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In this paper we propose a linear-time certifying algorithm for the single-source shortest-path problem capable of verifying graphs with positive, negative, and zero arc weights. Previously proposed linear-time approaches only work for…
Let $G$ be an edge-weighted directed graph with $n$ vertices embedded on an orientable surface of genus $g$. We describe a simple deterministic lexicographic perturbation scheme that guarantees uniqueness of minimum-cost flows and shortest…
Let $G$ be a simple, undirected graph on the vertex set $V=\{1,2,\ldots ,n\}$ and let $A$ be the adjacency matrix of $G.$ A non-empty subset $ \{i_{1},i_{2},\ldots ,i_{k}\}$ of $V$ is called a driver set for $G$ if the system…
We consider the parameterized complexity of the problem of tracking shortest s-t paths in graphs, motivated by applications in security and wireless networks. Given an undirected and unweighted graph with a source s and a destination t,…
A metro-line crossing minimization problem is to draw multiple lines on an underlying graph that models stations and rail tracks so that the number of crossings of lines becomes minimum. It has several variations by adding restrictions on…
We are interested in embedding trees T with maximum degree at most four in a rectangular grid, such that the vertices of T correspond to grid points, while edges of T correspond to non-intersecting straight segments of the grid lines. Such…
We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less…
Let G=(V,E)(|V|=n and |E|=m) be an undirected graph with positive edge weights. Let P_{G}(s, t) be a shortest s-t path in G. Let l be the number of edges in P_{G}(s, t). The \emph{Edge Replacement Path} problem is to compute a shortest s-t…
The solution of the shortest path problem on a surface is not only a theoretical problem to be solved in the field of mathematics, but also problems that need to be solved in very different fields such as medicine, defense and construction…
The distance of a graph from being triangle-free is a fundamental graph parameter, counting the number of edges that need to be removed from a graph in order for it to become triangle-free. Its corresponding computational problem is the…
Test suites are designed to validate the operation of a system against requirements. One important aspect of a test suite design is to ensure that system operation logic is tested completely. A test suite should drive a system through all…
Covering problems are classical computational problems concerning whether a certain combinatorial structure 'covers' another. For example, the minimum vertex covering problem aims to find the smallest set of vertices in a graph so that each…
We present an algorithm for tests generation tools based on symbolic execution. The algorithm is supposed to help in situations, when a tool is repeatedly failing to cover some code by tests. The algorithm then provides the tool a necessary…
Computing the shortest path between two given locations in a road network is an important problem that finds applications in various map services and commercial navigation products. The state-of-the-art solutions for the problem can be…
The overwhelming majority of survivable (fault-tolerant) network design models assume a uniform scenario set. Such a scenario set assumes that every subset of the network resources (edges or vertices) of a given cardinality $k$ comprises a…
We study the SHORTEST PATH problem with positive disjunctive constraints from the perspective of parameterized complexity. For positive disjunctive constraints, there are certain pair of edges such that any feasible solution must contain at…
We show that the following variation of the single-source shortest path problem is NP-complete. Let a weighted, directed, acyclic graph $G=(V,E,w)$ with source and sink vertices $s$ and $t$ be given. Let in addition a mapping $f$ on $E$ be…
In this paper we consider shortest path problems in a directed graph where the transitions between nodes are subject to uncertainty. We use a minimax formulation, where the objective is to guarantee that a special destination state is…
The crossing number of a graph is the minimum number of edge crossings that a graph can have when drawn in the plane. Determining this number, known as the Crossing Number problem, is a celebrated problem in combinatorial optimization. It…
In this paper we study minimum cut and maximum flow problems on planar graphs, both in static and in dynamic settings. First, we present an algorithm that given an undirected planar graph computes the minimum cut between any two given…