Cover Time and Broadcast Time

We introduce a new technique for bounding the cover time of random walks by relating it to the runtime of randomized broadcast. In particular, we strongly confirm for dense graphs the intuition of Chandra et al. (1997) that ``the cover time of the graph is an appropriate metric for the performance of certain kinds of randomized broadcast algorithms\u27\u27. In more detail, our results are as follows: begin{itemize} item For any graph $G=(V,E)$ of size $n$ and minimum degree $delta$, we have $mathcal{R}(G)= mathcal{O}(frac{|E|}{delta} cdot log n)$, where $mathcal{R}(G)$ denotes the quotient of the cover time and broadcast time. This bound is tight for binary trees and tight up to logarithmic factors for many graphs including hypercubes, expanders and lollipop graphs. item For any $delta$-regular (or almost $delta$-regular) graph $G$ it holds that $mathcal{R}(G) = Omega(frac{delta^2}{n} cdot frac{1}{log n})$. Together with our upper bound on $mathcal{R}(G)$, this lower bound strongly confirms the intuition of Chandra et al.~for graphs with minimum degree $Theta(n)$, since then the cover time equals the broadcast time multiplied by $n$ (neglecting logarithmic factors). item Conversely, for any $delta$ we construct almost $delta$-regular graphs that satisfy $mathcal{R}(G) = mathcal{O}(max { sqrt{n},delta } cdot log^2 n)$. Since any regular expander satisfies $mathcal{R}(G) = Theta(n)$, the strong relationship given above does not hold if $delta$ is polynomially smaller than $n$. end{itemize} Our bounds also demonstrate that the relationship between cover time and broadcast time is much stronger than the known relationships between any of them and the mixing time (or the closely related spectral gap)

Deterministic and Probabilistic Binary Search in Graphs

We consider the following natural generalization of Binary Search: in a given undirected, positively weighted graph, one vertex is a target. The algorithm's task is to identify the target by adaptively querying vertices. In response to querying a node $q$, the algorithm learns either that $q$ is the target, or is given an edge out of $q$ that lies on a shortest path from $q$ to the target. We study this problem in a general noisy model in which each query independently receives a correct answer with probability $p > \frac{1}{2}$ (a known constant), and an (adversarial) incorrect one with probability $1-p$. Our main positive result is that when $p = 1$ (i.e., all answers are correct), $\log_2 n$ queries are always sufficient. For general $p$, we give an (almost information-theoretically optimal) algorithm that uses, in expectation, no more than $(1 - \delta)\frac{\log_2 n}{1 - H(p)} + o(\log n) + O(\log^2 (1/\delta))$ queries, and identifies the target correctly with probability at leas $1-\delta$. Here, $H(p) = -(p \log p + (1-p) \log(1-p))$ denotes the entropy. The first bound is achieved by the algorithm that iteratively queries a 1-median of the nodes not ruled out yet; the second bound by careful repeated invocations of a multiplicative weights algorithm. Even for $p = 1$, we show several hardness results for the problem of determining whether a target can be found using $K$ queries. Our upper bound of $\log_2 n$ implies a quasipolynomial-time algorithm for undirected connected graphs; we show that this is best-possible under the Strong Exponential Time Hypothesis (SETH). Furthermore, for directed graphs, or for undirected graphs with non-uniform node querying costs, the problem is PSPACE-complete. For a semi-adaptive version, in which one may query $r$ nodes each in $k$ rounds, we show membership in $\Sigma_{2k-1}$ in the polynomial hierarchy, and hardness for $\Sigma_{2k-5}$

Modified Linear Programming and Class 0 Bounds for Graph Pebbling

Given a configuration of pebbles on the vertices of a connected graph $G$, a \emph{pebbling move} removes two pebbles from some vertex and places one pebble on an adjacent vertex. The \emph{pebbling number} of a graph $G$ is the smallest integer $k$ such that for each vertex $v$ and each configuration of $k$ pebbles on $G$ there is a sequence of pebbling moves that places at least one pebble on $v$. First, we improve on results of Hurlbert, who introduced a linear optimization technique for graph pebbling. In particular, we use a different set of weight functions, based on graphs more general than trees. We apply this new idea to some graphs from Hurlbert's paper to give improved bounds on their pebbling numbers. Second, we investigate the structure of Class 0 graphs with few edges. We show that every $n$-vertex Class 0 graph has at least $\frac53n - \frac{11}3$ edges. This disproves a conjecture of Blasiak et al. For diameter 2 graphs, we strengthen this lower bound to $2n - 5$, which is best possible. Further, we characterize the graphs where the bound holds with equality and extend the argument to obtain an identical bound for diameter 2 graphs with no cut-vertex.Comment: 19 pages, 8 figure

Safe solutions for walks on graphs

In this thesis we study the concept of “safe solutions” in different problems whose solutions are walks on graphs. A safe solution to a problem X can be understood as a partial solution common to all solutions to problem X. In problems whose solutions are walks on graphs, safe solutions refer to walks common to all walks which are solutions to the problem. In this thesis, we focused on formulating four main graph traversal problems and finding characterizations for those walks contained in all their solutions. We give formulations for these graph traversal problems, we prove some of their combinatorial and structural properties, and we give safe and complete algorithms for finding their safe solutions based on their characterizations. We use the genome assembly problem and its applications as our main motivating example for finding safe solutions in these graph traversal problems. We begin by motivating and exemplifying the notion of safe solutions through a problem on s-t paths in undirected graphs with at least two non-trivial biconnected components S and T and with s ∈ S, t ∈ T . We continue by reviewing similar and related notions in other fields, especially in combinatorial optimization and previous work on the bioinformatics problem of genome assembly. We then proceed to characterize the safe solutions to the Eulerian cycle problem, where one must find a circular walk in a graph G which traverses each edge exactly once. We suggest a characterization for them by improving on (Nagarajan, Pop, JCB 2009) and a polynomial-time algorithm for finding them. We then study edge-covering circular walks in a graph G. We look at the characterization from (Tomescu, Medvedev, JCB 2017) for their safe solutions and their suggested polynomial-time algorithm and we show an optimal O(mn)-time algorithm that we proposed in (Cairo et al. CPM 2017). Finally, we generalize this to edge-covering collections of circular walks. We characterize safe solutions in an edge-covering setting and provide a polynomial-time algorithm for computing them. We suggested these originally in (Obscura et al. ALMOB 2018)