Colored pebble motion on graphs

Fujita, Shinya; Nakamigawa, Tomoki; Sakuma, Tadashi

doi:10.1016/j.ejc.2011.09.019

Cited by 9 publications

(9 citation statements)

References 8 publications

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“…A polynomial-time algorithm has been found for the case with two colors, and a linear-time algorithm for trees [161]. In addition, it has been shown that the reconfiguration graph is connected when there are at least three colors, G is not a cycle and G has no (n − n r )-isthmus (where n r is the size of the largest color class), and the number of connected components has been determined for various cases [162]. In "parallel" variants, swapping is allowed on non-adjacent edges at the same time; the problem of determining the minimum number of swaps needed has been shown to be NP-complete, with a polynomial-time approximation algorithm for paths, and NP-complete for colored tokens, even with as few as two colors [163].…”

Section: Placing Tokens On All Verticesmentioning

confidence: 99%

Introduction to Reconfiguration

Nishimura¹

2017

Preprint

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Reconfiguration is concerned with relationships among solutions to a problem instance, where the reconfiguration of one solution to another is a sequence of steps such that each step produces an intermediate feasible solution. The solution space can be represented as a reconfiguration graph, where two vertices representing solutions are adjacent if one can be formed from the other in a single step. Work in the area encompasses both structural questions (Is the reconfiguration graph connected?) and algorithmic ones (How can one find the shortest sequence of steps between two solutions?) This survey discusses techniques, results, and future directions in the area.

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Section: Placing Tokens On All Verticesmentioning

confidence: 99%

Introduction to Reconfiguration

Nishimura¹

2017

Preprint

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“…This problem is a feasibility problem that starts with an initial distribution of colored pebbles on the vertices of a graph. The aim is to change the distribution of the pebbles into the target arrangement, given that each pebble can be moved to an adjacent vertex and each vertex is either empty or occupied by one pebble (Fujita et al 2012). Iordan (2019) addressed a generalization of the knight exchange problem with an To the best of the author's knowledge, there is neither any published work on the binary programming formulation nor any on the network flow formulation for this problem.…”

Section: Introductionmentioning

confidence: 99%

Knights Exchange Puzzle—Teaching the Efficiency of Modeling

Iranpoor

2021

INFORMS Transactions on Education

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Puzzles and games enhance the quality of teaching by creating an enjoyable, interactive, and playful atmosphere. The knight exchange is a famous, very old, and amusing game on the chessboard. This puzzle was used by the author to teach modeling in a mathematical programming course designed for graduate students. The aim was to teach the students the efficiency of the models. Accordingly, first, a binary programming formulation was developed. This formulation was, however, found to be inefficient, and tremendous time (i.e., more than four hours) and a large amount of processing memory were needed to solve the puzzle. The puzzle was subsequently formulated as a minimum cost network flow problem. The latter formulation outperformed the general binary formulation by solving the puzzle in less than a minute. The network formulation could also save the required processing memory. The results could help students to learn the value of modeling combinatorial optimization problems as network flows.

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“…the problem of determining whether all the configurations of the puzzle are rearrangeable from one another or not) for the case of |f −1 (0)| = 1 on general graphs, and it followed by the result of Kornhauser, Miller and Spirakis (FOCS '84) [9] for the case of |f −1 (0)| ≥ 2. In 2012, Fujita, Nakamigawa and Sakuma [5] generalized the problem to the case of "colored pebbles", where each pebble of P is distinguished by its color. They also completely solved the feasibility problem for their model.…”

Section: Introductionmentioning

confidence: 99%

“…They also completely solved the feasibility problem for their model. Note that Papadimitriou, Raghavan, Sudan and Tamaki (FOCS '94) [11] also treat a special case of this model in [5]. They consider the case that there exist two colors ("blue:robot" and "red:obstacle") of pebbles and that the number of blue colored pebbles is one (i.e.…”

Section: Introductionmentioning

confidence: 99%

Pebble Exchange Group of Graphs

Kato,

Nakamigawa,

Sakuma

2019

Preprint

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A graph puzzle Puz(G) of a graph G is defined as follows. A configuration of Puz(G) is a bijection from the set of vertices of a board graph G to the set of vertices of a pebble graph G. A move of pebbles is defined as exchanging two pebbles which are adjacent on both a board graph and a pebble graph. For a pair of configurations f and g, we say that f is equivalent to g if f can be transformed into g by a sequence of finite moves.Let Aut(G) be the automorphism group of G, and let 1 G be the unit element of Aut(G). The pebble exchange group of G, denoted by Peb(G), is defined as the set of all automorphisms f of G such that 1 G and f are equivalent to each other.In this paper, some basic properties of Peb(G) are studied. Among other results, it is shown that for any connected graph G, all automorphisms of G are contained in Peb(G 2 ), where G 2 is a square graph of G.

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Colored pebble motion on graphs

Cited by 9 publications

References 8 publications

Introduction to Reconfiguration

Introduction to Reconfiguration

Knights Exchange Puzzle—Teaching the Efficiency of Modeling

Pebble Exchange Group of Graphs

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