Bidimensionality and EPTAS

Bidimensionality theory is a powerful framework for the development of metaalgorithmic techniques. It was introduced by Demaine et al. as a tool to obtain sub-exponential time parameterized algorithms for problems on H-minor free graphs. Demaine and Hajiaghayi extended the theory to obtain PTASs for bidimensional problems, and subsequently improved these results to EPTASs. Fomin et. al related the theory to the existence of linear kernels for parameterized problems. In this paper we revisit bidimensionality theory from the perspective of approximation algorithms and redesign the framework for obtaining EPTASs to be more powerful, easier to apply and easier to understand. Two of the most widely used approaches to obtain PTASs on planar graphs are the Lipton-Tarjan separator based approach, and Baker's approach. Demaine and Hajiaghayi strengthened both approaches using bidimensionality and obtained EPTASs for a multitude of problems. We unify the two strenghtened approaches to combine the best of both worlds. At the heart of our framework is a decomposition lemma which states that for "most" bidimensional problems, there is a polynomial time algorithm which given an H-minor-free graph G as input and an e > 0 outputs a vertex set X of size e * OPT such that the treewidth of G n X is f(e). Here, OPT is the objective function value of the problem in question and f is a function depending only on e. This allows us to obtain EPTASs on (apex)-minor-free graphs for all problems covered by the previous framework, as well as for a wide range of packing problems, partial covering problems and problems that are neither closed under taking minors, nor contractions. To the best of our knowledge for many of these problems including cycle packing, vertex-h-packing, maximum leaf spanning tree, and partial r-dominating set no EPTASs on planar graphs were previously known

Bidimensionality and Geometric Graphs

In this paper we use several of the key ideas from Bidimensionality to give a new generic approach to design EPTASs and subexponential time parameterized algorithms for problems on classes of graphs which are not minor closed, but instead exhibit a geometric structure. In particular we present EPTASs and subexponential time parameterized algorithms for Feedback Vertex Set, Vertex Cover, Connected Vertex Cover, Diamond Hitting Set, on map graphs and unit disk graphs, and for Cycle Packing and Minimum-Vertex Feedback Edge Set on unit disk graphs. Our results are based on the recent decomposition theorems proved by Fomin et al [SODA 2011], and our algorithms work directly on the input graph. Thus it is not necessary to compute the geometric representations of the input graph. To the best of our knowledge, these results are previously unknown, with the exception of the EPTAS and a subexponential time parameterized algorithm on unit disk graphs for Vertex Cover, which were obtained by Marx [ESA 2005] and Alber and Fiala [J. Algorithms 2004], respectively. We proceed to show that our approach can not be extended in its full generality to more general classes of geometric graphs, such as intersection graphs of unit balls in R^d, d >= 3. Specifically we prove that Feedback Vertex Set on unit-ball graphs in R^3 neither admits PTASs unless P=NP, nor subexponential time algorithms unless the Exponential Time Hypothesis fails. Additionally, we show that the decomposition theorems which our approach is based on fail for disk graphs and that therefore any extension of our results to disk graphs would require new algorithmic ideas. On the other hand, we prove that our EPTASs and subexponential time algorithms for Vertex Cover and Connected Vertex Cover carry over both to disk graphs and to unit-ball graphs in R^d for every fixed d

