arXiv subjects

Dimitrios M. Thilikos

Publications and source records attributed to Dimitrios M. Thilikos.

101 records · Page 6Linked to original sources

arXiv2013

Bidimensionality of Geometric Intersection Graphs

Let B be a finite collection of geometric (not necessarily convex) bodies in the plane. Clearly, this class of geometric objects naturally generalizes the class of disks, lines, ellipsoids, and even convex polygons. We consider geometric intersection graphs GB where each body of the collection B is represented by a vertex, and two vertices of GB are adjacent if the intersection of the corresponding bodies is non-empty. For such graph classes and under natural restrictions on their maximum degree or subgraph exclusion, we prove that the relation between their treewidth and the maximum size of a grid minor is linear. These combinatorial results vastly extend the applicability of all the meta-algorithmic results of the bidimensionality theory to geometrically defined graph classes.

cs.DM↗

arXiv2013

Polynomial Gap Extensions of the Erdős-Pósa Theorem

Given a graph $H$, we denote by ${\cal M}(H)$ all graphs that can be contracted to $H$. The following extension of the Erdős-Pósa Theorem holds: for every $h$-vertex planar graph $H$, there exists a function $f_{H}$ such that every graph $G$, either contains $k$ disjoint copies of graphs in ${\cal M}(H)$, or contains a set of $f_{H}(k)$ vertices meeting every subgraph of $G$ that belongs in ${\cal M}(H)$. In this paper we prove that this is the case for every graph $H$ of pathwidth at most 2 and, in particular, that $f_{H}(k) = 2^{O(h^2)}\cdot k^{2}\cdot \log k$. As a main ingredient of the proof of our result, we show that for every graph $H$ on $h$ vertices and pathwidth at most 2, either $G$ contains $k$ disjoint copies of $H$ as a minor or the treewidth of $G$ is upper-bounded by $2^{O(h^2)}\cdot k^{2}\cdot \log k$. We finally prove that the exponential dependence on $h$ in these bounds can be avoided if $H=K_{2,r}$. In particular, we show that $f_{K_{2,r}}=O(r^2\cdot k^2)$

cs.DM↗

arXiv2013

Excluding Graphs as Immersions in Surface Embedded Graphs

We prove a structural characterization of graphs that forbid a fixed graph $H$ as an immersion and can be embedded in a surface of Euler genus $γ$. In particular, we prove that a graph $G$ that excludes some connected graph $H$ as an immersion and is embedded in a surface of Euler genus $γ$ has either "small" treewidth (bounded by a function of $H$ and $γ$) or "small" edge connectivity (bounded by the maximum degree of $H$). Using the same techniques we also prove an excluded grid theorem on bounded genus graphs for the immersion relation.

math.CO↗

arXiv2012

Forbidding Kuratowski Graphs as Immersions

The immersion relation is a partial ordering relation on graphs that is weaker than the topological minor relation in the sense that if a graph $G$ contains a graph $H$ as a topological minor, then it also contains it as an immersion but not vice versa. Kuratowski graphs, namely $K_{5}$ and $K_{3,3}$, give a precise characterization of planar graphs when excluded as topological minors. In this note we give a structural characterization of the graphs that exclude Kuratowski graphs as immersions. We prove that they can be constructed by applying consecutive $i$-edge-sums, for $i\leq 3$, starting from graphs that are planar sub-cubic or of branch-width at most 10.

math.CO↗

arXiv2011

Confronting Intractability via Parameters

One approach to confronting computational hardness is to try to understand the contribution of various parameters to the running time of algorithms and the complexity of computational tasks. Almost no computational tasks in real life are specified by their size alone. It is not hard to imagine that some parameters contribute more intractability than others and it seems reasonable to develop a theory of computational complexity which seeks to exploit this fact. Such a theory should be able to address the needs of practicioners in algorithmics. The last twenty years have seen the development of such a theory. This theory has a large number of successes in terms of a rich collection of algorithmic techniques both practical and theoretical, and a fine-grained intractability theory. Whilst the theory has been widely used in a number of areas of applications including computational biology, linguistics, VLSI design, learning theory and many others, knowledge of the area is highly varied. We hope that this article will show both the basic theory and point at the wide array of techniques available. Naturally the treatment is condensed, and the reader who wants more should go to the texts, Downey and Fellows, Flum and Grohe, Niedermeier, and the upcoming undergraduate text Downey and Fellows.

