NP-Hardness and Fixed-Parameter Tractability of Realizing Degree Sequences with Directed Acyclic Graphs

In graph realization problems one is given a degree sequence and the task is to decide whether there is a graph whose vertex degrees match to the given sequence. This realization problem is known to be polynomial-time solvable when the graph is directed or undirected. In contrary, we show NP-completeness for the problem of realizing a given sequence of pairs of positive integers (representing indegrees and outdegrees) with a directed acyclic graph, answering an open question of Berger and M\"uller-Hannemann [FCT 2011]. Furthermore, we classify the problem as fixed-parameter tractable with respect to the parameter "maximum degree".Comment: new author Sepp Hartung, new section with fixed-parameter tractability result; 25 pages, 4 figure

Parameterized Complexity of Diameter

Diameter -- the task of computing the length of a longest shortest path -- is a fundamental graph problem. Assuming the Strong Exponential Time Hypothesis, there is no $O(n^{1.99})$-time algorithm even in sparse graphs [Roditty and Williams, 2013]. To circumvent this lower bound we aim for algorithms with running time $f(k)(n+m)$ where $k$ is a parameter and $f$ is a function as small as possible. We investigate which parameters allow for such running times. To this end, we systematically explore a hierarchy of structural graph parameters

Improved Upper and Lower Bound Heuristics for Degree Anonymization in Social Networks

Motivated by a strongly growing interest in anonymizing social network data, we investigate the NP-hard Degree Anonymization problem: given an undirected graph, the task is to add a minimum number of edges such that the graph becomes k-anonymous. That is, for each vertex there have to be at least k-1 other vertices of exactly the same degree. The model of degree anonymization has been introduced by Liu and Terzi [ACM SIGMOD'08], who also proposed and evaluated a two-phase heuristic. We present an enhancement of this heuristic, including new algorithms for each phase which significantly improve on the previously known theoretical and practical running times. Moreover, our algorithms are optimized for large-scale social networks and provide upper and lower bounds for the optimal solution. Notably, on about 26 % of the real-world data we provide (provably) optimal solutions; whereas in the other cases our upper bounds significantly improve on known heuristic solutions

Win-Win Kernelization for Degree Sequence Completion Problems

We study provably effective and efficient data reduction for a class of NP-hard graph modification problems based on vertex degree properties. We show fixed-parameter tractability for NP-hard graph completion (that is, edge addition) cases while we show that there is no hope to achieve analogous results for the corresponding vertex or edge deletion versions. Our algorithms are based on transforming graph completion problems into efficiently solvable number problems and exploiting f-factor computations for translating the results back into the graph setting. Our core observation is that we encounter a win-win situation: either the number of edge additions is small or the problem is polynomial-time solvable. This approach helps in answering an open question by Mathieson and Szeider [JCSS 2012] concerning the polynomial kernelizability of Degree Constraint Edge Addition and leads to a general method of approaching polynomial-time preprocessing for a wider class of degree sequence completion problems.Comment: 24 pages. Conference version appeared at SWAT 2014. Journal version to appear in JCSS 201

Parameterized Algorithmics for Graph Modification Problems: On Interactions with Heuristics

In graph modification problems, one is given a graph G and the goal is to apply a minimum number of modification operations (such as edge deletions) to G such that the resulting graph fulfills a certain property. For example, the Cluster Deletion problem asks to delete as few edges as possible such that the resulting graph is a disjoint union of cliques. Graph modification problems appear in numerous applications, including the analysis of biological and social networks. Typically, graph modification problems are NP-hard, making them natural candidates for parameterized complexity studies. We discuss several fruitful interactions between the development of fixed-parameter algorithms and the design of heuristics for graph modification problems, featuring quite different aspects of mutual benefits.Comment: Invited Paper at the 41st International Workshop on Graph-Theoretic Concepts in Computer Science (WG 15

