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**Published in:** LIPIcs, Volume 285, 18th International Symposium on Parameterized and Exact Computation (IPEC 2023)

We study the classic Dominating Set problem with respect to several prominent parameters. Specifically, we present algorithmic results that sidestep time complexity barriers by the incorporation of either approximation or larger parameterization. Our results span several parameterization regimes, including: (i,ii,iii) time/ratio-tradeoff for the parameters treewidth, vertex modulator to constant treewidth and solution size; (iv,v) FPT-algorithms for the parameters vertex cover number and feedback edge set number; and (vi) compression for the parameter feedback edge set number.

Ioannis Koutis, Michał Włodarczyk, and Meirav Zehavi. Sidestepping Barriers for Dominating Set in Parameterized Complexity. In 18th International Symposium on Parameterized and Exact Computation (IPEC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 285, pp. 31:1-31:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{koutis_et_al:LIPIcs.IPEC.2023.31, author = {Koutis, Ioannis and W{\l}odarczyk, Micha{\l} and Zehavi, Meirav}, title = {{Sidestepping Barriers for Dominating Set in Parameterized Complexity}}, booktitle = {18th International Symposium on Parameterized and Exact Computation (IPEC 2023)}, pages = {31:1--31:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-305-8}, ISSN = {1868-8969}, year = {2023}, volume = {285}, editor = {Misra, Neeldhara and Wahlstr\"{o}m, Magnus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2023.31}, URN = {urn:nbn:de:0030-drops-194506}, doi = {10.4230/LIPIcs.IPEC.2023.31}, annote = {Keywords: Dominating Set, Parameterized Complexity, Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We initiate the study of spectral generalizations of the graph isomorphism problem.
b) The Spectral Graph Dominance (SGD) problem: On input of two graphs G and H does there exist a permutation pi such that G preceq pi(H)?
c) The Spectrally Robust Graph Isomorphism (kappa-SRGI) problem: On input of two graphs G and H, find the smallest number kappa over all permutations pi such that pi(H) preceq G preceq kappa c pi(H) for some c. SRGI is a natural formulation of the network alignment problem that has various applications, most notably in computational biology.
G preceq c H means that for all vectors x we have x^T L_G x <= c x^T L_H x, where L_G is the Laplacian G.
We prove NP-hardness for SGD. We also present a kappa^3-approximation algorithm for SRGI for the case when both G and H are bounded-degree trees. The algorithm runs in polynomial time when kappa is a constant.

Alexandra Kolla, Ioannis Koutis, Vivek Madan, and Ali Kemal Sinop. Spectrally Robust Graph Isomorphism. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 84:1-84:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{kolla_et_al:LIPIcs.ICALP.2018.84, author = {Kolla, Alexandra and Koutis, Ioannis and Madan, Vivek and Sinop, Ali Kemal}, title = {{Spectrally Robust Graph Isomorphism}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {84:1--84:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.84}, URN = {urn:nbn:de:0030-drops-90887}, doi = {10.4230/LIPIcs.ICALP.2018.84}, annote = {Keywords: Network Alignment, Graph Isomorphism, Graph Similarity} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We are motivated by a tantalizing open question in exact algorithms: can we detect whether an n-vertex directed graph G has a Hamiltonian cycle in time significantly less than 2^n?
We present new randomized algorithms that improve upon several previous works:
1. We show that for any constant 0<lambda<1 and prime p we can count the Hamiltonian cycles modulo p^((1-lambda)n/(3p)) in expected time less than c^n for a constant c<2 that depends only on p and lambda. Such an algorithm was previously known only for the case of
counting modulo two [Bj\"orklund and Husfeldt, FOCS 2013].
2. We show that we can detect a Hamiltonian cycle in O^*(3^(n-alpha(G))) time and polynomial space, where alpha(G) is the size of the maximum independent set in G. In particular, this yields an O^*(3^(n/2)) time algorithm for bipartite directed graphs, which is faster than the exponential-space algorithm in [Cygan et al., STOC 2013].
Our algorithms are based on the algebraic combinatorics of "incidence assignments" that we can capture through evaluation of determinants of Laplacian-like matrices, inspired by the Matrix--Tree Theorem for directed graphs. In addition to the novel algorithms for directed Hamiltonicity, we use the Matrix--Tree Theorem to derive simple algebraic algorithms for detecting out-branchings. Specifically, we give an O^*(2^k)-time randomized algorithm for detecting out-branchings with at least k internal vertices, improving upon the algorithms of [Zehavi, ESA 2015] and [Bj\"orklund et al., ICALP 2015]. We also present an algebraic algorithm for the directed k-Leaf problem, based on a non-standard monomial detection problem.

Andreas Björklund, Petteri Kaski, and Ioannis Koutis. Directed Hamiltonicity and Out-Branchings via Generalized Laplacians. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 91:1-91:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bjorklund_et_al:LIPIcs.ICALP.2017.91, author = {Bj\"{o}rklund, Andreas and Kaski, Petteri and Koutis, Ioannis}, title = {{Directed Hamiltonicity and Out-Branchings via Generalized Laplacians}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {91:1--91:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.91}, URN = {urn:nbn:de:0030-drops-74208}, doi = {10.4230/LIPIcs.ICALP.2017.91}, annote = {Keywords: counting, directed Hamiltonicity, graph Laplacian, independent set, k-internal out-branching} }

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**Published in:** LIPIcs, Volume 14, 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)

We present three spectral sparsification algorithms that, on input a graph G with n vertices and m edges, return a graph H with n vertices and O(n log n/epsilon^2) edges that provides a strong approximation of G. Namely, for all vectors x and any epsilon>0, we have (1-epsilon) x^T L_G x <= x^T L_H x <= (1+epsilon) x^T L_G x, where L_G and L_H are the Laplacians of the two graphs. The first algorithm is a simple modification of the fastest known algorithm and runs in tilde{O}(m log^2 n) time, an O(log n) factor faster than before. The second algorithm runs in tilde{O}(m log n) time and generates a sparsifier with tilde{O}(n log^3 n) edges. The third algorithm applies to graphs where m>n log^5 n and runs in tilde{O}(m log_{m/ n log^5 n} n time. In the range where m>n^{1+r} for some constant r this becomes softO(m). The improved sparsification algorithms are employed to accelerate linear system solvers and algorithms for computing fundamental eigenvectors of dense SDD matrices.

Ioannis Koutis, Alex Levin, and Richard Peng. Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices. In 29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012). Leibniz International Proceedings in Informatics (LIPIcs), Volume 14, pp. 266-277, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2012)

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@InProceedings{koutis_et_al:LIPIcs.STACS.2012.266, author = {Koutis, Ioannis and Levin, Alex and Peng, Richard}, title = {{Improved Spectral Sparsification and Numerical Algorithms for SDD Matrices}}, booktitle = {29th International Symposium on Theoretical Aspects of Computer Science (STACS 2012)}, pages = {266--277}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-35-4}, ISSN = {1868-8969}, year = {2012}, volume = {14}, editor = {D\"{u}rr, Christoph and Wilke, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2012.266}, URN = {urn:nbn:de:0030-drops-34348}, doi = {10.4230/LIPIcs.STACS.2012.266}, annote = {Keywords: Spectral sparsification, linear system solving} }