Section: New Results

Cross-fertilising different computational approaches

Tree matching

We considered the following problem related to tree matching, that we called the Tree-Constrained Bipartite Matching problem. Given a bipartite graph $G=(V_1,V_2,E)$ with edge weights $w:E\to R^{+}$, a rooted tree $T_1$ on the set $V_1$ and a rooted tree $T_2$ on the set $V_2$, find a maximum weight matching $M$ in $G$, such that none of the matched nodes is an ancestor of another matched node in either of the trees [8] . This generalisation of the classical bipartite matching problem appears, for example, in the computational analysis of live cell video data. We showed that the problem is APX-hard and thus, unless P = NP, disproved a previous claim that it is solvable in polynomial time. Furthermore, we gave a 2-approximation algorithm based on a combination of the local ratio technique and a careful use of the structure of basic feasible solutions of a natural LP-relaxation, which we also show to have an integrality gap of $2-o\left(1\right)$. We then considered a natural generalisation of the problem, where trees are replaced by partially ordered sets (posets). We showed that the local ratio technique gives a $2k\sigma$- approximation for the $k$-dimensional matching generalisation of the problem, in which the maximum number of incomparable elements below (or above) any given element in each poset is bounded by $\sigma$. We finally gave an almost matching integrality gap example, and an inapproximability result showing that the dependence on $\sigma$ is most likely unavoidable.

Graph measures

We proposed a new algorithm that computes the radius and the diameter of a weakly connected digraph $G=(V,E)$, by finding bounds through heuristics and improving them until they are validated [5] . Although the worst-case running time is $O\left(\right|V\left|\right|E\left|\right)$, we experimentally showed that it performs much better in the case of real-world networks, finding the radius and diameter values after 10-100 BFSs instead of $\left|V\right|$ BFSs (independently of the value of |V|), and thus having running time $O\left(\right|E\left|\right)$ in practice. As far as we know, this is the first algorithm able to compute the diameter of weakly connected digraphs, apart from the naive algorithm, which runs in time $O\left(\right|V\left|\right|E\left|\right)$ performing a BFS from each node. In the particular cases of strongly connected directed or connected undirected graphs, we compared our algorithm with known approaches by performing experiments on a dataset composed by several real-world networks of different kinds. These experiments showed that, despite its generality, the new algorithm outperforms all previous methods, both in the radius and in the diameter computation, both in the directed and in the undirected case, both in average running time and in robustness. Finally, as an application example, we used the new algorithm to determine the solvability over time of the “Six Degrees of Kevin Bacon” game, and of the “Six Degrees of Wikipedia” game. As a consequence, we computed for the first time the exact value of the radius and the diameter of the whole Wikipedia digraph.

The closeness and the betweenness centralities are two well-known measures of importance of a vertex within a given complex network. Having high closeness or betweenness centrality can have positive impact on the vertex itself: hence, we considered the problem of determining how much a vertex can increase its centrality by creating a limited amount of new edges incident to it [40] . We first proved that this problem does not admit a polynomial-time approximation scheme (unless P=NP), and we then proposed a simple greedy approximation algorithm (with an almost tight approximation ratio), whose performance is then tested on synthetic graphs and real-world networks.

The (Gromov) hyperbolicity is a topological property of a graph, which has been recently applied in several different contexts, such as the design of routing schemes, network security, computational biology, the analysis of graph algorithms, and the classification of complex networks. Computing the hyperbolicity of a graph can be very time consuming: indeed, the best available algorithm has running-time $O\left(n^{3.69}\right)$, which is clearly prohibitive for big graphs. We provided a new and more efficient algorithm: although its worst-case complexity is $O\left(n^4\right)$, in practice it is much faster, allowing, for the first time, the computation of the hyperbolicity of graphs with up to 200,000 nodes [36] . We experimentally showed that the new algorithm drastically outperforms the best previously available algorithms, by analyzing a big dataset of real-world networks. Finally, we applied the new algorithm to compute the hyperbolicity of random graphs generated with the Erdös-Renyi model, the Chung-Lu model, and the Configuration Model.

Hypergraph problems

It had been previously proved independently and with different techniques that there exists an incremental output polynomial algorithm for the enumeration of the minimal edge dominating sets in graphs, i.e., minimal dominating sets in line graphs. We provided the first polynomial delay and polynomial space algorithm for the problem [42] . We proposed a new technique to enlarge the applicability of Berge's algorithm that is based on skipping hard parts of the enumeration by introducing a new search strategy. The new search strategy is given by a strong use of the structure of line graphs.

We also studied some average properties of hypergraphs and the average complexity of algorithms applied to hypergraphs under different probabilistic models [14] . Our approach is both theoretical and experimental since our goal is to obtain a random model that is able to capture the real-data complexity. Starting from a model that generalizes the Erdös-Renyi model and we obtain asymptotic estimations on the average number of transversals, irredundants and minimal transversals in a random hypergraph. We use those results to obtain an upper bound on the average complexity of algorithms to generate the minimal transversals of a hypergraph. Then we make our random model more complex in order to bring it closer to real-data and identify cases where the average number of minimal transversals is at most polynomial, quasi-polynomial or exponential.

The hypergraph transversal problem has been intensively studied, both from a theoretical and a practical point of view. In particular, its incremental complexity is known to be quasi-polynomial in general and polynomial for bounded hypergraphs. Recent applications in computational biology however require to solve a generalisation of this problem, that we call bi-objective transversal problem. The instance is in this case composed of a pair of hypergraphs $(A,B)$, and the aim is to enumerate minimal sets which hit all the hyperedges of $A$ while intersecting a minimal set of hyperedges of $B$. We formalised this problem and related it to the enumeration of minimal hitting sets of bundles [32] . We showed cases when under degree or dimension contraints, these problems remain NP-hard, and gave a polynomial algorithm for the case when $A$ has bounded dimension, by building a hypergraph whose transversals are exactly the hitting sets of bundles.