Results 1  10
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17
Analysis of data reduction: Transformations give evidence for nonexistence of polynomial kernels
, 2008
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A c k n 5Approximation Algorithm for Treewidth
, 2013
"... We give an algorithm that for an input nvertex graph G and integer k> 0, in time O(c k n) either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for treewidth which ..."
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Cited by 11 (1 self)
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We give an algorithm that for an input nvertex graph G and integer k> 0, in time O(c k n) either outputs that the treewidth of G is larger than k, or gives a tree decomposition of G of width at most 5k + 4. This is the first algorithm providing a constant factor approximation for treewidth which runs in time singleexponential in k and linear in n. Treewidth based computations are subroutines of numerous algorithms. Our algorithm can be used to speed up many such algorithms to work in time which is singleexponential in the treewidth and linear in the input size.
Treelike structure in large social and information networks
, 2013
"... Although large social and information networks are often thought of as having hierarchical or treelike structure, this assumption is rarely tested. We have performed a detailed empirical analysis of the treelike properties of realistic informatics graphs using two very different notions of treel ..."
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Cited by 6 (2 self)
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Although large social and information networks are often thought of as having hierarchical or treelike structure, this assumption is rarely tested. We have performed a detailed empirical analysis of the treelike properties of realistic informatics graphs using two very different notions of treelikeness: Gromov’s δhyperbolicity, which is a notion from geometric group theory that measures how treelike a graph is in terms of its metric structure; and tree decompositions, tools from structural graph theory which measure how treelike a graph is in terms of its cut structure. Although realistic informatics graphs often do not have meaningful treelike structure when viewed with respect to the simplest and most popular metrics, e.g., the value of δ or the treewidth, we conclude that many such graphs do have meaningful treelike structure when viewed with respect to more refined metrics, e.g., a sizeresolved notion of δ or a closer analysis of the tree decompositions. We also show that, although these two rigorous notions of treelikeness capture very different treelike structures in worstcase, for realistic informatics graphs they empirically identify surprisingly similar structure. We interpret this treelike structure in terms of the recentlycharacterized “nested coreperiphery” property of large informatics graphs; and we show that the fast and scalable kcore heuristic can be used to identify this treelike structure.
Approximate Tree Decompositions of Planar Graphs in Linear Time
"... Many algorithms have been developed for NPhard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NPha ..."
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Cited by 5 (0 self)
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Many algorithms have been developed for NPhard problems on graphs with small treewidth k. For example, all problems that are expressible in linear extended monadic second order can be solved in linear time on graphs of bounded treewidth. It turns out that the bottleneck of many algorithms for NPhard problems is the computation of a tree decomposition of width O(k). In particular, by the bidimensional theory, there are many linear extended monadic second order problems that can be solved on nvertex planar graphs with treewidth k in a time linear in n and subexponential in k if a tree decomposition of width O(k) can be found in such a time. We present the first algorithm that, on nvertex planar graphs with treewidth k, finds a tree decomposition of width O(k) in such a time. In more detail, our algorithm has a running time of O(nk3 log k). The previous best algorithm with a running time subexponential in k was the algorithm of Gu and Tamaki [12] with a running time of O(n1+ɛ log n) and an approximation ratio 1.5 + 1/ɛ for any ɛ> 0. The running time of our algorithm is also better than the running time of O(f(k) · n log n) of Reed’s algorithm [18] for general graphs, where f is a function exponential in k. Key words: tree decomposition, treewidth, branchwidth, rankwidth, planar graph, ℓouterplanar, linear
Sparsest cut on bounded treewidth graphs: Algorithms and hardness results
 In 45th Annual ACM Symposium on Symposium on Theory of Computing
, 2013
"... We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness resul ..."
