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Reconstructing hv-Convex Polyominoes from Orthogonal Projections [article]

Christoph Durr, Marek Chrobak
1999 arXiv   pre-print
Tomography is the area of reconstructing objects from projections. Here we wish to reconstruct a set of cells in a two dimensional grid, given the number of cells in every row and column. The set is required to be an hv-convex polyomino, that is all its cells must be connected and the cells in every row and column must be consecutive. A simple, polynomial algorithm for reconstructing hv-convex polyominoes is provided, which is several orders of magnitudes faster than the best previously known
more » ... gorithm from Barcucci et al. In addition, the problem of reconstructing a special class of centered hv-convex polyominoes is addressed. (An object is centered if it contains a row whose length equals the total width of the object). It is shown that in this case the reconstruction problem can be solved in linear time.

Online Clique Clustering

Marek Chrobak, Christoph Dürr, Aleksander Fabijan, Bengt J. Nilsson
2019 Algorithmica
Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of
more » ... dges in the clusters, we propose an online algorithm based on the doubling technique. It has an asymptotic competitive ratio at most 15.646 and a strict competitive ratio at most 22.641. We also show that no deterministic algorithm can have an asymptotic competitive ratio better than 6. For the variant where we want to minimize the number of edges between clusters, we show that the deterministic competitive ratio of the problem is n − ω(1), where n is the number of vertices in the graph.

New Results on the Fault-Tolerant Facility Placement Problem [article]

Li Yan, Marek Chrobak
2011 arXiv   pre-print
We studied the Fault-Tolerant Facility Placement problem (FTFP) which generalizes the uncapacitated facility location problem (UFL). In FTFP, we are given a set F of sites at which facilities can be built, and a set C of clients with some demands that need to be satisfied by different facilities. A client j has demand r_j. Building one facility at a site i incurs a cost f_i, and connecting one unit of demand from client j to a facility at site i∈ costs d_ij. d_ij's are assumed to form a metric.
more » ... A feasible solution specifies the number of facilities to be built at each site and the way to connect demands from clients to facilities, with the restriction that demands from the same client must go to different facilities. Facilities at the same site are considered different. The goal is to find a solution with minimum total cost. We gave a 1.7245-approximation algorithm to the FTFP problem. Our technique is via a reduction to the Fault-Tolerant Facility Location problem, in which each client has demand r_j but each site can have at most one facility built.

Algorithms for Placing Monitors in a Flow Network [article]

Francis Chin, Marek Chrobak, Li Yan
2009 arXiv   pre-print
In the Flow Edge-Monitor Problem, we are given an undirected graph G=(V,E), an integer k > 0 and some unknown circulation \psi on G. We want to find a set of k edges in G, so that if we place k monitors on those edges to measure the flow along them, the total number of edges for which the flow can be uniquely determined is maximized. In this paper, we first show that the Flow Edge-Monitor Problem is NP-hard, and then we give two approximation algorithms: a 3-approximation algorithm with running
more » ... time O((m+n)^2) and a 2-approximation algorithm with running time O((m+n)^3), where n = |V| and m=|E|.

Finite automata and unary languages

Marek Chrobak
1986 Theoretical Computer Science
Chrobak For a~,..., ak such that god(a1,..., as) = 1 we denote by G(a~,..., ak) the greatest number b such that the Diophantine equation alx 1 -I-. • • -I-akXk = b has no solution in natural numbers.  ...  Chrobak For our purpose the bound in the corollary below will be sufficient. Corollary A. F(n) = O(H(n)). The second problem concerns linear Diophantine equations.  ...

Faster Information Gathering in Ad-Hoc Radio Tree Networks [article]

Marek Chrobak, Kevin P. Costello
2015 arXiv   pre-print
We study information gathering in ad-hoc radio networks. Initially, each node of the network has a piece of information called a rumor, and the overall objective is to gather all these rumors in the designated target node. The ad-hoc property refers to the fact that the topology of the network is unknown when the computation starts. Aggregation of rumors is not allowed, which means that each node may transmit at most one rumor in one step. We focus on networks with tree topologies, that is we
more » ... sume that the network is a tree with all edges directed towards the root, but, being ad-hoc, its actual topology is not known. We provide two deterministic algorithms for this problem. For the model that does not assume any collision detection nor acknowledgement mechanisms, we give an O(n n)-time algorithm, improving the previous upper bound of O(n n). We also show that this running time can be further reduced to O(n) if the model allows for acknowledgements of successful transmissions.

