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Counting Euler Tours in Undirected Bounded Treewidth Graphs [article]

Nikhil Balaji, Samir Datta, Venkatesh Ganesan
2015 arXiv   pre-print
We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a #SAC^1 upper bound.  ...  of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs.  ...  Euler tours in bounded tree-width directed and undirected graphs can be counted in Logspace but the approach had a serious flaw in the undirected version.  ... 
arXiv:1510.04035v2 fatcat:loqv3aauhjds5jmbo5nqcuzwye

Counting Euler Tours in Undirected Bounded Treewidth Graphs

Nikhil Balaji, Samir Datta, Venkatesh Ganesan
unpublished
We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel-by proving a #SAC 1 ⊆ NC 2 ⊆ P upper bound.  ...  number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs.  ...  McKenzie, Partha Mukhopadhyay, Ramprasad Saptharishi, Srikanth Srinivasan and V.Vinay for reading a follow-up work that resulted from this paper and for their comments, from which we discovered an error in  ... 
fatcat:4wm5c2ti4bgddpwb2xg4oaz63a

Tree-width and Logspace: Determinants and Counting Euler Tours [article]

Nikhil Balaji, Samir Datta
2013 arXiv   pre-print
tours in undirected graphs, all in L.  ...  Notice that undirected Euler tours are not known to be MSO-expressible and the corresponding counting problem is in fact #P-hard for general graphs.  ...  Counting the number of Euler tours in a directed graph where the underlying undirected graph is bounded treewidth. 5. Counting the number of Euler tours in a undirected bounded treewidth graph.  ... 
arXiv:1312.7468v2 fatcat:cfyu5jvox5hnzjyinydlsxethi

Exact counting of Euler Tours for Graphs of Bounded Treewidth [article]

Prasad Chebolu, Mary Cryan, Russell Martin
2013 arXiv   pre-print
In this paper we give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian graph of bounded treewidth.  ...  To date, no polynomial-time algorithm for counting Euler tours of any class of graphs is known except for the very special case of series-parallel graphs (which have treewidth 2).  ...  |ORB(G, r)|. (3) Therefore in order to count Euler tours of an undirected Eulerian multigraph, it suffices to count Orbs of that graph. This is the approach we will take in this paper.  ... 
arXiv:1310.0185v1 fatcat:7wth6zzcvvgexouqf3wn6yqnia

Bounded Treewidth and Space-Efficient Linear Algebra [article]

Nikhil Balaji, Samir Datta
2014 arXiv   pre-print
This technique yields Logspace algorithms for counting the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs.  ...  of the adjacency matrix of a bounded tree-width graph and as our main result prove that it is in Logspace.  ...  in a previous "proof" of Theorem 4; and to Raghav Kulkarni for suggesting proof strategies for Corollary 5 and Lemma 3; and to Sebastian Kuhnert for the proof of Proposition 2.  ... 
arXiv:1412.2470v1 fatcat:thrsngyfq5cydn6uphdmzxute4

Bounded Treewidth and Space-Efficient Linear Algebra [chapter]

Nikhil Balaji, Samir Datta
2015 Lecture Notes in Computer Science  
This technique yields Logspace algorithms for counting the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs.  ...  of the adjacency matrix of a bounded tree-width graph and as our main result prove that it is in Logspace.  ...  in a previous "proof" of Theorem 4; and to Raghav Kulkarni for suggesting proof strategies for Corollary 5 and Lemma 3; and to Sebastian Kuhnert for the proof of Proposition 2.  ... 
doi:10.1007/978-3-319-17142-5_26 fatcat:hv6yitji25drzd5mpmzer3mwcu

Page 3940 of Mathematical Reviews Vol. , Issue 96g [page]

1996 Mathematical Reviews  
In fact we show that, for permutation graphs, the treewidth and pathwidth are equal.  ...  The algorithm can be applied for a computation of connectivity, ear decomposition, biconnectivity, strong orientation, ST-numbering and Euler tours problems with the same complexity.  ... 

NC Algorithms for Computing a Perfect Matching and a Maximum Flow in One-Crossing-Minor-Free Graphs [article]

David Eppstein, Vijay V. Vazirani
2020 arXiv   pre-print
if so, finding one. ∙ Finding a minimum weight perfect matching in the graph, assuming that the edge weights are polynomially bounded. ∙ Finding a maximum st-flow in the network, with arbitrary capacities  ...  Building on recent NC algorithms for planar and bounded-genus perfect matching by Anari and Vazirani and later by Sankowski, we obtain NC algorithms for perfect matching in any minor-closed graph family  ...  Acknowledgements The research of David Eppstein was supported in part by NSF grants CCF-1618301 and CCF-1616248. The research of Vijay Vazirani was supported in part by NSF grant CCF-1815901.  ... 
arXiv:1802.00084v2 fatcat:zlzad5kkynhpvjyuhjjluroh7a

