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Subexponential time algorithms for finding small tree and path decompositions [article]

Hans L. Bodlaender, Jesper Nederlof
2016 arXiv   pre-print
The Minimum Size Tree Decomposition (MSTD) and Minimum Size Path Decomposition (MSPD) problems ask for a given n-vertex graph G and integer k, what is the minimum number of bags of a tree decomposition  ...  We present algorithms that solve both problems for fixed k in 2^O(n/ n) time and show that they cannot be solved in 2^o(n / n) time, assuming the Exponential Time Hypothesis.  ...  We thank the anonymous referees for their detailed and helpful comments.  ... 
arXiv:1601.02415v2 fatcat:iu2zhpgpjffmdl34vhkujzc4nm

Subexponential Time Algorithms for Finding Small Tree and Path Decompositions [chapter]

Hans L. Bodlaender, Jesper Nederlof
2015 Lecture Notes in Computer Science  
The Minimum Size Tree Decomposition (MSTD) and Minimum Size Path Decomposition (MSPD) problems ask for a given n-vertex graph G and integer k, what is the minimum number of bags of a tree decomposition  ...  We present algorithms that solve both problems for fixed k in 2 O(n/ log n) time and show that they cannot be solved in 2 o(n/ log n) time, assuming the Exponential Time Hypothesis.  ...  We thank the anonymous referees for their detailed and helpful comments.  ... 
doi:10.1007/978-3-662-48350-3_16 fatcat:74rujxgshnhqnmqwds4zk55x4m

Subexponential Parameterized Algorithms for Planar and Apex-Minor-Free Graphs via Low Treewidth Pattern Covering

Fedor V. Fomin, Daniel Lokshtanov, Daniel Marx, Marcin Pilipczuk, Michal Pilipczuk, Saket Saurabh
2016 2016 IEEE 57th Annual Symposium on Foundations of Computer Science (FOCS)  
for graphs of bounded treewidth, this result gives a versatile technique for obtaining (randomized) subexponential parameterized algorithms for problems on planar graphs, usually with running time bound  ...  k-PATH, VERTEX COVER LOCAL SEARCH, and SUBGRAPH ISOMORPHISM, among others.  ...  for pointing out the similarities between our clustering procedure and tools from the theory of metric embeddings.  ... 
doi:10.1109/focs.2016.62 dblp:conf/focs/FominLMPPS16 fatcat:dyyrhwi5yfgqrh2h2lnjn7pf2u

Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Jana Novotná, Karolina Okrasa, Michal Pilipczuk, Pawel Rzazewski, Erik Jan Van Leeuwen, Bartosz Walczak, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
This leads, for example, to the following corollaries for specific classes C and D: a largest induced forest in a Pt-free graph can be found in 2Õ (n 2/3 ) time, for every fixed t; and a largest induced  ...  ACM Subject Classification Theory of computation → Design and analysis of algorithms; Theory of computation → Problems, reductions and completeness  ...  Therefore, for the purpose of verifying property 3, we can assume that a tree decomposition of width O(w) is additionally provided 23:6 Subexponential Algorithms Finding Large Sparse Subgraphs on input  ... 
doi:10.4230/lipics.ipec.2019.23 dblp:conf/iwpec/NovotnaOPRLW19 fatcat:lcgp6expunh4rlp3jjysmw6lsa

Faster approximation schemes and parameterized algorithms on (odd-)H-minor-free graphs

Siamak Tazari
2012 Theoretical Computer Science  
We use these algorithms to derive faster 2-approximations; furthermore, we present the first PTASes and subexponential FPTalgorithms for independent set and vertex cover on these graph classes using a  ...  Our technique applies, in particular, to problems such as Steiner tree, (directed) subgraph with a property, (directed) longest path, and (connected/independent) dominating set, on some or all proper minor-closed  ...  We improved the best known FPT-algorithms for k-Steiner Tree and Directed k-Path in H-minor-free graphs that were previously only known to be in EPT even on planar graphs (in the case of Steiner Tree).  ... 
doi:10.1016/j.tcs.2011.09.014 fatcat:253avdexpzhetkbpp24762fsnm

Subexponential-time algorithms for finding large induced sparse subgraphs [article]

Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, Erik Jan van Leeuwen, Bartosz Walczak
2019 arXiv   pre-print
This leads, for example, to the following corollaries for specific classes C and D: * a largest induced forest in a P_t-free graph can be found in 2^Õ(n^2/3) time, for every fixed t; and * a largest induced  ...  ; and * the considered problem admits a single-exponential fixed-parameter algorithm when parameterized by the treewidth of the input graph.  ...  It is known that small balanced separators can be used to construct tree decompositions of small width, as made explicit in the following lemma. Lemma 1 ([11] ).  ... 
arXiv:1910.01082v1 fatcat:25wctiirxbdn7iuuwgrltscxeq

Subexponential-Time Algorithms for Finding Large Induced Sparse Subgraphs

Jana Novotná, Karolina Okrasa, Michał Pilipczuk, Paweł Rzążewski, Erik Jan van Leeuwen, Bartosz Walczak
2020 Algorithmica  
This leads, for example, to the following corollaries for specific classes C and D :a largest induced forest in a P t -free graph can be found in 2 O ~ ( n 2 / 3 ) time, for every fixed t; anda largest  ...  Let C and D be hereditary graph classes. Consider the following problem: given a graph G ∈ D , find a largest, in terms of the number of vertices, induced subgraph of G that belongs to C .  ...  It is known that small balanced separators can be used to construct tree decompositions of small width, as made explicit in the following lemma.  ... 
doi:10.1007/s00453-020-00745-z pmid:34720297 pmcid:PMC8550324 fatcat:3xx4rhvhbvaq3ewppedpfhmuqu

Faster Approximation Schemes and Parameterized Algorithms on H-Minor-Free and Odd-Minor-Free Graphs [chapter]

Siamak Tazari
2010 Lecture Notes in Computer Science  
We use these algorithms to derive faster 2-approximations; furthermore, we present the first PTASes and subexponential FPT-algorithms for independent set and vertex cover on these graph classes using a  ...  Our technique applies, in particular, to problems such as Steiner tree, (directed) subgraph with a property, (directed) longest path, and (connected/independent) dominating set, on some or all proper minor-closed  ...  I would like to thank Holger Dell, Martin Grohe, and Matthias Mnich for helpful discussions and comments on earlier versions of this work.  ... 
doi:10.1007/978-3-642-15155-2_56 fatcat:zno6rurjkre3dasjvjchdcbizu

The Square Root Phenomenon in Planar Graphs [chapter]

Dániel Marx
2013 Lecture Notes in Computer Science  
Subexponential algorithm for 3-Coloring Can we achieve even better running time (e.g., 2 O( 3 √ n) ) on planar graphs? There is no 2 o(m) -time algorithm for m-clause 3SAT.  ...  We use this observation to find a path of length at least k on planar graphs: Set w := 5 √ k. Find an O(1)-approximate tree decomposition.  ...  Input: A k × k matrix and a set of pairs S i,j ⊆ [D] × [D] for each cell. Find: A pair s i,j ∈ S i,j for each cell such that Vertical neighbors agree in the 1st coordinate.  ... 
doi:10.1007/978-3-642-38756-2_1 fatcat:ldsbn2zrdje4tlferdy2z7cg5u

The Square Root Phenomenon in Planar Graphs [chapter]

Dániel Marx
2013 Lecture Notes in Computer Science  
Subexponential algorithm for 3-Coloring Can we achieve even better running time (e.g., 2 O( 3 √ n) ) on planar graphs? There is no 2 o(m) -time algorithm for m-clause 3SAT.  ...  We use this observation to find a path of length at least k on planar graphs: Set w := 5 √ k. Find an O(1)-approximate tree decomposition.  ...  Input: A k × k matrix and a set of pairs S i,j ⊆ [D] × [D] for each cell. Find: A pair s i,j ∈ S i,j for each cell such that Vertical neighbors agree in the 1st coordinate.  ... 
doi:10.1007/978-3-642-39212-2_4 fatcat:aoetw6pginbj3gkfymdj3f54fq