On Structural Parameterizations for the 2-Club Problem

The NP-hard 2-Club problem is, given an undirected graph G=(V,E) and l\in N, to decide whether there is a vertex set S\subseteq V of size at least l such that the induced subgraph G[S] has diameter at most two. We make progress towards a systematic classification of the complexity of 2-Club with respect to a hierarchy of prominent structural graph parameters. First, we present the following tight NP-hardness results: 2-Club is NP-hard on graphs that become bipartite by deleting one vertex, on graphs that can be covered by three cliques, and on graphs with domination number two and diameter three. Then, we consider the parameter h-index of the input graph. This parameter is motivated by real-world instances and the fact that 2-Club is fixed-parameter tractable with respect to the larger parameter maximum degree. We present an algorithm that solves 2-Club in |V|^{f(k)} time with k being the h-index. By showing W[1]-hardness for this parameter, we provide evidence that the above algorithm cannot be improved to a fixed-parameter algorithm. Furthermore, the reduction used for this hardness result can be modified to show that 2-Club is NP-hard if the input graph has constant degeneracy. Finally, we show that 2-Club is fixed-parameter tractable with respect to distance to cographs.Comment: An extended abstract of this paper appeared in Proceedings of the 39th International Conference on Current Trends in Theory and Practice of Computer Science (SOFSEM'13), Jan. 2013, volume 7741 of LNCS, pages 233-243, Springer, 201

Finding Points in General Position

We study computational aspects of the General Position Subset Selection problem defined as follows: Given a set of points in the plane, find a maximum-cardinality subset of points in general position. We prove that General Position Subset Selection is NP-hard, APX-hard, and give several fixed-parameter tractability results as well as a subexponential running time lower bound based on the Exponential Time Hypothesis.Comment: 17 pages, improved problem kernel wrt. dual parameter h, added a figur

Fractals for Kernelization Lower Bounds

The composition technique is a popular method for excluding polynomial-size problem kernels for NP-hard parameterized problems. We present a new technique exploiting triangle-based fractal structures for extending the range of applicability of compositions. Our technique makes it possible to prove new no-polynomial-kernel results for a number of problems dealing with length-bounded cuts. In particular, answering an open question of Golovach and Thilikos [Discrete Optim. 2011], we show that, unless NP $\subseteq$ coNP / poly, the NP-hard Length-Bounded Edge-Cut (LBEC) problem (delete at most $k$ edges such that the resulting graph has no $s$-$t$ path of length shorter than $\ell$) parameterized by the combination of $k$ and $\ell$ has no polynomial-size problem kernel. Our framework applies to planar as well as directed variants of the basic problems and also applies to both edge and vertex deletion problems. Along the way, we show that LBEC remains NP-hard on planar graphs, a result which we believe is interesting in its own right.Comment: An extended abstract appeared in Proc. of the 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). A full version will appear in SIAM Journal on Discrete Mathematics (SIDMA

A More Fine-Grained Complexity Analysis of Finding the Most Vital Edges for Undirected Shortest Paths

We study the NP-hard Shortest Path Most Vital Edges problem arising in the context of analyzing network robustness. For an undirected graph with positive integer edge lengths and two designated vertices $s$ and $t$, the goal is to delete as few edges as possible in order to increase the length of the (new) shortest $st$-path as much as possible. This scenario has been studied from the viewpoint of parameterized complexity and approximation algorithms. We contribute to this line of research by providing refined computational tractability as well as hardness results. We achieve this by a systematic investigation of various problem-specific parameters and their influence on the computational complexity. Charting the border between tractability and intractability, we also identify numerous challenges for future research

Prices Matter for the Parameterized Complexity of Shift Bribery

In the Shift Bribery problem, we are given an election (based on preference orders), a preferred candidate $p$, and a budget. The goal is to ensure that $p$ wins by shifting $p$ higher in some voters' preference orders. However, each such shift request comes at a price (depending on the voter and on the extent of the shift) and we must not exceed the given budget. We study the parameterized computational complexity of Shift Bribery with respect to a number of parameters (pertaining to the nature of the solution sought and the size of the election) and several classes of price functions. When we parameterize Shift Bribery by the number of affected voters, then for each of our voting rules (Borda, Maximin, Copeland) the problem is W[2]-hard. If, instead, we parameterize by the number of positions by which $p$ is shifted in total,then the problem is fixed-parameter tractable for Borda and Maximin,and is W[1]-hard for Copeland. If we parameterize by the budget, then the results depend on the price function class. We also show that Shift Bribery tends to be tractable when parameterized by the number of voters, but that the results for the number of candidates are more enigmatic