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Reflections on multivariate algorithmics and problem parameterization
 PROC. 27TH STACS
, 2010
"... Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and e ..."
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Cited by 37 (21 self)
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Research on parameterized algorithmics for NPhard problems has steadily grown over the last years. We survey and discuss how parameterized complexity analysis naturally develops into the field of multivariate algorithmics. Correspondingly, we describe how to perform a systematic investigation and exploitation of the “parameter space” of computationally hard problems.
PlanarF Deletion: Approximation, Kernelization and Optimal FPT Algorithms
"... Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selectin ..."
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Cited by 18 (8 self)
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Let F be a finite set of graphs. In the FDeletion problem, we are given an nvertex graph G and an integer k as input, and asked whether at most k vertices can be deleted from G such that the resulting graph does not contain a graph from F as a minor. FDeletion is a generic problem and by selecting different sets of forbidden minors F, one can obtain various fundamental problems such as Vertex Cover, Feedback Vertex Set or Treewidth ηDeletion. In this paper we obtain a number of generic algorithmic results about FDeletion, when F contains at least one planar graph. The highlights of our work are • A constant factor approximation algorithm for the optimization version of FDeletion; • A linear time and single exponential parameterized algorithm, that is, an algorithm running in time O(2 O(k) n), for the parameterized version of FDeletion where all graphs in F are connected; • A polynomial kernel for parameterized FDeletion. These algorithms unify, generalize, and improve a multitude of results in the literature. Our main results have several direct applications, but also the methods we develop on the way have applicability beyond the scope of this paper. Our results – constant factor approximation, polynomial kernelization and FPT algorithms – are stringed together by a common theme of polynomial time preprocessing.
Weak Compositions and Their Applications to Polynomial Lower Bounds for Kernelization
"... Abstract. We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lowerbounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1} ∗ × N be two parameterized problems where the unparameterized versi ..."
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Cited by 18 (2 self)
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Abstract. We introduce a new form of composition called weak composition that allows us to obtain polynomial kernelization lowerbounds for several natural parameterized problems. Let d ≥ 2 be some constant and let L1, L2 ⊆ {0, 1} ∗ × N be two parameterized problems where the unparameterized version of L1 is NPhard. Assuming coNP ̸ ⊆ NP/poly, our framework essentially states that composing t L1instances each with parameter k, to an L2instance with parameter k ′ ≤ t 1/d k O(1) , implies that L2 does not have a kernel of size O(k d−ε) for any ε> 0. We show two examples of weak composition and derive polynomial kernelization lower bounds for dBipartite Regular Perfect Code and dDimensional Matching, parameterized by the solution size k. By reduction, using linear parameter transformations, we then derive the following lowerbounds for kernel sizes when the parameter is the solution size k (assuming coNP ̸ ⊆ NP/poly): – dSet Packing, dSet Cover, dExact Set Cover, Hitting Set with dBounded Occurrences, and Exact Hitting Set with dBounded Occurrences have no kernels of size O(k d−3−ε) for any ε> 0. – Kd Packing and Induced K1,d Packing have no kernels of size O(k d−4−ε) for any ε> 0. – dRedBlue Dominating Set and dSteiner Tree have no kernels of sizes O(k d−3−ε) and
Pushing the power of stochastic greedy ordering schemes for inference in graphical models
 IN AAAI 2011
, 2011
"... We study iterative randomized greedy algorithms for generating (elimination) orderings with small induced width and state space size two parameters known to bound the complexity of inference in graphical models. We propose and implement the Iterative Greedy Variable Ordering (IGVO) algorithm, a new ..."
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Cited by 15 (10 self)
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We study iterative randomized greedy algorithms for generating (elimination) orderings with small induced width and state space size two parameters known to bound the complexity of inference in graphical models. We propose and implement the Iterative Greedy Variable Ordering (IGVO) algorithm, a new variant within this algorithm class. An empirical evaluation using different ranking functions and conditions of randomness, demonstrates that IGVO finds significantly better orderings than standard greedy ordering implementations when evaluated within an anytime framework. Additional order of magnitude improvements are demonstrated on a multicore system, thus further expanding the set of solvable graphical models. The experiments also confirm the superiority of the MinFill heuristic within the iterative scheme.
GraphBased Data Clustering with Overlaps
 TO APPEAR IN DISCRETE OPTIMIZATION,
, 2010
"... We introduce overlap cluster graph modification problems where, other than in most previous work, the clusters of the target graph may overlap. More precisely, the studied graph problems ask for a minimum number of edge modifications such that the resulting graph consists of clusters (that is, maxim ..."
