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Sorting by Swaps with Noisy Comparisons [article]

Tomáš Gavenčiak, Barbara Geissmann, Johannes Lengler
2018 arXiv   pre-print
We study sorting of permutations by random swaps if each comparison gives the wrong result with some fixed probability $p<1/2$. We use this process as prototype for the behaviour of randomized, comparison-based optimization heuristics in the presence of noisy comparisons. As quality measure, we compute the expected fitness of the stationary distribution. To measure the runtime, we compute the minimal number of steps after which the average fitness approximates the expected fitness of the
more » ... tness of the stationary distribution. We study the process where in each round a random pair of elements at distance at most $r$ are compared. We give theoretical results for the extreme cases $r=1$ and $r=n$, and experimental results for the intermediate cases. We find a trade-off between faster convergence (for large $r$) and better quality of the solution after convergence (for small $r$).
arXiv:1803.04509v1 fatcat:mgsdjecrsnbznauru3l6f4k7ia

Cops and Robbers on Intersection Graphs [article]

Tomáš Gavenčiak, Przemysław Gordinowicz, Vít Jelínek, Pavel Klavík, Jan Kratochvíl
2016 arXiv   pre-print
Email addresses: gavento@kam.mff.cuni.cz (Tomáš Gavenčiak), pgordin@p.lodz.pl (Przemys law Gordinowicz), jelinek@iuuk.mff.cuni.cz (Vít Jelínek), klavik@iuuk.mff.cuni.cz (Pavel Klavík), honza@kam.mff.cuni.cz  ... 
arXiv:1607.08058v1 fatcat:yyeeioogmbde5hjqtd2tb7proy

Deciding first order logic properties of matroids [article]

Tomas Gavenciak and Daniel Kral and Sang-il Oum
2011 arXiv   pre-print
Frick and Grohe [J. ACM 48 (2006), 1184-1206] introduced a notion of graph classes with locally bounded tree-width and established that every first order logic property can be decided in almost linear time in such a graph class. Here, we introduce an analogous notion for matroids (locally bounded branch-width) and show the existence of a fixed parameter algorithm for first order logic properties in classes of regular matroids with locally bounded branch-width. To obtain this result, we show
more » ... result, we show that the problem of deciding the existence of a circuit of length at most k containing two given elements is fixed parameter tractable for regular matroids.
arXiv:1108.5457v1 fatcat:sizqdul2nzechhgavbtpknxmby

Firefighting on square, hexagonal, and triangular grids [article]

Tomas Gavenciak, Jan Kratochvil, Pawel Pralat
2014 arXiv   pre-print
In this paper, we consider the \emph{firefighter problem} on a graph $G=(V,E)$ that is either finite or infinite. Suppose that a fire breaks out at a given vertex $v \in V$. In each subsequent time unit, a firefighter protects one vertex which is not yet on fire, and then the fire spreads to all unprotected neighbors of the vertices on fire. The objective of the firefighter is to save as many vertices as possible (if $G$ is finite) or to stop the fire from spreading (for an infinite case). The
more » ... nfinite case). The surviving rate $\rho(G)$ of a finite graph $G$ is defined as the expected percentage of vertices that can be saved when a fire breaks out at a vertex of $G$ that is selected uniformly random. For a finite square grid $P_n \square P_n$, we show that $5/8 + o(1) \le \rho(P_n \square P_n) \le 67243/105300 + o(1)$ (leaving the gap smaller than 0.014) and conjecture that the surviving rate is asymptotic to 5/8. We define the surviving rate for infinite graphs and prove it to be 1/4 for the infinite square grid, even in the case of finitely many initial fires. For the infinite hexagonal grid we provide a winning strategy if two additional vertices can be protected at any point of the process, and we conjecture that the firefighter has no strategy to stop the fire without additional help. We also show how the speed of the spreading fire can be reduced by a constant factor.
arXiv:1305.7076v2 fatcat:qfr5gsnbjrexbo4tuqe4w5gaga