cs.CC↗

arXiv2011

Dynamic Programming for Graphs on Surfaces

We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2^{O(k log k)} n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called "surface cut decomposition", generalizing sphere cut decompositions of planar graphs introduced by Seymour and Thomas, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of non-crossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2^{O(k)} n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction.

cs.DS↗

arXiv2011

Asymptotic Enumeration of Non-crossing Partitions on Surfaces

We generalize the notion of non-crossing partition on a disk to general surfaces with boundary. For this, we consider a surface $Σ$ and introduce the number $C_Σ(n)$ of non-crossing partitions of a set of $n$ points laying on the boundary of $Σ$. Our proofs use bijective techniques arising from map enumeration, joint with the symbolic method and singularity analysis on generating functions. An outcome of our results is that the exponential growth of $C_Σ(n)$ is the same as the one of the $n$-th Catalan number, i.e., does not change when we move from the case where $Σ$ is a disk to general surfaces with boundary.

math.CO↗

arXiv2011

Optimizing the Graph Minors Weak Structure Theorem

One of the major results of [N. Robertson and P. D. Seymour. Graph minors. XIII. The disjoint paths problem. J. Combin. Theory Ser. B, 63(1):65--110, 1995], also known as the weak structure theorem, revealed the local structure of graphs excluding some graph as a minor: each such graph $G$ either has small treewidth or contains the subdivision of a wall that can be arranged "bidimensionally" inside $G$, given that some small set of vertices are removed. We prove an optimized version of that theorem where (i) the relation between the treewidth of the graph and the height of the wall is linear (thus best possible) and (ii) the number of vertices to be removed is minimized.

math.CO↗

arXiv2010

Contracting planar graphs to contractions of triangulations

For every graph $H$, there exists a polynomial-time algorithm deciding if a planar input graph $G$ can be contracted to~$H$. However, the degree of the polynomial depends on the size of $H$. In this paper, we identify a class of graphs $\cal C$ such that for every $H \in \cal C$, there exists an algorithm deciding in time $f(|V(H)|) \cdot |V(G)|^{\bigO{1}}$ whether a planar graph $G$ can be contracted to~$H$. (The function $f(\cdot)$ does not depend on $G$.) The class $\cal C$ is the closure of planar triangulated graphs under taking of contractions. In fact, we prove that a graph $H \in \cal C$ if and only if there exists a constant $c_H$ such that if the tree-width of a graph is at least $c_H$, it contains $H$ as a contraction. We also provide a characterization of $\cal C$ in terms of minimal forbidden contractions.

math.CO↗

arXiv2009

Rank-width and Tree-width of H-minor-free Graphs

We prove that for any fixed r>=2, the tree-width of graphs not containing K_r as a topological minor (resp. as a subgraph) is bounded by a linear (resp. polynomial) function of their rank-width. We also present refinements of our bounds for other graph classes such as K_r-minor free graphs and graphs of bounded genus.

math.CO↗

arXiv2008

Approximating acyclicity parameters of sparse hypergraphs

The notions of hypertree width and generalized hypertree width were introduced by Gottlob, Leone, and Scarcello in order to extend the concept of hypergraph acyclicity. These notions were further generalized by Grohe and Marx, who introduced the fractional hypertree width of a hypergraph. All these width parameters on hypergraphs are useful for extending tractability of many problems in database theory and artificial intelligence. In this paper, we study the approximability of (generalized, fractional) hyper treewidth of sparse hypergraphs where the criterion of sparsity reflects the sparsity of their incidence graphs. Our first step is to prove that the (generalized, fractional) hypertree width of a hypergraph H is constant-factor sandwiched by the treewidth of its incidence graph, when the incidence graph belongs to some apex-minor-free graph class. This determines the combinatorial borderline above which the notion of (generalized, fractional) hypertree width becomes essentially more general than treewidth, justifying that way its functionality as a hypergraph acyclicity measure. While for more general sparse families of hypergraphs treewidth of incidence graphs and all hypertree width parameters may differ arbitrarily, there are sparse families where a constant factor approximation algorithm is possible. In particular, we give a constant factor approximation polynomial time algorithm for (generalized, fractional) hypertree width on hypergraphs whose incidence graphs belong to some H-minor-free graph class.

cs.DS↗