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Cited by 14 (5 self)
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We introduce overlap cluster graph modification problems where, other than in most previous work, the clusters of the target graph may overlap. More precisely, the studied graph problems ask for a minimum number of edge modifications such that the resulting graph consists of clusters (that is, maximal cliques) that may overlap up to a certain amount specified by the overlap number s. In the case of svertexoverlap, each vertex may be part of at most s maximal cliques; sedgeoverlap is analogously defined in terms of edges. We provide a complexity dichotomy (polynomialtime solvable versus NPhard) for the underlying edge modification problems, develop forbidden subgraph characterizations of “cluster graphs with overlaps”, and study the parameterized complexity in terms of the number of allowed edge modifications, achieving fixedparameter tractability (in case of constant svalues) and parameterized hardness (in case of unbounded svalues).
Average Parameterization and Partial Kernelization for Computing Medians
 PROC. 9TH LATIN
, 2010
"... We propose an effective polynomialtime preprocessing strategy for intractable median problems. Developing a new methodological framework, we show that if the input instances of generally intractable problems exhibit a sufficiently high degree of similarity between each other on average, then there ..."
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Cited by 13 (9 self)
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We propose an effective polynomialtime preprocessing strategy for intractable median problems. Developing a new methodological framework, we show that if the input instances of generally intractable problems exhibit a sufficiently high degree of similarity between each other on average, then there are efficient exact solving algorithms. In other words, we show that the median problems Swap Median Permutation, Consensus Clustering, Kemeny Score, and Kemeny Tie Score all are fixedparameter tractable with respect to the parameter “average distance between input objects”. To this end, we develop the new concept of “partial kernelization” and identify interesting polynomialtime solvable special cases for the considered problems.
Every Ternary Permutation Constraint Satisfaction Problem Parameterized Above Average Has a Kernel with a Quadratic Number of Variables
, 2010
"... A ternary PermutationCSP is specified by a subset Π of the symmetric group S3. An instance of such a problem consists of a set of variables V and a multiset of constraints, which are ordered triples of distinct variables of V. The objective is to find a linear ordering α of V that maximizes the num ..."
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Cited by 12 (7 self)
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A ternary PermutationCSP is specified by a subset Π of the symmetric group S3. An instance of such a problem consists of a set of variables V and a multiset of constraints, which are ordered triples of distinct variables of V. The objective is to find a linear ordering α of V that maximizes the number of triples whose rearrangement (under α) follows a permutation in Π. We prove that every ternary PermutationCSP parameterized above average has a kernel with a quadratic number of variables.
Constant thresholds can make target set selection tractable
 In MedAlg
"... Abstract. Target Set Selection, which is a prominent NPhard problem occurring in social network analysis and distributed computing, is notoriously hard both in terms of achieving useful approximation as well as fixedparameter algorithms. The task is to select a minimum number of vertices into a “t ..."
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Cited by 9 (3 self)
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Abstract. Target Set Selection, which is a prominent NPhard problem occurring in social network analysis and distributed computing, is notoriously hard both in terms of achieving useful approximation as well as fixedparameter algorithms. The task is to select a minimum number of vertices into a “target set ” such that all other vertices will become active in course of a dynamic process (which may go through several activation rounds). A vertex, which is equipped with a threshold value t, becomes active once at least t of its neighbors are active; initially, only the target set vertices are active. We contribute further insights into islands of tractability for Target Set Selection by spotting new parameterizations characterizing some sparse graphs as well as some “cliquish ” graphs and developing corresponding fixedparameter tractability and (parameterized) hardness results. In particular, we demonstrate that upperbounding the thresholds by a constant may significantly alleviate the search for efficiently solvable, but still meaningful special cases of Target Set Selection. 1
On Tractable Cases of Target Set Selection
"... We study the NPcomplete TARGET SET SELECTION (TSS) problem occurring in social network analysis. Complementing results on its approximability and extending results for its restriction to trees and bounded treewidth graphs, we classify the influence of the parameters “diameter”, “cluster edge delet ..."
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Cited by 7 (3 self)
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We study the NPcomplete TARGET SET SELECTION (TSS) problem occurring in social network analysis. Complementing results on its approximability and extending results for its restriction to trees and bounded treewidth graphs, we classify the influence of the parameters “diameter”, “cluster edge deletion number”, “vertex cover number”, and “feedback edge set number ” of the underlying graph on the problem’s complexity, revealing both tractable and intractable cases. For instance, even for diametertwo split graphs TSS remains very hard. TSS can be efficiently solved on graphs with small feedback edge set number and also turns out to be fixedparameter tractable when parameterized by the vertex cover number, both results contrasting known parameterized intractability results for the parameter treewidth. While these tractability results are relevant for sparse networks, we also show efficient fixedparameter algorithms for the parameter cluster edge deletion number, yielding tractability for certain dense networks.