Compact I/O-Efficient Representation of Separable Graphs and Optimal Tree Layouts [article]

Tomáš Gavenčiak, Jakub Tětek
2018 arXiv   pre-print
Compact and I/O-efficient data representations play an important role in efficient algorithm design, as memory bandwidth and latency can present a significant performance bottleneck, slowing the computation by orders of magnitude. While this problem is very well explored in e.g. uniform numerical data processing, structural data applications (e.g. on huge graphs) require different algorithm-dependent approaches. Separable graph classes (i.e. graph classes with balanced separators of size
more » ... tors of size $\mathcal{O}(n^c)$ with $c < 1$) include planar graphs, bounded genus graphs, and minor-free graphs. In this article we present two generalizations of the separator theorem, to partitions with small regions only on average and to weighted graphs. Then we propose I/O-efficient succinct representation and memory layout for random walks in(weighted) separable graphs in the pointer machine model, including an efficient algorithm to compute them. Finally, we present a worst-case I/O-optimal tree layout algorithm for root-leaf path traversal, show an additive (+1)-approximation of optimal compact layout and contrast this with NP-completeness proof of finding an optimal compact layout.
arXiv:1811.06749v1 fatcat:dm6fuo7ktjdlhfgrxh5q7qxxjy

Catching a Fast Robber on Interval Graphs [chapter]

Tomáš Gavenčiak
2011 Lecture Notes in Computer Science  
We analyse the Cops and ∞-fast Robber game on the class of interval graphs and prove it to be polynomially decidable on such graphs. This solves an open problem posed in paper "Pursuing a fast robber on a graph" by Fomin et al. [4] The game is known to be already NP-hard on chordal graphs and split-graphs. The game is played by two players, one controlling k cops, the other a robber. The players alternate in turns, all the cops move at once to distance at most one, the robber moves along any
more » ... moves along any cop-free path. Cops win by capturing the robber, the robber by avoiding capture. The analysis relies on the properties of an interval representation of the graph. We show that while the game-state graph is generally exponential, every cops' move can be decomposed into simple moves of three types, and the states are reduced to those defined by certain cuts of the interval representation. This gives a restricted game equivalent to the original one together with a winning strategy computable in polynomial time.
doi:10.1007/978-3-642-20877-5_35 fatcat:wvhy7yi7d5ckhcqhj6svupbaoe

Hypertree-depth and minors in hypergraphs

Isolde Adler, Tomáš Gavenčiak, Tereza Klimošová
2012 Theoretical Computer Science  
We introduce two new notions for hypergraphs, hypertree-depth and minors in hypergraphs. We characterise hypergraphs of bounded hypertree-depth by the ultramonotone robber and marshals game, by vertex-hyperrankings and by centred hypercolourings. Furthermore, we show that minors in hypergraphs are 'well-behaved' with respect to hypertree-depth and other hypergraph invariants, such as generalised hypertree-depth and generalised hyperpath-width. We work in the framework of hypergraph pairs (G,
more » ... rgraph pairs (G, H), consisting of a graph G and a hypergraph H that share the same vertex set. This general framework allows us to obtain hypergraph minors, graph minors and induced graph minors as special cases.
doi:10.1016/j.tcs.2012.09.007 fatcat:tc53ms3r3ng6zpe7sa75wydvcq

Cop-win graphs with maximum capture-time

Tomáš Gavenčiak
2010 Discrete Mathematics  
We present an upper bound n − 4 for the maximum length of a cop and robber game (the capture-time) on a cop-win graph of order n. This bound matches the known lower bound. We analyze the structure of the class of all graphs attaining this maximum and describe an inductive construction of the entire class. A cop and robber game is a two-player vertex-to-vertex pursuit combinatorial game where the players stand on the vertices of a graph and alternate in moving to adjacent vertices. Cop's goal is
more » ... ices. Cop's goal is to capture the robber by occupying the same vertex as the robber, robber's goal is to avoid capture.
doi:10.1016/j.disc.2010.01.015 fatcat:sj73zkbfijcmtiglugbulnv6ge