Subexponential Time Algorithms for Embedding H-Minor Free Graphs

Hans L. Bodlaender, Jesper Nederlof, Tom C. Van Der Zanden, Marc Herbstritt
2016 International Colloquium on Automata, Languages and Programming  
We show that, for any fixed graph H and > 0, if P is H-Minor Free and G has treewidth tw, We also show that this is optimal, in the sense that the existence of an algorithm for one of these problems running  ...  in 2 o(n/ log n) time would contradict the Exponential Time Hypothesis.  ...  They give a 2 O(n/ log n) -time algorithm for finding tree decompositions with few bags and a matching lower bound (based on the Exponential Time Hypothesis), and a 2 O(n/ log n) algorithm for determining  ... 
doi:10.4230/lipics.icalp.2016.9 dblp:conf/icalp/BodlaenderNZ16 fatcat:msf7om5zsvht3hflwrweoewfmq

H-colouring P_t-free graphs in subexponential time [article]

Carla Groenland and Karolina Okrasa and Pawel Rzążewski and Alex Scott and Paul Seymour and Sophie Spirkl
2019 arXiv   pre-print
On the other hand, no subexponential time algorithm exists for 4-colourability of P_t-free graphs assuming the Exponential Time Hypothesis.  ...  We show that the generating function for (list) graph homomorphisms from G to H can be calculated in subexponential time 2^O(√(tn(n))) for n=|V(G)| in the class of P_t-free graphs G.  ...  The pathwidth of G is defined as the minimum of max i |X i | − 1 over the path-decompositions of G. Let T be a rooted tree with root r. For a vertex v, let T v denote the path of T between r and v.  ... 
arXiv:1803.05396v3 fatcat:jkbqftbapzfc3lbhfuvucc52oq

Max Weight Independent Set in graphs with no long claws: An analog of the Gyárfás' path argument [article]

Konrad Majewski, Tomáš Masařík, Jana Novotná, Karolina Okrasa, Marcin Pilipczuk, Paweł Rzążewski, Marek Sokołowski
2022 arXiv   pre-print
] and provide a subexponential-time algorithm with improved running time 2^𝒪(√(n)log n) and a quasipolynomial-time approximation scheme with improved running time 2^𝒪(ε^-1log^5 n).  ...  The Gyárfás' path argument, a powerful tool that is the main building block for many algorithms in P_t-free graphs, ensures that given an n-vertex P_t-free graph, in polynomial time we can find a set P  ...  We expect it is a first step to get a quasipolynomial-time algorithm for MWIS in S t,t,t -free graphs, similarly as Theorem 1 is an essential ingredient of the algorithms for P t -free graphs [15, 23]  ... 
arXiv:2203.04836v1 fatcat:aoalxcrsc5gfnf2ijr6rk63kfq

Parameterized Algorithms and Data Reduction for Safe Convoy Routing

René Van Bevern, Till Fluschnik, Oxana Yu. Tsidulko, Michael Wagner
2018 Algorithmic Approaches for Transportation Modeling, Optimization, and Systems  
For graphs with constant crossing number, we provide a subexponential 2 O( √ n) -time algorithm and prove a matching lower bound.  ...  Regarding tree-like graphs, we obtain a 2 O(tw) • 2 • n-time algorithm for graphs of treewidth tw, show that there is no problem kernel with size polynomial in tw, yet show a problem kernel with size polynomial  ...  the path length, and so can our subexponential-time algorithms in planar graphs (Theorem 2.1).  ... 
doi:10.4230/oasics.atmos.2018.10 dblp:conf/atmos/BevernFT18 fatcat:zyneram6v5h6jkujxn277vufhy

Subexponential Parameterized Algorithms and Kernelization on Almost Chordal Graphs

Fedor V. Fomin, Petr A. Golovach, Peter Sanders, Grzegorz Herman, Fabrizio Grandoni
2020 European Symposium on Algorithms  
of subexponential parameterized algorithms for most of the structural parameters and optimization problems is highly unlikely.  ...  Besides subexponential time algorithms, the class of Chordal-ke graphs appears to be appealing from the perspective of kernelization (with parameter k).  ...  Acknowledgements We thank Torstein Strømme, Daniel Lokshtanov, and Pranabendu Misra for fruitful discussions on the topic of this paper.  ... 
doi:10.4230/lipics.esa.2020.49 dblp:conf/esa/FominG20a fatcat:dh4jo3kb5rc3xgw7rqfted5doi
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