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**Published in:** LIPIcs, Volume 181, 31st International Symposium on Algorithms and Computation (ISAAC 2020)

Graph parameters such as the clique number, the chromatic number, and the independence number are central in many areas, ranging from computer networks to linguistics to computational neuroscience to social networks. In particular, the chromatic number of a graph (i.e., the smallest number of colors needed to color all vertices such that no two adjacent vertices are of the same color) can be applied in solving practical tasks as diverse as pattern matching, scheduling jobs to machines, allocating registers in compiler optimization, and even solving Sudoku puzzles. Typically, however, the underlying graphs are subject to (often minor) changes. To make these applications of graph parameters robust, it is important to know which graphs are stable for them in the sense that adding or deleting single edges or vertices does not change them. We initiate the study of stability of graphs for such parameters in terms of their computational complexity. We show that, for various central graph parameters, the problem of determining whether or not a given graph is stable is complete for Θ₂ᵖ, a well-known complexity class in the second level of the polynomial hierarchy, which is also known as "parallel access to NP."

Fabian Frei, Edith Hemaspaandra, and Jörg Rothe. Complexity of Stability. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 19:1-19:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{frei_et_al:LIPIcs.ISAAC.2020.19, author = {Frei, Fabian and Hemaspaandra, Edith and Rothe, J\"{o}rg}, title = {{Complexity of Stability}}, booktitle = {31st International Symposium on Algorithms and Computation (ISAAC 2020)}, pages = {19:1--19:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-173-3}, ISSN = {1868-8969}, year = {2020}, volume = {181}, editor = {Cao, Yixin and Cheng, Siu-Wing and Li, Minming}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2020.19}, URN = {urn:nbn:de:0030-drops-133631}, doi = {10.4230/LIPIcs.ISAAC.2020.19}, annote = {Keywords: Stability, Robustness, Complexity, Local Modifications, Colorability, Vertex Cover, Clique, Independent Set, Satisfiability, Unfrozenness, Criticality, DP, coDP, Parallel Access to NP} }

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**Published in:** LIPIcs, Volume 138, 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)

Can we efficiently compute optimal solutions to instances of a hard problem from optimal solutions to neighboring (i.e., locally modified) instances? For example, can we efficiently compute an optimal coloring for a graph from optimal colorings for all one-edge-deleted subgraphs? Studying such questions not only gives detailed insight into the structure of the problem itself, but also into the complexity of related problems; most notably graph theory’s core notion of critical graphs (e.g., graphs whose chromatic number decreases under deletion of an arbitrary edge) and the complexity-theoretic notion of minimality problems (also called criticality problems, e.g., recognizing graphs that become 3-colorable when an arbitrary edge is deleted).
We focus on two prototypical graph problems, Colorability and Vertex Cover. For example, we show that it is NP-hard to compute an optimal coloring for a graph from optimal colorings for all its one-vertex-deleted subgraphs, and that this remains true even when optimal solutions for all one-edge-deleted subgraphs are given. In contrast, computing an optimal coloring from all (or even just two) one-edge-added supergraphs is in P. We observe that Vertex Cover exhibits a remarkably different behavior, demonstrating the power of our model to delineate problems from each other more precisely on a structural level.
Moreover, we provide a number of new complexity results for minimality and criticality problems. For example, we prove that Minimal-3-UnColorability is complete for DP (differences of NP sets), which was previously known only for the more amenable case of deleting vertices rather than edges. For Vertex Cover, we show that recognizing beta-vertex-critical graphs is complete for Theta_2^p (parallel access to NP), obtaining the first completeness result for a criticality problem for this class.

Elisabet Burjons, Fabian Frei, Edith Hemaspaandra, Dennis Komm, and David Wehner. Finding Optimal Solutions With Neighborly Help. In 44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 138, pp. 78:1-78:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{burjons_et_al:LIPIcs.MFCS.2019.78, author = {Burjons, Elisabet and Frei, Fabian and Hemaspaandra, Edith and Komm, Dennis and Wehner, David}, title = {{Finding Optimal Solutions With Neighborly Help}}, booktitle = {44th International Symposium on Mathematical Foundations of Computer Science (MFCS 2019)}, pages = {78:1--78:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-117-7}, ISSN = {1868-8969}, year = {2019}, volume = {138}, editor = {Rossmanith, Peter and Heggernes, Pinar and Katoen, Joost-Pieter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2019.78}, URN = {urn:nbn:de:0030-drops-110221}, doi = {10.4230/LIPIcs.MFCS.2019.78}, annote = {Keywords: Critical Graphs, Computational Complexity, Structural Self-Reducibility, Minimality Problems, Colorability, Vertex Cover, Satisfiability, Reoptimization, Advice} }

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**Published in:** LIPIcs, Volume 117, 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)

