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##### GANG - 2019

Application Domains
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## Section: New Results

### Graph and Combinatorial Algorithms

#### Fast Diameter Computation within Split Graphs

When can we compute the diameter of a graph in quasi linear time? In [22], we address this question for the class of split graphs, that we observe to be the hardest instances for deciding whether the diameter is at most two. We stress that although the diameter of a non-complete split graph can only be either 2 or 3, under the Strong Exponential-Time Hypothesis (SETH) we cannot compute the diameter of a split graph in less than quadratic time. Therefore it is worth to study the complexity of diameter computation on subclasses of split graphs, in order to better understand the complexity border. Specifically, we consider the split graphs with bounded clique-interval number and their complements, with the former being a natural variation of the concept of interval number for split graphs that we introduce in this paper. We first discuss the relations between the clique-interval number and other graph invariants such as the classic interval number of graphs, the treewidth, the VC-dimension and the stabbing number of a related hypergraph. Then, in part based on these above relations, we almost completely settle the complexity of diameter computation on these subclasses of split graphs:

• For the $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k=𝒪\left(1\right)$, and even in quasi linear time if $k=o\left(logn\right)$ and in addition a corresponding ordering is given. However, under SETH this cannot be done in truly subquadratic time for any $k=\omega \left(logn\right)$.

• For the complements of $k$-clique-interval split graphs, we can compute their diameter in truly subquadratic time if $k=𝒪\left(1\right)$, and even in time $𝒪\left(km\right)$ if a corresponding ordering is given. Again this latter result is optimal under SETH up to polylogarithmic factors.

Our findings raise the question whether a $k$-clique interval ordering can always be computed in quasi linear time. We prove that it is the case for $k=1$ and for some subclasses such as bounded-treewidth split graphs, threshold graphs and comparability split graphs. Finally, we prove that some important subclasses of split graphs – including the ones mentioned above – have a bounded clique-interval number.

#### Diameter computation on H-minor free graphs and graphs of bounded (distance) VC-dimension

Under the Strong Exponential-Time Hypothesis, the diameter of general unweighted graphs cannot be computed in truly subquadratic time. Nevertheless there are several graph classes for which this can be done such as bounded-treewidth graphs, interval graphs and planar graphs, to name a few. We propose to study unweighted graphs of constant distance VC-dimension as a broad generalization of many such classes – where the distance VC-dimension of a graph $G$ is defined as the VC-dimension of its ball hypergraph: whose hyperedges are the balls of all possible radii and centers in $G$. In particular for any fixed $H$, the class of $H$-minor free graphs has distance VC-dimension at most $|V\left(H\right)|-1$. In [23], we show the following.

• Our first main result is a Monte Carlo algorithm that on graphs of distance VC-dimension at most $d$, for any fixed $k$, either computes the diameter or concludes that it is larger than $k$ in time $\stackrel{˜}{𝒪}\left(k·m{n}^{1-{\epsilon }_{d}}\right)$, where ${\epsilon }_{d}\in \left(0;1\right)$ only depends on $d$. We thus obtain a truly subquadratic-time parameterized algorithm for computing the diameter on such graphs.

• Then as a byproduct of our approach, we get the first truly subquadratic-time randomized algorithm for constant diameter computation on all the nowhere dense graph classes. The latter classes include all proper minor-closed graph classes, bounded-degree graphs and graphs of bounded expansion.

• Finally, we show how to remove the dependency on $k$ for any graph class that excludes a fixed graph $H$ as a minor. More generally, our techniques apply to any graph with constant distance VC-dimension and polynomial expansion (or equivalently having strongly sublinear balanced separators). As a result for all such graphs one obtains a truly subquadratic-time randomized algorithm for computing their diameter.

We note that all our results also hold for radius computation. Our approach is based on the work of Chazelle and Welzl who proved the existence of spanning paths with strongly sublinear stabbing number for every hypergraph of constant VC-dimension. We show how to compute such paths efficiently by combining known algorithms for the stabbing number problem with a clever use of $\epsilon$-nets, region decomposition and other partition techniques.

#### Approximation of eccentricites and distance using $\delta$-hyperbolicity

In [9], we show that the eccentricities of all vertices of a $\delta$-hyperbolic graph $G=\left(V,E\right)$ can be computed in linear time with an additive one-sided error of at most $c·\delta$, i.e., after a linear time preprocessing, for every vertex $v$ of $G$ one can compute in $O\left(1\right)$ time an estimate $\overline{ec{c}_{G}\left(v\right)}$ of its eccentricity $ec{c}_{G}\left(v\right):=max\left\{{d}_{G}\left(u,v\right):u\in V\right\}$ such that $ec{c}_{G}\left(v\right)\le \overline{ec{c}_{G}\left(v\right)}\le ec{c}_{G}\left(v\right)+c·\delta$ for a small constant $c$. We prove that every $\delta$-hyperbolic graph $G$ has a shortest path tree $T$, constructible in linear time, such that for every vertex $v$ of $G$, $ec{c}_{G}\left(v\right)\le ec{c}_{T}\left(v\right)\le ec{c}_{G}\left(v\right)+c·\delta$, where $ec{c}_{T}\left(v\right):=max\left\{{d}_{T}\left(u,v\right):u\in V\right\}$. These results are based on an interesting monotonicity property of the eccentricity function of hyperbolic graphs: the closer a vertex is to the center of G, the smaller its eccentricity is. We also show that the distance matrix of G with an additive one-sided error of at most ${c}^{\text{'}}·\delta$ can be computed in $O\left(|V{|}^{2}lo{g}^{2}|V|\right)$ time, where ${c}^{\text{'}} is a small constant. Recent empirical studies show that many real world graphs (including Internet application networks, web networks, collaboration networks, social networks, biological networks, and others) have small hyperbolicity. So, we analyze the performance of our algorithms for approximating eccentricities and distance matrix on a number of real-world networks. Our experimental results show that the obtained estimates are even better than the theoretical bounds.

#### Graph and Hypergraph Decompositions

In [26], we study modular decomposition of hypergraphs and propose some polynomial algorithms to this aim. We also study several notions of approximation of modular decomposition of graphs, by relaxing the definition of modules introducing a tolerance ($ϵ$ edges can miss) this will be presented at CALDAM 2020, Hyderabad. Both topics can be seen as the search for new models of regularity in discrete structures, as in particular bipartite graphs. In both references our polynomial algorithms have to be improved before being applied on real-world data.