Markov-Chain-Based Heuristics for the Feedback Vertex Set Problem for Digraphs

A feedback vertex set (FVS) of an undirected or directed graph G=(V, A) is a set F such that G-F is acyclic. The minimum feedback vertex set problem asks for a FVS of G of minimum cardinality whereas the weighted minimum feedback vertex set problem consists of determining a FVS F of minimum weight w(F) given a real-valued weight function w. Both problems are NP-hard [Karp72]. Nethertheless, they have been found to have applications in many fields. So one is naturally interested in approximation algorithms. While most of the existing approximation algorithms for feedback vertex set problems rely on local properties of G only, this thesis explores strategies that use global information about G in order to determine good solutions. The pioneering work in this direction has been initiated by Speckenmeyer [Speckenmeyer89]. He demonstrated the use of Markov chains for determining low cardinality FVSs. Based on his ideas, new approximation algorithms are developed for both the unweighted and the weighted minimum feedback vertex set problem for digraphs. According to the experimental results presented in this thesis, these new algorithms outperform all other existing approximation algorithms. An additional contribution, not related to Markov chains, is the identification of a new class of digraphs G=(V, A) which permit the determination of an optimum FVS in time O(|V|^4). This class strictly encompasses the completely contractible graphs [Levy/Low88]

Structural Rounding: Approximation Algorithms for Graphs Near an Algorithmically Tractable Class

We develop a framework for generalizing approximation algorithms from the structural graph algorithm literature so that they apply to graphs somewhat close to that class (a scenario we expect is common when working with real-world networks) while still guaranteeing approximation ratios. The idea is to edit a given graph via vertex- or edge-deletions to put the graph into an algorithmically tractable class, apply known approximation algorithms for that class, and then lift the solution to apply to the original graph. We give a general characterization of when an optimization problem is amenable to this approach, and show that it includes many well-studied graph problems, such as Independent Set, Vertex Cover, Feedback Vertex Set, Minimum Maximal Matching, Chromatic Number, (l-)Dominating Set, Edge (l-)Dominating Set, and Connected Dominating Set. To enable this framework, we develop new editing algorithms that find the approximately-fewest edits required to bring a given graph into one of a few important graph classes (in some cases these are bicriteria algorithms which simultaneously approximate both the number of editing operations and the target parameter of the family). For bounded degeneracy, we obtain an O(r log{n})-approximation and a bicriteria (4,4)-approximation which also extends to a smoother bicriteria trade-off. For bounded treewidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w}))-approximation, and for bounded pathwidth, we obtain a bicriteria (O(log^{1.5} n), O(sqrt{log w} * log n))-approximation. For treedepth 2 (related to bounded expansion), we obtain a 4-approximation. We also prove complementary hardness-of-approximation results assuming P != NP: in particular, these problems are all log-factor inapproximable, except the last which is not approximable below some constant factor 2 (assuming UGC)

Clearing Contamination in Large Networks

In this work, we study the problem of clearing contamination spreading through a large network where we model the problem as a graph searching game. The problem can be summarized as constructing a search strategy that will leave the graph clear of any contamination at the end of the searching process in as few steps as possible. We show that this problem is NP-hard even on directed acyclic graphs and provide an efficient approximation algorithm. We experimentally observe the performance of our approximation algorithm in relation to the lower bound on several large online networks including Slashdot, Epinions and Twitter. The experiments reveal that in most cases our algorithm performs near optimally

Kernels for Deletion to Classes of Acyclic Digraphs

In the Directed Feedback Vertex Set (DFVS) problem, we are given a digraph D on n vertices and a positive integer k and the objective is to check whether there exists a set of vertices S of size at most k such that F = D - S is a directed acyclic digraph. In a recent paper, Mnich and van Leeuwen [STACS 2016] considered the kernelization complexity of DFVS with an additional restriction on F, namely that F must be an out-forest (Out-Forest Vertex Deletion Set), an out-tree (Out-Tree Vertex Deletion Set), or a (directed) pumpkin (Pumpkin Vertex Deletion Set). Their objective was to shed some light on the kernelization complexity of the DFVS problem, a well known open problem in the area of Parameterized Complexity. In this article, we improve the kernel sizes of Out-Forest Vertex Deletion Set from O(k^3) to O(k^2) and of Pumpkin Vertex Deletion Set from O(k^18) to O(k^3). We also prove that the former kernel size is tight under certain complexity theoretic assumptions

Universal hypotrochoidic law for random matrices with cyclic correlations

The celebrated elliptic law describes the distribution of eigenvalues of random matrices with correlations between off-diagonal pairs of elements, having applications to a wide range of physical and biological systems. Here, we investigate the generalization of this law to random matrices exhibiting higher-order cyclic correlations between k tuples of matrix entries. We show that the eigenvalue spectrum in this ensemble is bounded by a hypotrochoid curve with k -fold rotational symmetry. This hypotrochoid law applies to full matrices as well as sparse ones, and thereby holds with remarkable universality. We further extend our analysis to matrices and graphs with competing cycle motifs, which are described more generally by polytrochoid spectral boundaries

Spectral characterizations of complex unit gain graphs

While eigenvalues of graphs are well studied, spectral analysis of complex unit gain graphs is still in its infancy. This thesis considers gain graphs whose gain groups are gradually less and less restricted, with the ultimate goal of classifying gain graphs that are characterized by their spectra. In such cases, the eigenvalues of a gain graph contain sufficient structural information that it might be uniquely (up to certain equivalence relations) constructed when only given its spectrum. First, the first infinite family of directed graphs that is – up to isomorphism – determined by its Hermitian spectrum is obtained. Since the entries of the Hermitian adjacency matrix are complex units, these objects may be thought of as gain graphs with a restricted gain group. It is shown that directed graphs with the desired property are extremely rare. Thereafter, the perspective is generalized to include signs on the edges. By encoding the various edge-vertex incidence relations with sixth roots of unity, the above perspective can again be taken. With an interesting mix of algebraic and combinatorial techniques, all signed directed graphs with degree at most 4 or least multiplicity at most 3 are determined. Subsequently, these characterizations are used to obtain signed directed graphs that are determined by their spectra. Finally, an extensive discussion of complex unit gain graphs in their most general form is offered. After exploring their various notions of symmetry and many interesting ties to complex geometries, gain graphs with exactly two distinct eigenvalues are classified