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Sorting by Swaps with Noisy Comparisons
[article]

2018
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arXiv
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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

arXiv:1803.04509v1
fatcat:mgsdjecrsnbznauru3l6f4k7ia
## 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 $r$).##
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Cops and Robbers on Intersection Graphs
[article]

2016
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arXiv
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pre-print

Email addresses: gavento@kam.mff.cuni.cz (

arXiv:1607.08058v1
fatcat:yyeeioogmbde5hjqtd2tb7proy
*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 ...##
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Deciding first order logic properties of matroids
[article]

2011
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arXiv
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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

arXiv:1108.5457v1
fatcat:sizqdul2nzechhgavbtpknxmby
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... 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.##
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Firefighting on square, hexagonal, and triangular grids
[article]

2014
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arXiv
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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

arXiv:1305.7076v2
fatcat:qfr5gsnbjrexbo4tuqe4w5gaga
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... 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.##
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Compact I/O-Efficient Representation of Separable Graphs and Optimal Tree Layouts
[article]

2018
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arXiv
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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

arXiv:1811.06749v1
fatcat:dm6fuo7ktjdlhfgrxh5q7qxxjy
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... 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.##
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Catching a Fast Robber on Interval Graphs
[chapter]

2011
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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

doi:10.1007/978-3-642-20877-5_35
fatcat:wvhy7yi7d5ckhcqhj6svupbaoe
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... 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.##
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Hypertree-depth and minors in hypergraphs

2012
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Theoretical Computer Science
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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,

doi:10.1016/j.tcs.2012.09.007
fatcat:tc53ms3r3ng6zpe7sa75wydvcq
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... 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.##
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Cop-win graphs with maximum capture-time

2010
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Discrete Mathematics
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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

doi:10.1016/j.disc.2010.01.015
fatcat:sj73zkbfijcmtiglugbulnv6ge
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... ices. Cop's goal is to capture the robber by occupying the same vertex as the robber, robber's goal is to avoid capture.##
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Cops and Robbers on Intersection Graphs
[chapter]

2013
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Lecture Notes in Computer Science
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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

doi:10.1007/978-3-642-45030-3_17
fatcat:k7ojs6vguvhslmropbob6iwnna
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... 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.##
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Seasonal variation in SARS-CoV-2 transmission in temperate climates
[article]

2021
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medRxiv
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pre-print

Brauner, Sören Mindermann, Mrinank Sharma, David Johnston, John Salvatier,

doi:10.1101/2021.06.10.21258647
fatcat:jjsbfchn3fcihkvcykmdclfvi4
*Tomáš**Gavenčiak*, Anna B. ...##
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Integer Programming in Parameterized Complexity: Three Miniatures
[article]

2018
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arXiv
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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

arXiv:1711.02032v3
fatcat:zzn6cumyqrcifpk7kevo5kaafm
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... 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.##
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Performance of Bounded-Rational Agents With the Ability to Self-Modify
[article]

2021
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arXiv
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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

arXiv:2011.06275v2
fatcat:aautuxhym5d5ppptrbt4ibcd34
## 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.##
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Integer Programming in Parameterized Complexity: Three Miniatures

2019
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International Symposium on Parameterized and Exact Computation
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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

doi:10.4230/lipics.ipec.2018.21
dblp:conf/iwpec/GavenciakKK18
fatcat:sem7iagkrzco7b5klxrguketre
## 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##
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LemmaTag: Jointly Tagging and Lemmatizing for Morphologically-Rich Languages with BRNNs
[article]

2018
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arXiv
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pre-print

*Tomáš*

*Gavenčiak*has been supported by Czech Science Foundation (GACR) project 17-10090Y "Network optimization". ...

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

2015
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arXiv
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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

arXiv:1507.00640v2
fatcat:2xlaquhqxbfc5gedz5hqjc4v6e
## 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.
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