Papers by Panos Giannopoulos
ACM Transactions on Algorithms, 2011
We study the parameterized complexity of the k -center problem on a given n -point set P in ℝ d ,... more We study the parameterized complexity of the k -center problem on a given n -point set P in ℝ d , with the dimension d as the parameter. We show that the rectilinear 3-center problem is fixed-parameter tractable, by giving an algorithm that runs in O ( n log n ) time for any fixed dimension d . On the other hand, we show that this is unlikely to be the case with both the Euclidean and rectilinear k -center problems for any k ≥ 2 and k ≥ 4 respectively. In particular, we prove that deciding whether P can be covered by the union of 2 balls of given radius or by the union of 4 cubes of given side length is W[1]-hard with respect to d , and thus not fixed-parameter tractable unless FPT=W[1]. For the Euclidean case, we also show that even an n o ( d ) -time algorithm does not exist, unless there is a 2 o ( n ) -time algorithm for n -variable 3SAT, that is, the Exponential Time Hypothesis fails.
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Siam Journal on Computing, 2008
Page 1. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. COM... more Page 1. Copyright © by SIAM. Unauthorized reproduction of this article is prohibited. SIAM J. COMPUT. c 2008 Society for Industrial and Applied Mathematics Vol. 38, No. 1, pp. 226240 IMPROVING THE STRETCH FACTOR ...
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Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m≥ n. We consider the... more Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1–ε)-approximation algorithms for translations and for rigid motions, which run in O((nm/ε 2)log (m/ε)) and O((n 2 m 2/ε 3)log m)) time, respectively. For rigid motions, we can also compute a (1–ε)-approximation in O((m 2 n 4/3 δ 1/3 / ε 3)) time, where Δ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1–ε)-approximation algorithm for rigid motions that runs in O((m 2/ε 4)log (m/ε)log2 m) time. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.
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The Discrete Milling problem is a natural and quite general graph-theoretic model for geometric m... more The Discrete Milling problem is a natural and quite general graph-theoretic model for geometric milling problems: Given a graph, one asks for a walk that covers all its vertices with a minimum number of turns, as specified in the graph model by a 0/1 turncost function f x at each vertex x giving, for each ordered pair of edges (e,f) incident at x, the turn cost at x of a walk that enters the vertex on edge e and departs on edge f. We describe an initial study of the parameterized complexity of the problem.
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Computing Research Repository, 2011
Further, we derive that testing whether a given set is an {\epsilon}-net with respect to half-spa... more Further, we derive that testing whether a given set is an {\epsilon}-net with respect to half-spaces takes {n^\Omega(d)} time under the same assumption. As intermediate results, we discover the W[1]-hardness of other well known problems, such as determining the largest empty star inside the unit cube. For this, we show that it is even hard to approximate within a factor of {2^n}.
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The Computer Journal, 2008
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International Journal of Computational Geometry and Applications, 2004
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Computing Research Repository, 2009
The Abstract Milling problem is a natural and quite general graph-theoretic model for geometric m... more The Abstract Milling problem is a natural and quite general graph-theoretic model for geometric milling problems. Given a graph, one asks for a walk that covers all its vertices with a minimum number of turns, as specified in the graph model by a 0/1 turncost function fx at each vertex x giving, for each ordered pair of edges (e,f) incident at x, the turn cost at x of a walk that enters the vertex on edge e and departs on edge f. We describe an initial study of the parameterized complexity of the problem. Our main positive result shows that Abstract Milling, parameterized by: number of turns, treewidth and maximum degree, is fixed-parameter tractable, We also show that Abstract Milling parameterized by (only) the number of turns and the pathwidth, is hard for W[1] -- one of the few parameterized intractability results for bounded pathwidth.
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Deciding whether two n-point sets A, B ∈ℝ d are congruent is a fundamental problem in geometric p... more Deciding whether two n-point sets A, B ∈ℝ d are congruent is a fundamental problem in geometric pattern matching. When the dimension d is unbounded, the problem is equivalent to graph isomorphism and is conjectured to be in FPT. When |A|=mB|=n, the problem becomes that of deciding whether A is congruent to a subset of B and is known to be NP-complete. We show that point subset congruence, with d as a parameter, is W[1]-hard, and that it cannot be solved in O(mn o(d))-time, unless SNP⊂DTIME(2 o(n)). This shows that, unless FPT=W[1], the problem of finding an isometry of A that minimizes its directed Hausdorff distance, or its Earth Mover’s Distance, to B, is not in FPT.
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ACM Transactions on Algorithms, 2009
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Computing Research Repository, 2011
We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most no... more We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a linear-time algorithm for a variant of problem (i) where the path connecting $a$ to $b$ must stay inside a given polygon $P$ with a constant number of holes, the segments are contained in $P$, and the endpoints of the segments are on the boundary of $P$. For problem (iii) we provide a cubic-time algorithm.
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Computational Geometry: Theory and Applications, 2009
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Journal of Complexity
ABSTRACT Discrepancy measures how uniformly distributed a point set is with respect to a given se... more ABSTRACT Discrepancy measures how uniformly distributed a point set is with respect to a given set of ranges. Depending on the ranges, several variants arise, including star discrepancy, box discrepancy, and discrepancy of halfspaces. These problems are solvable in time nO(d), where d is the dimension of the underlying space. As such a dependency on d becomes intractable for high-dimensional data, we ask whether it can be moderated. We answer this question negatively by proving that the canonical decision problems are W[1]-hard with respect to the dimension, implying that no f(d)⋅nO(1)-time algorithm is possible for any function f(d) unless FPT=W[1]. We also discover the W[1]-hardness of other well known problems, such as determining the largest empty box that contains the origin and is inside the unit cube. This is shown to be hard even to approximate within a factor of 2n.
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Papers by Panos Giannopoulos