Linear matrix period in max-plus algebra

Gavalec, Martin

doi:10.1016/s0024-3795(00)00020-3

Cited by 30 publications

(29 citation statements)

References 12 publications

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“…There are l 2 t edges in K, so there are at most O((l 2 t) It is likely that a fixed-parameter tractable solution can also be described by the use of min-max algebra for shortest paths, see [5] and [2], [8].…”

Section: Dealing With Bridge Vectorsmentioning

confidence: 99%

Minimizing the Number of Label Transitions Around a Nonseparating Vertex of a Planar Graph

Mohar¹,

Škoda²

2012

JGAA

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We study the minimum number of label transitions around a given vertex v0 in a planar multigraph G, in which the edges incident with v0 are labelled with integers 1, . . . , l, and the minimum is taken over all embeddings of G in the plane. For a fixed number of labels, a lineartime fixed-parameter tractable algorithm that computes the minimum number of label transitions around v0 is presented. If the number of labels is unconstrained, then the problem of deciding whether the minimum number of label transitions is at most k is NP-complete.

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Section: Dealing With Bridge Vectorsmentioning

confidence: 99%

Minimizing the Number of Label Transitions Around a Nonseparating Vertex of a Planar Graph

Mohar¹,

Škoda²

2012

JGAA

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“…Namely, the period of A is the lcm of all its orbit periods [9] and an O(n 4 ) algorithm for computing the period of an orbit was given in [24]. However, the problem to decide whether for a given matrix A a vector x exists, such that per(A) = per(A; x), is an NP-complete problem already in the binary case [21].…”

Section: Powers Of Matrices Orbits and Eigenvectorsmentioning

confidence: 99%

“…In particular, for the description of the behaviour of fuzzy systems, whose states are expressed using fuzzy values taken from the real interval 0; 1 or an arbitrary distributive lattice, the powers of lattice matrices are of special importance. Among the structures, over which the powers of matrices are computed, are max-addition (linear systems with synchronization [3,11,24,39,45]) max-multiplication [12,13], max-min (fuzzy systems, [9,16,17,18,20,35,36,53]), and their generalizations to max-t-norm [14,15] and to general sup-inf in a distributive lattice [52]. Results include proving convergence for special types of matrices [16,29,53], proving upper bounds [36] or computing the length of the oscillation period of the matrix power sequence [20], estimating its exponent, investigating connections between power sequence and eigenvectors [8,51], etc.…”

Section: Introductionmentioning

confidence: 99%

“…An interested reader is invited to study the relevant references, e.g. [3,11,24,38,39,45,54], for max-plus algebra, [12,13,41] for max-times algebra, [14,15] for products using t-norms and [54] for a general algebraic framework.) We would like to stress the signiÿcance of the graph-theoretical approach in the study of this topic, which was applied as early as in 1957 [43].…”

Section: Introductionmentioning

confidence: 99%

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Powers of matrices over distributive lattices?a review

Cechlárová

2003

Fuzzy Sets and Systems

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“…It was shown in [1] by Baccelli et al that a matrix A is almost linear periodic if the corresponding digraph G(A) is strongly connected. Gavalec described in [5] an O(n 3 ) algorithm for computing the linear period of a matrix, in this case. Further it was proved in [5], that in general case, when G(A) is not necessarily strongly connected, the problem of deciding whether a given matrix in max-plus algebra is not almost linear periodic, is NP-complete.…”

Section: Introductionmentioning

confidence: 99%