Exact counting of Euler tours for generalized series-parallel graphs

Prasad Chebolu, Mary Cryan, Russell Martin
2012 Journal of Discrete Algorithms  
We give a simple polynomial-time algorithm to exactly count the number of Euler tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random  ...  Note that the class of generalized series-parallel graphs includes all outerplanar graphs. We can perform the counting in time O(m 3 ), where is the maximum degree of the graph with m edges.  ...  More recently, Creed [3] showed that counting the number of Euler tours in undirected graphs remains P -complete if G is restricted to be a planar graph.  ... 
doi:10.1016/j.jda.2011.03.011 fatcat:spcew5omnzdutphbo63ljxpmie

Exact counting of Euler Tours for generalized series-parallel graphs [article]

Prasad Chebolu, Mary Cryan, Russell Martin
2010 arXiv   pre-print
We give a simple polynomial-time algorithm to exactly count the number of Euler Tours (ETs) of any Eulerian generalized series-parallel graph, and show how to adapt this algorithm to exactly sample a random  ...  Note that the class of generalized seriesparallel graphs includes all outerplanar graphs. We can perform the counting in time O(mΔ^3), where Δ is the maximum degree of the graph with m edges.  ...  In fact, a recent result by Steve Noble [9] shows that (i),(ii) and (iii) could be counted exactly for a larger class of graphs, namely, the class of bounded treewidth graphs.  ... 
arXiv:1005.3477v1 fatcat:wcckfcxszndxnmumjaqi5nxqpa

Subexponential parameterized algorithms on bounded-genus graphs and H-minor-free graphs

Erik D. Demaine, Fedor V. Fomin, Mohammadtaghi Hajiaghayi, Dimitrios M. Thilikos
2005 Journal of the ACM  
In a parallel development A preliminary version of this article appeared in 867 of combinatorial results, we establish an upper bound on the treewidth (or branchwidth) of a boundedgenus graph that excludes  ...  In particular, this general category of graphs includes planar graphs, bounded-genus graphs, single-crossing-minorfree graphs, and any class of graphs that is closed under taking minors.  ...  Seymour for many discussions that led to combinatorial results of this article and for providing a portal into the Graph Minor Theory.  ... 
doi:10.1145/1101821.1101823 fatcat:3ooif6ldsvd3pm4fw4a3tgonym

Parallel Algorithms with Optimal Speedup for Bounded Treewidth

Hans L. Bodlaender, Torben Hagerup
1998 SIAM journal on computing (Print)  
bounded treewidth, including all decision problems expressible in monadic second-order logic.  ...  We describe the first parallel algorithm with optimal speedup for constructing minimum-width tree decompositions of graphs of bounded treewidth.  ...  Then we construct an Euler tour of the modified tree (see [40] ) and root the tree by breaking the Euler tour at an arbitrary node of degree at most 2, declared to be the root, computing the distance  ... 
doi:10.1137/s0097539795289859 fatcat:3yz74oi4fff7rgzve56ikammcq

Parallel algorithms with optimal speedup for bounded treewidth [chapter]

Hans L. Bodlaender, Torben Hagerup
1995 Lecture Notes in Computer Science  
bounded treewidth, including all decision problems expressible in monadic second-order logic.  ...  We describe the first parallel algorithm with optimal speedup for constructing minimum-width tree decompositions of graphs of bounded treewidth.  ...  Then we construct an Euler tour of the modified tree (see [40] ) and root the tree by breaking the Euler tour at an arbitrary node of degree at most 2, declared to be the root, computing the distance  ... 
doi:10.1007/3-540-60084-1_80 fatcat:4dajycrm5jdidbuwaoctyztd6m

Ubiquitous Parameterization — Invitation to Fixed-Parameter Algorithms [chapter]

Rolf Niedermeier
2004 Lecture Notes in Computer Science  
In every area of computer science, we find all kinds of "special aspects" to the problems encountered.  ...  The purpose of this article is to stir the reader's interest in this field by providing a gentle introduction to the rewarding field of fixed-parameter algorithms.  ...  : Input: An undirected graph G = (V, E).  ... 
doi:10.1007/978-3-540-28629-5_4 fatcat:vf5gtr3u7jceblnz2ucj77hmky

A Fast Algorithm for the Product Structure of Planar Graphs [article]

Pat Morin
2020 arXiv   pre-print
Dujmović et al (FOCS2019) recently proved that every planar graph G is a subgraph of H⊠ P, where ⊠ denotes the strong graph product, H is a graph of treewidth 8 and P is a path.  ...  In this note, we show that this algorithm can be made to run in O(nlog n) time.  ...  Ivana implemented the algorithm described in Section 2 and the algorithm described in Section 3.  ... 
arXiv:2004.02530v3 fatcat:4vqbxl2be5fjrfqj3hutcvkni4
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