Cops and Robbers on Intersection Graphs [chapter]

Tomás Gavenčiak, Vít Jelínek, Pavel Klavík, Jan Kratochvíl
2013 Lecture Notes in Computer Science  
The game of cops and robber, introduced by Nowakowski and Winkler in 1983, is played by two players on a graph G, one controlling k cops and the other one robber, all positioned on VG. The players alternate in moving their pieces to distance at most 1 each. The cops win if they capture the robber, the robber wins by escaping indefinitely. The copnumber of G, that is the smallest k such that k cops win the game, has recently been a widely studied parameter. Intersection graph classes are defined
more » ... classes are defined by their geometric representations: the vertices are represented by certain geometrical shapes and two vertices are adjacent if and only if their representations intersect. Some well-known intersection classes include interval and string graphs. Various properties of many of these classes have been studied recently, including an interest in their game-theoretic properties. In this paper we show an upper bound on the cop-number of string graphs and sharp bounds on the cop-number of interval filament graphs, circular graphs, circular arc graphs and function graphs. These results also imply polynomial algorithms determining cop-number for all these classes and their sub-classes.
doi:10.1007/978-3-642-45030-3_17 fatcat:k7ojs6vguvhslmropbob6iwnna

Seasonal variation in SARS-CoV-2 transmission in temperate climates [article]

Tomas Gavenciak, Joshua Teperowski Monrad, Gavin Leech, Mrinank Sharma, Soren Mindermann, Jan Markus Brauner, Samir Bhatt, Jan Kulveit
2021 medRxiv   pre-print
Brauner, Sören Mindermann, Mrinank Sharma, David Johnston, John Salvatier, Tomáš Gavenčiak, Anna B.  ... 
doi:10.1101/2021.06.10.21258647 fatcat:jjsbfchn3fcihkvcykmdclfvi4

Integer Programming in Parameterized Complexity: Three Miniatures [article]

Tomáš Gavenčiak and Dušan Knop and Martin Koutecký
2018 arXiv   pre-print
Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra's algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP
more » ... etween FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime $f(k)poly(n)$. We focus on: * Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used. * Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups. * Minding the poly(n): reducing $f(k)$ often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds. Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, and indefinite quadratic programs in fixed dimension.
arXiv:1711.02032v3 fatcat:zzn6cumyqrcifpk7kevo5kaafm

Performance of Bounded-Rational Agents With the Ability to Self-Modify [article]

Jakub Tětek, Marek Sklenka, Tomáš Gavenčiak
2021 arXiv   pre-print
Self-modification of agents embedded in complex environments is hard to avoid, whether it happens via direct means (e.g. own code modification) or indirectly (e.g. influencing the operator, exploiting bugs or the environment). It has been argued that intelligent agents have an incentive to avoid modifying their utility function so that their future instances work towards the same goals. Everitt et al. (2016) formally show that providing an option to self-modify is harmless for perfectly
more » ... r perfectly rational agents. We show that this result is no longer true for agents with bounded rationality. In such agents, self-modification may cause exponential deterioration in performance and gradual misalignment of a previously aligned agent. We investigate how the size of this effect depends on the type and magnitude of imperfections in the agent's rationality (1-4 below). We also discuss model assumptions and the wider problem and framing space. We examine four ways in which an agent can be bounded-rational: it either (1) doesn't always choose the optimal action, (2) is not perfectly aligned with human values, (3) has an inaccurate model of the environment, or (4) uses the wrong temporal discounting factor. We show that while in the cases (2)-(4) the misalignment caused by the agent's imperfection does not increase over time, with (1) the misalignment may grow exponentially.
arXiv:2011.06275v2 fatcat:aautuxhym5d5ppptrbt4ibcd34

