Exact exponential algorithms to find tropical connected sets of minimum size

Chapelle, Mathieu; Cochefert, Manfred; Kratsch, Dieter; Letourneur, Romain; Liedloff, Mathieu

doi:10.1016/j.tcs.2017.03.003

Cited by 2 publications

(2 citation statements)

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“…For example tropical dominating sets were studied in [16], tropical matchings in [10], tropical paths in [9], tropical vertex-disjoint cycles in [31]. However, it seems that tropical connected sets attract the greatest attention [7,8,17,27]. Most probably that can be explained by close connection of connected tropical sets with a Graph Motif problem that was motivated by applications in biological network analysis [29] and later found applications in social networks [2] and in the context of mass spectrometry [5].…”

Section: Introductionmentioning

confidence: 99%

“…Angles d'Auriac et al [17] proved that finding a minimum tropical connected set is NP-complete even on trees of height three as well as on split and interval graphs. An exact exponential-time algorithm for Minimum Tropical Connected Set was presented by Chapelle et al [8]. In the case of a general input graph, they provide a O(1.5359 n ) algorithm while in the case of trees they give a O(1.2721 n ) algorithm.…”

Section: Introductionmentioning

confidence: 99%

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Enumeration of Minimal Tropical Connected Sets

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2023

Lecture Notes in Computer Science

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A subset of vertices in a vertex-colored graph is called tropical if vertices of each color present in the subset. This paper is dedicated to the enumeration of all minimal tropical connected sets in various classes of graphs. We show that all minimal tropical connected sets can be enumerated in O( 1.7142 n ) time on n-vertex interval graph which improves previous O(1.8613 n ) upper bound obtained by Kratsch et al. Moreover, for chordal and general class of graphs we present algorithms with running times in O(1.937 n ) and O(1.999958 n ), respectively. The last two algorithms answer question implicitly asked in the paper [Kratsch et al. SOFSEM 2017]: «Is the number of tropical sets significantly smaller than the trivial upper bound 2 n ?».

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