Related papers: Popular conjectures imply strong lower bounds for …
For many algorithmic problems on graphs of treewidth $t$, a standard dynamic programming approach gives an algorithm with time and space complexity $2^{\mathcal{O}(t)}\cdot n^{\mathcal{O}(1)}$. It turns out that when one considers the more…
We revisit classic string problems considered in the area of parameterized complexity, and study them through the lens of dynamic data structures. That is, instead of asking for a static algorithm that solves the given instance efficiently,…
We extend the well known bottleneck paths problem in two directions for directed unweighted (unit edge cost) graphs with positive real edge capacities. Firstly we narrow the problem domain and compute the bottleneck of the entire network in…
This paper presents a quantum algorithm for triangle finding over sparse graphs that improves over the previous best quantum algorithm for this task by Buhrman et al. [SIAM Journal on Computing, 2005]. Our algorithm is based on the recent…
Detecting if a graph contains a $k$-Clique is one of the most fundamental problems in computer science. The asymptotically fastest algorithm runs in time $O(n^{\omega k/3})$, where $\omega$ is the exponent of Boolean matrix multiplication.…
We revisit the classic problem of dynamically maintaining shortest paths between all pairs of nodes of a directed weighted graph. The allowed updates are insertions and deletions of nodes and their incident edges. We give worst-case…
The maximum matching problem in dynamic graphs subject to edge updates (insertions and deletions) has received much attention over the last few years; a multitude of approximation/time tradeoffs were obtained, improving upon the folklore…
Estimating the number of triangles in graph streams using a limited amount of memory has become a popular topic in the last decade. Different variations of the problem have been studied, depending on whether the graph edges are provided in…
We present new lower bounds that show that a polynomial number of passes are necessary for solving some fundamental graph problems in the streaming model of computation. For instance, we show that any streaming algorithm that finds a…
Connectivity related concepts are of fundamental interest in graph theory. The area has received extensive attention over four decades, but many problems remain unsolved, especially for directed graphs. A directed graph is 2-edge-connected…
We consider the following two algorithmic problems: given a graph $G$ and a subgraph $H\subseteq G$, decide whether $H$ is an isometric or a geodesically convex subgraph of $G$. It is relatively easy to see that the problems can be solved…
Finding, counting and/or listing triangles (three vertices with three edges) in large graphs are natural fundamental problems, which received recently much attention because of their importance in complex network analysis. We provide here a…
We revisit the 3SUM problem in the \emph{preprocessed universes} setting. We present an algorithm that, given three sets $A$, $B$, $C$ of $n$ integers, preprocesses them in quadratic time, so that given any subsets $A' \subseteq A$, $B'…
Here the All-pairs shortest path problem on weighted undirected sparse graphs is being considered. For the problem considered, we propose ``disassembly and assembly of a graph'' algorithm which uses a solution of the problem on a…
In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input…
We study the fully dynamic All-Pairs Shortest Paths (APSP) problem in undirected edge-weighted graphs. Given an $n$-vertex graph $G$ with non-negative edge lengths, that undergoes an online sequence of edge insertions and deletions, the…
A semi-streaming algorithm in dynamic graph streams processes any $n$-vertex graph by making one or multiple passes over a stream of insertions and deletions to edges of the graph and using $O(n \cdot \mbox{polylog}(n))$ space.…
General factors are a generalization of matchings. Given a graph $G$ with a set $\pi(v)$ of feasible degrees, called a degree constraint, for each vertex $v$ of $G$, the general factor problem is to find a (spanning) subgraph $F$ of $G$…
In this note, we consider the complexity of maintaining the longest increasing subsequence (LIS) of an array under (i) inserting an element, and (ii) deleting an element of an array. We show that no algorithm can support queries and updates…
We present a general technique, based on parametric search with some twist, for solving a variety of optimization problems on a set of semi-algebraic geometric objects of constant complexity. The common feature of these problems is that…