arXiv subjects

André Nichterlein

Publications and source records attributed to André Nichterlein.

47 records · Page 3Linked to original sources

arXiv2016

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.

cs.MA↗

arXiv2016

Fixed-Parameter Algorithms for DAG Partitioning

Finding the origin of short phrases propagating through the web has been formalized by Leskovec et al. [ACM SIGKDD 2009] as DAG Partitioning: given an arc-weighted directed acyclic graph on $n$ vertices and $m$ arcs, delete arcs with total weight at most $k$ such that each resulting weakly-connected component contains exactly one sink---a vertex without outgoing arcs. DAG Partitioning is NP-hard. We show an algorithm to solve DAG Partitioning in $O(2^k \cdot (n+m))$ time, that is, in linear time for fixed $k$. We complement it with linear-time executable data reduction rules. Our experiments show that, in combination, they can optimally solve DAG Partitioning on simulated citation networks within five minutes for $k\leq190$ and $m$ being $10^7$ and larger. We use our obtained optimal solutions to evaluate the solution quality of Leskovec et al.'s heuristic. We show that Leskovec et al.'s heuristic works optimally on trees and generalize this result by showing that DAG Partitioning is solvable in $2^{O(w^2)}\cdot n$ time if a width-$w$ tree decomposition of the input graph is given. Thus, we improve an algorithm and answer an open question of Alamdari and Mehrabian [WAW 2012]. We complement our algorithms by lower bounds on the running time of exact algorithms and on the effectivity of data reduction.

cs.DS↗

arXiv2016

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.

cs.DS↗

arXiv2016

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.

cs.DM↗

arXiv2015

Parameterized Inapproximability of Target Set Selection and Generalizations

In this paper, we consider the Target Set Selection problem: given a graph and a threshold value $thr(v)$ for any vertex $v$ of the graph, find a minimum size vertex-subset to "activate" s.t. all the vertices of the graph are activated at the end of the propagation process. A vertex $v$ is activated during the propagation process if at least $thr(v)$ of its neighbors are activated. This problem models several practical issues like faults in distributed networks or word-to-mouth recommendations in social networks. We show that for any functions $f$ and $ρ$ this problem cannot be approximated within a factor of $ρ(k)$ in $f(k) \cdot n^{O(1)}$ time, unless FPT = W[P], even for restricted thresholds (namely constant and majority thresholds). We also study the cardinality constraint maximization and minimization versions of the problem for which we prove similar hardness results.

cs.CC↗

arXiv2014

Parameterized Approximability of Maximizing the Spread of Influence in Networks

In this paper, we consider the problem of maximizing the spread of influence through a social network. Given a graph with a threshold value~$thr(v)$ attached to each vertex~$v$, the spread of influence is modeled as follows: A vertex~$v$ becomes "active" (influenced) if at least $thr(v)$ of its neighbors are active. In the corresponding optimization problem the objective is then to find a fixed number of vertices to activate such that the number of activated vertices at the end of the propagation process is maximum. We show that this problem is strongly inapproximable in fpt-time with respect to (w.r.t.) parameter $k$ even for very restrictive thresholds. In the case that the threshold of each vertex equals its degree, we prove that the problem is inapproximable in polynomial time and it becomes $r(n)$-approximable in fpt-time w.r.t. parameter $k$ for any strictly increasing function $r$. Moreover, we show that the decision version is W[1]-hard w.r.t. parameter $k$ but becomes fixed-parameter tractable on bounded degree graphs.

cs.DS↗

arXiv2014

Constant-factor approximations for Capacitated Arc Routing without triangle inequality

Given an undirected graph with edge costs and edge demands, the Capacitated Arc Routing problem (CARP) asks for minimum-cost routes for equal-capacity vehicles so as to satisfy all demands. Constant-factor polynomial-time approximation algorithms were proposed for CARP with triangle inequality, while CARP was claimed to be NP-hard to approximate within any constant factor in general. Correcting this claim, we show that any factor α approximation for CARP with triangle inequality yields a factor α approximation for the general CARP.

cs.DS↗

arXiv2014

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.

cs.SI↗

arXiv2013

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.

cs.CC↗

arXiv2012

On the Parameterized and Approximation Hardness of Metric Dimension

The NP-hard Metric Dimension problem is to decide for a given graph G and a positive integer k whether there is a vertex subset of size at most k that separates all vertex pairs in G. Herein, a vertex v separates a pair {u,w} if the distance (length of a shortest path) between v and u is different from the distance of v and w. We give a polynomial-time computable reduction from the Bipartite Dominating Set problem to Metric Dimension on maximum degree three graphs such that there is a one-to-one correspondence between the solution sets of both problems. There are two main consequences of this: First, it proves that Metric Dimension on maximum degree three graphs is W[2]-complete with respect to the parameter k. This answers an open question concerning the parameterized complexity of Metric Dimension posed by Díaz et al. [ESA'12] and already mentioned by Lokshtanov [Dagstuhl seminar, 2009]. Additionally, it implies that Metric Dimension cannot be solved in n^{o(k)} time, unless the assumption FPT \neq W[1] fails. This proves that a trivial n^{O(k)} algorithm is probably asymptotically optimal. Second, as Bipartite Dominating Set is inapproximable within o(log n), it follows that Metric Dimension on maximum degree three graphs is also inapproximable by a factor of o(log n), unless NP=P. This strengthens the result of Hauptmann et al. [JDA 2012] who proved APX-hardness on bounded-degree graphs.

cs.CC↗

arXiv2012

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 Müller-Hannemann [FCT 2011]. Furthermore, we classify the problem as fixed-parameter tractable with respect to the parameter "maximum degree".

cs.CC↗