Non-crossing shortest paths in planar graphs with applications to max flow, and path graphs

This thesis is concerned with non-crossing shortest paths in planar graphs with applications to st-max flow vitality and path graphs. In the first part we deal with non-crossing shortest paths in a plane graph G, i.e., a planar graph with a fixed planar embedding, whose extremal vertices lie on the same face of G. The first two results are the computation of the lengths of the non-crossing shortest paths knowing their union, and the computation of the union in the unweighted case. Both results require linear time and we use them to describe an efficient algorithm able to give an additive guaranteed approximation of edge and vertex vitalities with respect to the st-max flow in undirected planar graphs, that is the max flow decrease when the edge/vertex is removed from the graph. Indeed, it is well-known that the st-max flow in an undirected planar graph can be reduced to a problem of non-crossing shortest paths in the dual graph. We conclude this part by showing that the union of non-crossing shortest paths in a plane graph can be covered with four forests so that each path is contained in at least one forest. In the second part of the thesis we deal with path graphs and directed path graphs, where a (directed) path graph is the intersection graph of paths in a (directed) tree. We introduce a new characterization of path graphs that simplifies the existing ones in the literature. This characterization leads to a new list of local forbidden subgraphs of path graphs and to a new algorithm able to recognize path graphs and directed path graphs. This algorithm is more intuitive than the existing ones and does not require sophisticated data structures

A New Algorithm to Recognize Path Graphs

A Path Graph is the intersection graph of vertex paths in an undirected tree. We present a new algorithm to recognize Path Graphs. It has the same worst-case time complexity as the faster recognition algorithm known so far [A.A. Sch\"affer, A faster algorithm to recognize undirected path graphs, Discrete Applied Mathematics, 43 (1993), pp. 261-295.] but it has an easier and more intuitive implementation based on a new characterization of Path Graphs [N. Apollonio and L. Balzotti, A New Characterization of Path Graphs, CoRR, abs/1911.09069, (2019).].Comment: 10 pages, 2 figure

Two New Characterizations of Path Graphs

Path graphs are intersection graphs of paths in a tree. We start from the characterization of path graphs by Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection Graphs of Paths in a Tree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181] and we reduce it to some 2-colorings subproblems, obtaining the first characterization that directly leads to a polynomial recognition algorithm. Then we introduce the collection of the attachedness graphs of a graph and we exhibit a list of minimal forbidden 2-edge colored subgraphs in each of the attachedness graph.Comment: 18 pages, 6 figure

Computing Lengths of Shortest Non-Crossing Paths in Planar Graphs

Given a plane undirected graph $G$ with non-negative edge weights and a set of $k$ terminal pairs on the external face, it is shown in Takahashi et al., (Algorithmica, 16, 1996, pp. 339-357) that the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs (if they exist) can be computed in $O(n \log n)$ worst-case time, where $n$ is the number of vertices of $G$. This technique only applies when the union $U$ of the computed shortest paths is a forest. We show that given a plane undirected weighted graph $U$ and a set of $k$ terminal pairs on the external face, it is always possible to compute the lengths of $k$ non-crossing shortest paths joining the $k$ terminal pairs in linear worst-case time, provided that the graph $U$ is the union of $k$ shortest paths, possibly containing cycles. Moreover, each shortest path $\pi$ can be listed in $O(\ell+\ell\log\lceil{\frac{k}{\ell}}\rceil)$, where $\ell$ is the number of edges in $\pi$. As a consequence, the problem of computing multi-terminal distances in a plane undirected weighted graph can always be solved in $O(n \log k)$ worst-case time in the general case.Comment: 17 pages, 11 figure

INTEND: Intent-Based Data Operation in the Computing Continuum

The European Commission (EC) Digital Decade strategy to gain by 2030 autonomy in the digital economy requires more and more data to be processed in the Cloud-Edge-IoT computing continuum, instead of only in the central cloud. This requires advanced automation and intelligence of the continuum. At the same time, recent breakthroughs in Artificial Intelligence (AI) research have shown unprecedented results in handling creative tasks. Such human-like intelligence will eventually disrupt how people use the cloud and continuum. The European Union (EU) -funded project INTEND aims at bringing such human-like intelligence into the cognitive continuum, to achieve the novel concept of intent-based data operation. The project will deliver 11 novel software tools, which integrate into an INTEND toolbox. The outputs pave the way of migrating EU’s data industry from cloud to the continuum, and implement EC’s strategy of human-centric AI in the domain of data processing and computing continuum

A New Algorithm to Recognize Path Graphs and Directed Path Graphs

A path graph is the intersection graph of paths in a tree. A directed path graph is the intersection graph of paths in a directed tree. We present a new algorithm to recognize path graphs and directed path graphs. It has the same worst-case time complexity as the faster recognition algorithms known so far but it does not require complex data structures and it has an easy and intuitive implementation based on a new characterization of path graphs [N. Apollonio and L. Balzotti, A New Characterization of Path Graphs, arXiv:1911.09069, (2019).]

A New Characterization of Path Graphs

In this paper we give a "good characterization" of path graphs, namely, we prove that path graph membership is in $NPcap CoNP$ without resorting to existing polynomial time algorithms. The characterization is given in terms of the collection of the emph{attachedness graphs} of a graph, a novel device to deal with the connected components of a graph after the removal of clique separators. On the one hand, the characterization refines and simplifies the characterization of path graphs due to Monma and Wei [C.L.~Monma,~and~V.K.~Wei, Intersection {G}raphs of {P}aths in a {T}ree, J. Combin. Theory Ser. B, 41:2 (1986) 141--181], which we build on, by reducing a constrained vertex coloring problem defined on the emph{attachedness graphs} to a vertex 2-coloring problem on the same graphs. On the other hand, the characterization allows us to exhibit two exhaustive lists of obstructions to path graph membership in the form of minimal forbidden induced/partial 2-edge colored subgraphs in each of the emph{attachedness graphs}

Max Flow Vitality of Edges and Vertices in Undirected Planar Graphs

We study the problem of computing the vitality of edges and vertices with respect to $st$-max flow in undirected planar graphs, where the vitality of an edge/vertex in a graph with respect to max flow between two fixed vertices $s,t$ is defined as the max flow decrease when the edge/vertex is removed from the graph. We show that a $delta$ additive approximation of the vitality of all edges with capacity at most $c$ can be computed in $O(rac{c}{delta}n +n log log n)$ time, where $n$ is the size of the graph. A similar result is given for the vitality of vertices. All our algorithms work in $O(n)$ space

How Vulnerable is an Undirected Planar Graph with Respect to Max Flow

We study the problem of computing the vitality of edges and vertices with respect to the st-max flow in undirected planar graphs, where the vitality of an edge/vertex is the st-max flow decrease when the edge/vertex is removed from the graph. This allows us to establish the vulnerability of the graph with respect to the st-max flow. We give efficient algorithms to compute an additive guaranteed approximation of the vitality of edges and vertices in planar undirected graphs. We show that in the general case high vitality values are well approximated in time close to the time currently required to compute st-max flow O(n log log n). We also give improved, and sometimes optimal, results in the case of integer capacities. All our algorithms work in O(n) space