Generalized Set and Graph Packing Problems

Many complex systems that exist in nature and society can be expressed in terms of networks (e.g., social networks, communication networks, biological networks, Web graph, among others). Usually a node represents an entity while an edge represents an interaction between two entities. A community arises in a network when two or more entities have common interests, e.g., related proteins, industrial sectors, groups of people, documents of a collection. There exist applications that model a community as a fixed graph H [98, 10, 119, 2, 142, 136]. Additionally, it is not expected that an entity of the network belongs to only one community; that is, communities tend to share their members. The community discovering or community detection problem consists on finding all communities in a given network. This problem has been extensively studied from a practical perspective [61, 137, 122, 116]. However, we believe that this problem also brings many interesting theoretical questions. Thus in this thesis, we will address this problem using a more rigorous approach. To that end, we first introduce graph problems that we consider capture well the community discovering problem. These graph problems generalize the classical H-Packing problem [88] in two different ways. In the H-Packing with t-Overlap problem, the goal is to find in a given graph G (the network) at least k subgraphs (the communities) isomorphic to a member of a family of graphs H (the community models) such that each pair of subgraphs overlaps in at most t vertices (the shared members). On the other hand, in the H-Packing with t-Membership problem instead of limiting the pairwise overlap, each vertex of G is contained in at most t subgraphs of the solution. For both problems each member of H has at most r vertices and m edges. An instance of the H-Packing with t-Overlap and t-Membership problems corresponds to an instance of the H-Packing problem for t = 0 and t = 1, respectively. We also restrict the overlap between the edges of the subgraphs in the solution instead of the vertices (called H-Packing with t-Edge Overlap and t-Edge Membership problems). Given the closeness of the r-Set Packing problem [87] to the H-Packing problem, we also consider overlap in the problem of packing disjoint sets of size at most r. As usual for set packing problems, given a collection S drawn from a universe U, we seek a sub-collection S'⊆S consisting of at least k sets subject to certain disjointness restrictions. In the r-Set Packing with t-Membership, each element of U belongs to at most t sets of S' while in the r-Set Packing with t-Overlap each pair of sets in S' overlaps in at most t elements. For both problems, each set of S has at most r elements. We refer to all the problems introduced in this thesis simply as packing problems with overlap. Also, we group as the family of t-Overlap problems: H-Packing with t-Overlap, H-Packing with t-Edge Overlap, and r-Set Packing with t-Overlap. While we call the family of t-Membership problems: H-Packing with t-Membership, H-Packing with t-Edge Membership, and r-Set Packing with t-Membership. The classical H-Packing and r-Set Packing problems are NP-complete [87, 88]. We will show in this thesis that allowing overlap in a packing does not make the problems "easier". More precisely, we show that the H-Packing with t-Membership and the r-Set Packing with t-Membership are NP-complete when H = {H'} and H' is an arbitrary connected graph with at least three vertices and r≥3, respectively. Parameterized complexity, introduced by Downey and Fellows [44], is an exciting and interesting approach to deal with NP-complete problems. The underlying idea of this approach is to isolate some aspects or parts of the input (known as the parameters) to investigate whether these parameters make the problem tractable or intractable. The main goal of this thesis is to study the parameterized complexity of our packing problems with overlap. We set up as a parameter k the size of the solution (number of communities), and we consider as fixed-constants r, m and t. We show that our problems admit polynomial kernels via two types of techniques: polynomial parametric transformations (PPTs) [16] and classical reduction algorithms [43]. PPTs are mainly used to show lower bounds and as far as we know they have not been used as extensively to obtain kernel results as classical kernelization techniques [96, 42]. Thus, we believe that employing PPTs is a promising approach to obtain kernel reductions for other problems as well. On the other hand, with non-trivial generalizations of kernelization algorithms for the classical H-Packing problem [114], we are able to improve our kernel sizes obtained via PPTs. These improved kernel sizes are equivalent to the kernel sizes for the disjoint version when t = 0 and t = 1 for the t-Overlap and t-Membership problems, respectively. We also obtain fixed-parameter algorithms for our packing problems with overlap (other than running brute force on the kernel). Our algorithms combine a search tree and a greedy localization technique and generalize a fixed-parameter algorithm for the problem of packing disjoint triangles [54]. In addition, we obtain faster FPT-algorithms by transforming our overlapping problems into an instance of the disjoint version of our problems. Finally, we introduce the Π-Packing with α()-Overlap problem to allow for more complex overlap constraints than the ones considered by the t-Overlap and t-Membership problems and also to include more general communities definitions. This problem seeks at least k induced subgraphs in a graph G subject to: each subgraph has at most r vertices and obeys a property Π (a community definition) and for any pair of subgraphs Hi,Hj, with i≠j, we have that α(Hi,Hj) = 0 holds (an overlap constraint). We show that the Π-Packing with α()-Overlap problem is fixed-parameter tractable provided that Π is computable in polynomial time in n and α() obeys some natural conditions. Motivated by practical applications we give several examples of α() functions which meet those conditions

Searching for network modules

When analyzing complex networks a key target is to uncover their modular structure, which means searching for a family of modules, namely node subsets spanning each a subnetwork more densely connected than the average. This work proposes a novel type of objective function for graph clustering, in the form of a multilinear polynomial whose coefficients are determined by network topology. It may be thought of as a potential function, to be maximized, taking its values on fuzzy clusterings or families of fuzzy subsets of nodes over which every node distributes a unit membership. When suitably parametrized, this potential is shown to attain its maximum when every node concentrates its all unit membership on some module. The output thus is a partition, while the original discrete optimization problem is turned into a continuous version allowing to conceive alternative search strategies. The instance of the problem being a pseudo-Boolean function assigning real-valued cluster scores to node subsets, modularity maximization is employed to exemplify a so-called quadratic form, in that the scores of singletons and pairs also fully determine the scores of larger clusters, while the resulting multilinear polynomial potential function has degree 2. After considering further quadratic instances, different from modularity and obtained by interpreting network topology in alternative manners, a greedy local-search strategy for the continuous framework is analytically compared with an existing greedy agglomerative procedure for the discrete case. Overlapping is finally discussed in terms of multiple runs, i.e. several local searches with different initializations.Comment: 10 page

Stabbing line segments with disks: complexity and approximation algorithms

Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii $r>0$ where the set of segments forms a straight line drawing $G=(V,E)$ of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for $r\in [d_{\min},\eta d_{\max}]$ and some constant $\eta$ where $d_{\max}$ and $d_{\min}$ are Euclidean lengths of the longest and shortest graph edges respectively. Fast $O(|E|\log|E|)$-time $O(1)$-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality $r\geq \eta d_{\max}$ holds uniformly for some constant $\eta>0,$ i.e. when lengths of edges of $G$ are uniformly bounded from above by some linear function of $r.$Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International Conference on Analysis of Images, Social Networks and Texts (AIST-2017