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Cited by 4 (0 self)
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We give a 2approximation algorithm for NonUniform Sparsest Cut that runs in time nO(k), where k is the treewidth of the graph. This improves on the previous 22 kapproximation in time poly(n)2O(k) due to Chlamtác ̌ et al. [CKR10]. To complement this algorithm, we show the following hardness results: If the NonUniform Sparsest Cut problem has a ρapproximation for seriesparallel graphs (where ρ ≥ 1), then the MaxCut problem has an algorithm with approximation factor arbitrarily close to 1/ρ. Hence, even for such restricted graphs (which have treewidth 2), the Sparsest Cut problem is NPhard to approximate better than 17/16 − ε for ε> 0; assuming the Unique Games Conjecture the hardness becomes 1/αGW − ε. For graphs with large (but constant) treewidth, we show a hardness result of 2 − ε assuming the Unique Games Conjecture. Our algorithm rounds a linear program based on (a subset of) the SheraliAdams lift of the standard Sparsest Cut LP. We show that even for treewidth2 graphs, the LP has an integrality gap close to 2 even after polynomially many rounds of SheraliAdams. Hence our approach cannot be improved even on such restricted graphs without using a stronger relaxation. 1
LargeTreewidth Graph Decompositions and Applications
"... Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph G into nodedisjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number ..."
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Cited by 4 (1 self)
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Treewidth is a graph parameter that plays a fundamental role in several structural and algorithmic results. We study the problem of decomposing a given graph G into nodedisjoint subgraphs, where each subgraph has sufficiently large treewidth. We prove two theorems on the tradeoff between the number of the desired subgraphs h, and the desired lower bound r on the treewidth of each subgraph. The theorems assert that, given a graph G with treewidth k, a decomposition with parameters h, r is feasible whenever hr 2 ≤ k/poly log(k), or h 3 r ≤ k/poly log(k) holds. We then show a framework for using these theorems to bypass the wellknown GridMinor Theorem of Robertson and Seymour in some applications. In particular, this leads to substantially improved parameters in some ErdosPósatype results, and faster running times for algorithms for some fixedparameter tractable problems.
Computing and exploiting treedecomposition for (Max)CSP
, 2005
"... Methods exploiting the treedecomposition notion seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. Nevertheless, they have not shown a real practical interest yet. So, in this paper, we first study several methods for computing an approximate o ..."
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Cited by 2 (1 self)
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Methods exploiting the treedecomposition notion seem to provide the best approach for solving constraint networks w.r.t. the theoretical time complexity. Nevertheless, they have not shown a real practical interest yet. So, in this paper, we first study several methods for computing an approximate optimal treedecomposition before assessing their relevance for solving CSPs. Then, we propose and compare several strategies to achieve the best depthfirst traversal of the associated cluster tree w.r.t. CSP solving. These strategies concern the choice of the root cluster (i.e. the first visited cluster) and the order according to which we visit the sons of a given cluster. Finally, we propose a new decomposition strategy and heuristics which both rely on probabilistic criteria and which significantly improve the runtime.
On the Subexponential Time Complexity of CSP
 In Proc. of the 27th AAAI Conference on Artificial Intelligence
, 2013
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Quantum Inf Process DOI 10.1007/s1112801306839 Adiabatic quantum programming: minor embedding with hard faults
, 2012
"... Abstract Adiabatic quantum programming defines the timedependent mapping of a quantum algorithm into an underlying hardware or logical fabric. An essential step is embedding problemspecific information into the quantum logical fabric. We present algorithms for embedding arbitrary instances of the ..."
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Abstract Adiabatic quantum programming defines the timedependent mapping of a quantum algorithm into an underlying hardware or logical fabric. An essential step is embedding problemspecific information into the quantum logical fabric. We present algorithms for embedding arbitrary instances of the adiabatic quantum optimization algorithm into a square lattice of specialized unit cells. These methods extend with fabric growth while scaling linearly in time and quadratically in footprint. We also provide methods for handling hard faults in the logical fabric without invoking approximations to the original problem and illustrate their versatility through numerical studies of embeddability versus fault rates in square lattices of complete bipartite unit cells. The studies show that these algorithms are more resilient to faulty fabrics than naive embedding approaches, a feature which should prove useful in benchmarking the adiabatic quantum optimization algorithm on existing faulty hardware.
Graph Minors and Parameterized Algorithm Design
"... Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic ..."
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Abstract. The Graph Minors Theory, developed by Robertson and Seymour, has been one of the most influential mathematical theories in parameterized algorithm design. We present some of the basic algorithmic techniques and methods that emerged from this theory. We discuss its direct metaalgorithmic consequences, we present the algorithmic applications of core theorems such as the gridexclusion theorem, and we give a brief description of the irrelevant vertex technique.