Better Bounds for Incremental Frequency Allocation in Bipartite Graphs [article]

Marek Chrobak, Łukasz Jeż, Jiří Sgall
2011 arXiv   pre-print
As shown by Chrobak and Sgall [6] , this dynamic version is NP-hard even for the special case when the input graph is a path.  ...  A lower bound of 4/3 for paths was given in [4] , and later Chrobak and Sgall [6] gave an incremental algorithm with the same ratio.  ...

SIGACT news online algorithms column 2

Marek Chrobak
2004 ACM SIGACT News
This problem has recently been studied by Lin et al [31] and by Chrobak et al [15] , who, independently, improved the first constant ratio of ≈ 30 from [32, 33] to 8, in the deterministic case, and  ...

Coordination mechanisms for congestion games

Marek Chrobak, Elias Koutsoupias
2004 ACM SIGACT News

Modeling Fluid Mixing in Microfluidic Grids [article]

Huong Luu, Marek Chrobak
2019 arXiv   pre-print
We describe an approach for modeling fluid concentration profiles in grid-based microfluidic chips for fluid mixing. This approach provides an algorithm that predicts fluid concentrations at the chip outlets. Our algorithm significantly outperforms COMSOL finite element simulations in term of runtime while still producing results that closely approximate those of COMSOL.

Online Clique Clustering [article]

Marek Chrobak, Christoph Durr, Aleksander Fabijan, Bengt Nilsson
2016 arXiv   pre-print
Clique clustering is the problem of partitioning the vertices of a graph into disjoint clusters, where each cluster forms a clique in the graph, while optimizing some objective function. In online clustering, the input graph is given one vertex at a time, and any vertices that have previously been clustered together are not allowed to be separated. The goal is to maintain a clustering with an objective value close to the optimal solution. For the variant where we want to maximize the number of
more » ... dges in the clusters, we propose an online strategy based on the doubling technique. It has an asymptotic competitive ratio at most 15.646 and an absolute competitive ratio at most 22.641. We also show that no deterministic strategy can have an asymptotic competitive ratio better than 6. For the variant where we want to minimize the number of edges between clusters, we show that the deterministic competitive ratio of the problem is n-ω(1), where n is the number of vertices in the graph.

Computing simple paths among obstacles

Qi Cheng, Marek Chrobak, Gopalakrishnan Sundaram
2000 Computational geometry
Given a set X of points in the plane, two distinguished points s, t ∈ X, and a set Φ of obstacles represented by line segments, we wish to compute a simple polygonal path from s to t that uses only points in X as vertices and avoids the obstacles in Φ. We present two results: (1) we show that finding such simple paths among arbitrary obstacles is NP-complete, and (2) we give a polynomial-time algorithm that computes simple paths when the obstacles form a simple polygon P and X is inside P . Our
more » ... algorithm runs in time O(m 2 n 2 ), where m is the number of vertices of P and n is the number of points in X.

Better bounds for incremental medians

Marek Chrobak, Mathilde Hurand
2011 Theoretical Computer Science
[5] and, independently, by Chrobak et al. [4] . In [5] , the authors also show that a 16-competitive incremental median sequence can be computed in polynomial time. Our results.  ...

Prywatyzacja wykonania zadania ochrony przeciwpożarowej przez gminę

Mateusz Paplicki, Paulina Chrobak, Marek Poniatowski
2018 Prawo
Kocowski, Prywatyzacja zarządzania majątkiem…, s. 56. 114 MATEUSZ PAPLICKI, PAULINA CHROBAK, MAREK PONIATOWSKI III.  ...  116 MATEUSZ PAPLICKI, PAULINA CHROBAK, MAREK PONIATOWSKI OSP jest nietypowym stowarzyszeniem, ponieważ można zakwalifikować ją do społecznych organizacji ratowniczych 45 .  ...  Artykuł 11c u.d.p.w. przewiduje pominięcie otwartego 118 MATEUSZ PAPLICKI, PAULINA CHROBAK, MAREK PONIATOWSKI konkursu ofert przez ministra właściwego do spraw wewnętrznych w przypadkach dotyczących zadań  ...

Hierarchies of one-way multihead automata languages

Marek Chrobak
1986 Theoretical Computer Science
Their method was used in Chrobak [1] to prove that 2DCA~2NCA. Except well-known crossing sequence arguments [6, 17] , there are two more new techniques which should be mentioned here.  ...  Chrobak In this section we will investigate properties of certain subsets of N 2 called seminets and semisectors. Both seminets and semisectors are semilinear sets of special form.  ...
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