We show that the counting class LWPP [S. Fenner et al., 1994] remains unchanged even if one allows a polynomial number of gap values rather than one. On the other hand, we show that it is impossible to improve this from polynomially many gap values to a superpolynomial number of gap values by relativizable proof techniques.
The first of these results implies that the Legitimate Deck Problem (from the study of graph reconstruction) is in LWPP (and thus low for PP, i.e., PP^{Legitimate Deck} = PP) if the weakened version of the Reconstruction Conjecture holds in which the number of nonisomorphic preimages is assumed merely to be polynomially bounded. This strengthens the 1992 result of Köbler, Schöning, and Torán [J. Köbler et al., 1992] that the Legitimate Deck Problem is in LWPP if the Reconstruction Conjecture holds, and provides strengthened evidence that the Legitimate Deck Problem is not NP-hard.
We additionally show on the one hand that our main LWPP robustness result also holds for WPP, and also holds even when one allows both the rejection- and acceptance- gap-value targets to simultaneously be polynomial-sized lists; yet on the other hand, we show that for the #P-based analog of LWPP the behavior much differs in that, in some relativized worlds, even two target values already yield a richer class than one value does.

Edith Hemaspaandra, Lane A. Hemaspaandra, Holger Spakowski, and Osamu Watanabe. The Robustness of LWPP and WPP, with an Application to Graph Reconstruction. In 43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 117, pp. 51:1-51:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{hemaspaandra_et_al:LIPIcs.MFCS.2018.51, author = {Hemaspaandra, Edith and Hemaspaandra, Lane A. and Spakowski, Holger and Watanabe, Osamu}, title = {{The Robustness of LWPP and WPP, with an Application to Graph Reconstruction}}, booktitle = {43rd International Symposium on Mathematical Foundations of Computer Science (MFCS 2018)}, pages = {51:1--51:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-086-6}, ISSN = {1868-8969}, year = {2018}, volume = {117}, editor = {Potapov, Igor and Spirakis, Paul and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2018.51}, URN = {urn:nbn:de:0030-drops-96330}, doi = {10.4230/LIPIcs.MFCS.2018.51}, annote = {Keywords: structural complexity theory, robustness of counting classes, the legitimate deck problem, PP-lowness, the Reconstruction Conjecture} }

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**Published in:** LIPIcs, Volume 20, 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)

Most theoretical definitions about the complexity of manipulating elections focus on the decision problem of recognizing which instances can be successfully manipulated, rather than the search problem of finding the successful manipulative actions. Since the latter is a far more natural goal for manipulators, that definitional focus may be misguided if these two complexities can differ. Our main result is that they probably do differ: If integer factoring is hard, then for election manipulation, election bribery, and some types of election control, there are election systems for which recognizing which instances can be successfully manipulated is in polynomial time but producing the successful manipulations cannot be done in polynomial time.

Edith Hemaspaandra, Lane A. Hemaspaandra, and Curtis Menton. Search versus Decision for Election Manipulation Problems. In 30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013). Leibniz International Proceedings in Informatics (LIPIcs), Volume 20, pp. 377-388, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2013)

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@InProceedings{hemaspaandra_et_al:LIPIcs.STACS.2013.377, author = {Hemaspaandra, Edith and Hemaspaandra, Lane A. and Menton, Curtis}, title = {{Search versus Decision for Election Manipulation Problems}}, booktitle = {30th International Symposium on Theoretical Aspects of Computer Science (STACS 2013)}, pages = {377--388}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-50-7}, ISSN = {1868-8969}, year = {2013}, volume = {20}, editor = {Portier, Natacha 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.2013.377}, URN = {urn:nbn:de:0030-drops-39498}, doi = {10.4230/LIPIcs.STACS.2013.377}, annote = {Keywords: Search vs. decision, application of structural complexity theory} }

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**Published in:** LIPIcs, Volume 1, 25th International Symposium on Theoretical Aspects of Computer Science (2008)

Modal logics are widely used in computer science. The complexity
of modal satisfiability problems has been investigated since the
1970s, usually proving results on a case-by-case basis. We prove a
very general classification for a wide class of relevant logics:
Many important subclasses of modal logics can be obtained by
restricting the allowed models with first-order Horn formulas. We
show that the satisfiability problem for each of these logics is
either NP-complete or PSPACE-hard, and exhibit a simple
classification criterion. Further, we prove matching PSPACE upper
bounds for many of the PSPACE-hard logics.

Edith Hemaspaandra and Henning Schnoor. On the Complexity of Elementary Modal Logics. In 25th International Symposium on Theoretical Aspects of Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 1, pp. 349-360, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{hemaspaandra_et_al:LIPIcs.STACS.2008.1356, author = {Hemaspaandra, Edith and Schnoor, Henning}, title = {{On the Complexity of Elementary Modal Logics}}, booktitle = {25th International Symposium on Theoretical Aspects of Computer Science}, pages = {349--360}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-06-4}, ISSN = {1868-8969}, year = {2008}, volume = {1}, editor = {Albers, Susanne and Weil, Pascal}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2008.1356}, URN = {urn:nbn:de:0030-drops-13561}, doi = {10.4230/LIPIcs.STACS.2008.1356}, annote = {Keywords: } }