic variant is the special case in which $S$ and $T$ partition the vertex set. In this paper we present a comp…
For a graph class $Π$, the $Π$-Vertex Deletion problem has as input an undirected graph $G=(V,E)$ and an integer $k$ and asks whether there is a set of at most $k$ vertices that can be deleted from $G$ such that the resulting graph is a member of $Π$. By a classic result of Lewis and Yannakakis [J. Comput. Syst. Sci. '80], $Π$-Vertex Deletion is NP-hard for all hereditary properties $Π$. We adapt the original NP-hardness construction to show that under the Exponential Time Hypothesis (ETH) tight complexity results can be obtained. We show that $Π$-Vertex Deletion does not admit a $2^{o(n)}$-ti…
(see paper for full abstract) We show that the Edge-Disjoint Paths problem is W[1]-hard parameterized by the number $k$ of terminal pairs, even when the input graph is a planar directed acyclic graph (DAG). This answers a question of Slivkins (ESA '03, SIDMA '10). Moreover, under the Exponential Time Hypothesis (ETH), we show that there is no $f(k)\cdot n^{o(k)}$ algorithm for Edge-Disjoint Paths on planar DAGs, where $k$ is the number of terminal pairs, $n$ is the number of vertices and $f$ is any computable function. Our hardness holds even if both the maximum in-degree and maximum out-degre…
The Cheeger inequality for undirected graphs, which relates the conductance of an undirected graph and the second smallest eigenvalue of its normalized Laplacian, is a cornerstone of spectral graph theory. The Cheeger inequality has been extended to directed graphs and hypergraphs using normalized Laplacians for those, that are no longer linear but piecewise linear transformations. In this paper, we introduce the notion of a submodular transformation $F:\{0,1\}^n \to \mathbb{R}^m$, which applies $m$ submodular functions to the $n$-dimensional input vector, and then introduce the notions of its…
This paper is about the problem of finding a shortest $s$-$t$ path using at most $h$ edges in edge-weighted graphs. The Bellman--Ford algorithm solves this problem in $O(hm)$ time, where $m$ is the number of edges. We show that this running time is optimal, up to subpolynomial factors, under popular fine-grained complexity assumptions. More specifically, we show that under the APSP Hypothesis the problem cannot be solved faster already in undirected graphs with non-negative edge weights. This lower bound holds even restricted to graphs of arbitrary density and for arbitrary $h \in O(\sqrt{m})$…
Flows over time are used to model many real-world logistic and routing problems. The networks underlying such problems -- streets, tracks, etc. -- are inherently undirected and directions are only imposed on them to reduce the danger of colliding vehicles and similar problems. Thus the question arises, what influence the orientation of the network has on the network flow over time problem that is being solved on the oriented network. In the literature, this is also referred to as the contraflow or lane reversal problem. We introduce and analyze the price of orientation: How much flow is lost i…
We prove essentially tight lower bounds, conditionally to the Exponential Time Hypothesis, for two fundamental but seemingly very different cutting problems on surface-embedded graphs: the Shortest Cut Graph problem and the Multiway Cut problem. A cut graph of a graph $G$ embedded on a surface $S$ is a subgraph of $G$ whose removal from $S$ leaves a disk. We consider the problem of deciding whether an unweighted graph embedded on a surface of genus $g$ has a cut graph of length at most a given value. We prove a time lower bound for this problem of $n^{Ω(g/\log g)}$ conditionally to ETH. In oth…
We introduce a set of techniques that allow for efficiently generating many independent random walks in the Massive Parallel Computation (MPC) model with space per machine strongly sublinear in the number of vertices. In this space-per-machine regime, many natural approaches to graph problems struggle to overcome the $Θ(\log n)$ MPC round complexity barrier. Our techniques enable breaking this barrier for PageRank---one of the most important applications of random walks---even in more challenging directed graphs, and for approximate bipartiteness and expansion testing. In the undirected case, …
It is $\mathsf{NP}$-hard to determine the minimum number of branching vertices needed in a single-source distance-preserving subgraph of an undirected graph. We show that this problem can be solved in polynomial time if the input graph is an interval graph. In earlier work, it was shown that every interval graph with $k$ terminal vertices admits an all-pairs distance-preserving subgraph with $O(k\log k)$ branching vertices. We consider graphs that can be expressed as the strong product of two interval graphs, and present a polynomial time algorithm that takes such a graph with $k$ terminals as…
We give almost tight conditional lower bounds on the running time of the kHyperPath problem. Given an $r$-uniform hypergraph for some integer $r$, kHyperPath seeks a tight path of length $k$. That is, a sequence of $k$ nodes such that every consecutive $r$ of them constitute a hyperedge in the graph. This problem is a natural generalization of the extensively-studied kPath problem in graphs. We show that solving kHyperPath in time $O^*(2^{(1-γ)k})$ where $γ>0$ is independent of $r$ is probably impossible. Specifically, it implies that Set Cover on $n$ elements can be solved in time $O^*(2^{(1 …
We consider the problem of estimating the graph size, where one is given only local access to the graph. We formally define a query model in which one starts with a \emph{seed} node and is allowed to make queries about neighbours of nodes that have already been seen. In the case of undirected graphs, an estimator of Katzir et al. (2014) based on a sample from the stationary distribution $π$ uses $O\left(\frac{1}{\|π\|_2} + \text{davg}\right)$ queries, we prove that this is tight. In addition, we establish this as a lower bound even when the algorithm is allowed to crawl the graph arbitrarily, …
Given an {\em unweighted} undirected graph $G = (V,E)$, and a pair of parameters $ε> 0$, $β= 1,2,\ldots$, a subgraph $G' =(V,H)$, $H \subseteq E$, of $G$ is a {\em $(1+ε,β)$-spanner} (aka, a {\em near-additive spanner}) of $G$ if for every $u,v \in V$, $$d_{G'}(u,v) \le (1+ε)d_G(u,v) + β~.$$ It was shown in \cite{EP01} that for any $n$-vertex $G$ as above, and any $ε> 0$ and $κ= 1,2,\ldots$, there exists a $(1+ε,β)$-spanner $G'$ with $O_{ε,κ}(n^{1+1/κ})$ edges, with $$β= β_{EP} = \left({{\log κ} \over ε}\right)^{\log κ- 2}~.$$ This bound remains state-of-the-art, and its dependence on $ε$ (for…