Integer Programming in Parameterized Complexity: Three Miniatures

Tomás Gavenciak, Dusan Knop, Martin Koutecký, Michael Wagner
2019 International Symposium on Parameterized and Exact Computation  
Powerful results from the theory of integer programming have recently led to substantial advances in parameterized complexity. However, our perception is that, except for Lenstra's algorithm for solving integer linear programming in fixed dimension, there is still little understanding in the parameterized complexity community of the strengths and limitations of the available tools. This is understandable: it is often difficult to infer exact runtimes or even the distinction between FPT and XP
more » ... etween FPT and XP algorithms, and some knowledge is simply unwritten folklore in a different community. We wish to make a step in remedying this situation. To that end, we first provide an easy to navigate quick reference guide of integer programming algorithms from the perspective of parameterized complexity. Then, we show their applications in three case studies, obtaining FPT algorithms with runtime f (k) poly(n). We focus on: Modeling: since the algorithmic results follow by applying existing algorithms to new models, we shift the focus from the complexity result to the modeling result, highlighting common patterns and tricks which are used. Optimality program: after giving an FPT algorithm, we are interested in reducing the dependence on the parameter; we show which algorithms and tricks are often useful for speed-ups. Minding the poly(n): reducing f (k) often has the unintended consequence of increasing poly(n); so we highlight the common trade-offs and show how to get the best of both worlds. Specifically, we consider graphs of bounded neighborhood diversity which are in a sense the simplest of dense graphs, and we show several FPT algorithms for Capacitated Dominating Set, Sum Coloring, and Max-q-Cut by modeling them as convex programs in fixed dimension, n-fold integer programs, bounded dual treewidth programs, and indefinite quadratic programs in fixed dimension. ACM Subject Classification Theory of computation → Parameterized complexity and exact algorithms, Theory of computation → Graph algorithms analysis
doi:10.4230/lipics.ipec.2018.21 dblp:conf/iwpec/GavenciakKK18 fatcat:sem7iagkrzco7b5klxrguketre

LemmaTag: Jointly Tagging and Lemmatizing for Morphologically-Rich Languages with BRNNs [article]

Daniel Kondratyuk, Tomáš Gavenčiak, Milan Straka, Jan Hajič
2018 arXiv   pre-print
Tomáš Gavenčiak has been supported by Czech Science Foundation (GACR) project 17-10090Y "Network optimization".  ... 
arXiv:1808.03703v2 fatcat:ujvw6dc7zbdw7p7kcypanmrja4

Fixed parameter complexity of distance constrained labeling and uniform channel assignment problems [article]

Jiří Fiala and Tomáš Gavenčiak and Dušan Knop and Martin Koutecký and Jan Kratochvíl
2015 arXiv   pre-print
We study computational complexity of the class of distance-constrained graph labeling problems from the fixed parameter tractability point of view. The parameters studied are neighborhood diversity and clique width. We rephrase the distance constrained graph labeling problem as a specific uniform variant of the Channel Assignment problem and show that this problem is fixed parameter tractable when parameterized by the neighborhood diversity together with the largest weight. Consequently, every
more » ... onsequently, every $L(p_1, p_2, \dots, p_k)$-labeling problem is FPT when parameterized by the neighborhood diversity, the maximum $p_i$ and $k.$ Our results yield also FPT algorithms for all $L(p_1, p_2, \dots, p_k)$-labeling problems when parameterized by the size of a minimum vertex cover, answering an open question of Fiala et al.: Parameterized complexity of coloring problems: Treewidth versus vertex cover. The same consequence applies on Channel Assignment when the maximum weight is additionally included among the parameters. Finally, we show that the uniform variant of the Channel Assignment problem becomes NP-complete when generalized to graphs of bounded clique width.
arXiv:1507.00640v2 fatcat:2xlaquhqxbfc5gedz5hqjc4v6e
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