Sweeping Graphs with Large Clique Number

Yang, Boting; Dyer, Danny; Alspach, Brian

doi:10.1007/978-3-540-30551-4_77

Cited by 33 publications

(25 citation statements)

References 4 publications

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“…Monotonicity results for other versions of the problem can be found in [2,18,19,21,30], among others. Nonmonotonicity results can be found in, for instance, [11,32].…”

Section: Graph Searchingmentioning

confidence: 99%

Submodular Percolation

Brightwell¹,

Winkler²

2009

SIAM J. Discrete Math.

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Abstract. Let f : L → R be a submodular function on a modular lattice L; we show that there is a maximal chain C in L on which the sequence of values of f is minimal among all paths from 0 to 1 in the Hasse diagram of L, in a certain well-behaved partial order on sequences of reals. One consequence is that the maximum value of f on C is minimized over all such paths-i.e., if one can percolate from 0 to 1 on lattice points X with f (X) ≤ b, then one can do so along a maximal chain. The partial order on real sequences is defined by putting a(0), a(1), . . . , a(m) b (0), . . . , b(n) if there is a way to "schedule" the sequences starting at (a(0), b (0)) and ending at (a(m), b(n)) so that always a(i) ≤ b(j). Putting a ≡ b if a b a, each equivalence class has a unique shortest sequence which we call a worm. We use the properties of worms to give an efficient method to schedule many real sequences in parallel. The results in the paper are applied in a number of other settings, including obstacle navigation, graph search, coordinate percolation, and finding a lost child in a field.

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“…Monotonicity results for other versions of the problem can be found in [2,18,19,21,30], among others. Nonmonotonicity results can be found in, for instance, [11,32].…”

Section: Graph Searchingmentioning

confidence: 99%

Submodular Percolation

Brightwell¹,

Winkler²

2009

SIAM J. Discrete Math.

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“…imposing that no re-contamination is allowed when computing connected strategies can lead to strategies that require more agents. An example of this is constructed in [49]. Yet, for practical applications both of these properties, monotonicity and connectedness, are very useful.…”

Section: Related Workmentioning

confidence: 99%

Hierarchical visibility for guaranteed search in large-scale outdoor terrain

Kleiner

Kolling

Lewis

et al. 2011

Auton Agent Multi-Agent Syst

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Searching for moving targets in large environments is a challenging task that is relevant in several problem domains, such as capturing an invader in a camp, guarding security facilities, and searching for victims in large-scale search and rescue scenarios. The guaranteed search problem is to coordinate the search of a team of agents to guarantee the discovery of all targets. In this paper we present a self-contained solution to this problem in 2.5D real-world domains represented by digital elevation models (DEMs). We introduce hierarchical sampling on DEMs for selecting heuristically the close to minimal set of locations from which the entire surface of the DEM can be guarded. Locations are utilized to form a search graph on which search strategies for mobile agents are computed. For these strategies schedules are derived which include agent paths that are directly executable in the terrain. Presented experimental results demonstrate the performance of the method. The practical feasibility of our approach has been validated during a field experiment at the Gascola robot training site where teams of humans equipped with iPads successfully searched for adversarial and omniscient evaders. The field demonstration is the largest-scale implementation of a guaranteed search algorithm to date.

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“…Hence, the problem to decide the mixed-search number of a graph belongs to NP. Instead, connected graph searching, in which the set of clear edges must always induce a connected subgraph, is not monotone in general [27] and it is not known if connected search is in NP. Connected search is monotone in trees [2].…”

Section: Introductionmentioning

confidence: 99%

Exclusive Graph Searching

Blin

Burman²,

Nisse

2013

Lecture Notes in Computer Science

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Abstract. This paper tackles the well known graph searching problem, where a team of searchers aims at capturing an intruder in a network, modeled as a graph. All variants of this problem assume that any node can be simultaneously occupied by several searchers. This assumption may be unrealistic, e.g., in the case of searchers modeling physical searchers, or may require each individual node to provide additional resources, e.g., in the case of searchers modeling software agents. We thus investigate exclusive graph searching, in which no two or more searchers can occupy the same node at the same time, and, as for the classical variants of graph searching, we study the minimum number of searchers required to capture the intruder. This number is called the exclusive search number of the considered graph. Exclusive graph searching appears to be considerably more complex than classical graph searching, for at least two reasons: (1) it does not satisfy the monotonicity property, and (2) it is not closed under minor. Nevertheless, we design a polynomial-time algorithm which, given any tree T , computes the exclusive search number of T . Moreover, for any integer k, we provide a characterization of the trees T with exclusive search number at most k. This characterization allows us to describe a special type of exclusive search strategies, that can be executed in a distributed environment, i.e., in a framework in which the searchers are limited to cooperate in a distributed manner.

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Sweeping Graphs with Large Clique Number

Cited by 33 publications

References 4 publications

Submodular Percolation

Submodular Percolation

Hierarchical visibility for guaranteed search in large-scale outdoor terrain

Exclusive Graph Searching

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