Algorithmic aspects of distance constrained labeling: a survey

Hasunuma, Toru; Ishii, Toshimasa; Ono, Hiroyuki; Uno, Yushi

doi:10.15803/ijnc.4.2_251

Cited by 7 publications

(3 citation statements)

References 65 publications

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“…L(p)-Labeling is fixed-parameter tractable for the neighborhood diversity, p max , plus k [12]. Further related work for L(p)-Labeling can be found in the following surveys [7,22].…”

Section: Distance-constrained Labelingmentioning

confidence: 99%

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

Hanaka,

Ono,

Sugiyama

2024

IJNC

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For an undirected graph G = (V, E) and a k-non-negative integer vector p = (p1, . . . , p k ), a mapping l : V → N ∪ {0} is called an L(p)-labeling of G if |l(u) − l(v)| ≥ p d for any two distinct vertices u, v ∈ V with distance d, and the maximum value of {l(v) | v ∈ V } is called the span of l. Originally, L(p)-labeling of G for p = (2, 1) is introduced in the context of frequency assignment in radio networks, where 'close' transmitters must receive different frequencies and 'very close' transmitters must receive frequencies that are at least two frequencies apart so that they can avoid interference. L(p)-Labeling is the problem of finding the minimum span λp among L(p)labelings of G, which is NP-hard for every non-zero p. L(p)-Labeling is well studied for specific p's; in particular, many (exact or approximation) algorithms for general graphs or restricted classes of graphs are proposed for p = (2, 1) or more generally p = (p, q). Unfortunately, most algorithms strongly depend on the values of p, and it is not apparent to extend algorithms for p to ones for another p ′ in general. In this paper, we give a simple polynomial-time reduction of L(p)-Labeling on graphs with a small diameter to Metric (Path) TSP, which enables us to use numerous results on (Metric) TSP. On the practical side, we can utilize various high-performance heuristics for TSP, such as Concordo and LKH, to solve our problem. On the theoretical side, we can see that the problem for any p under this framework is 1.5-approximable, and it can be solved by the Held-Karp algorithm in O(2 n n 2 ) time, where n is the number of vertices, and so on.

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“…L(p)-Labeling is fixed-parameter tractable for the neighborhood diversity, p max , plus k [12]. Further related work for L(p)-Labeling can be found in the following surveys [7,22].…”

Section: Distance-constrained Labelingmentioning

confidence: 99%

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

Hanaka,

Ono,

Sugiyama

2024

IJNC

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“…, p l )-labeling if for all i P [l] and all v, w P V we have |c(v) ´c(w)| ě p i whenever dist G (v, w) = i, where dist G (v, w) is the graphtheoretical distance between v and w, i. e., the length of a shortest v-w path in G. The notion of L(1)-labeling coincides with that of proper coloring. The case of L(p, q)-labelings, that is, where l = 2, has received special attention, in particular in connection with frequency assignment, see, e. g., [8,18,37]. As a variation of this, not the distance |c(v) ´c(w)| between colors is considered, but instead it is required that min |c(v) ´c(w)|, k ´|c(v) ´c(w)| ě p i , whenever dist G (v, w) = i, as considered in [35].…”

Section: Previous and Related Workmentioning

confidence: 99%

Price of anarchy for graph coloring games with concave payoff

Kliemann¹,

Sheykhdarabadi²,

Srivastav³

2017

Journal of Dynamics &Amp; Games

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We study the price of anarchy in a class of graph coloring games (a subclass of polymatrix common-payoff games). In those games, players are vertices of an undirected, simple graph, and the strategy space of each player is the set of colors from 1 to k. A tight bound on the price of anarchy of k k´1 is known (Hoefer 2007, Kun et al. 2013, for the case that each player's payoff is the number of her neighbors with different color than herself. The study of more complex payoff functions was left as an open problem.We compute payoff for a player by determining the distance of her color to the color of each of her neighbors, applying a non-negative, real-valued, concave function f to each of those distances, and then summing up the resulting values. This includes the payoff functions suggested by Kun et al. (2013) for future work as special cases.Denote f˚the maximum value that f attains on the possible distances 0, . . . , k´1. We prove an upper bound of 2 on the price of anarchy for concave functions f that are non-decreasing or which assume f˚at a distance on or below k 2 . Matching lower bounds are given for the monotone case and for the case that f˚is assumed in k 2 for even k. For general concave functions, we prove an upper bound of 3. We use a simple but powerful technique: we obtain an upper bound of λ ě 1 on the price of anarchy if we manage to give a splitting λ 1 + . . .The discovery of working splittings can be supported by computer experiments. We show how, once we have an idea what kind of splittings work, this technique helps in giving simple proofs, which mainly work by case distinctions, algebraic manipulations, and real calculus.Graph coloring, especially in distributed and game-theoretic settings, is often used to model spectrum sharing scenarios, such as the selection of WLAN frequencies. For such an application, our framework would allow to express a dependence of the degree of radio interference on the distance between two frequencies. Model, Notation, Basic NotionsLet G = (V, E) be an undirected, simple graph without any isolated vertices (n := |V| and m := |E|) and k P N ě2 . For v P V denote N(v) := {w P V ; {v, w} P E} the set of its neighbors and deg(v) := |N(v)| its degree. A function c : V ÝÑ [k] is called a k-coloring or coloring, where [k] = {1, . . . , k}. Vertices of the graph represent players, with the set of colors [k] being the strategy space for each player. We sometimes call the set [k] the spectrum. Clearly, k-colorings are exactly the strategy profiles of this game. 1 Given a coloring c, define the payoff for player v aswhere f is a non-negative, real-valued function defined on [0, k] (choosing this domain instead of {0, . . . , k´1} is technically easier). Given such f , we denote f˚:= max iPD f (i) the maximum value that f attains on the possible distances D := {0, . . . , k´1} between two colors, and D˚( f ) := {i P D ; f (i) = f˚} the set of distances where f˚is attained. We call f (|c(v)´c(w)|) the contribution of edge {v, w}. So f˚is an upper bound on the contribution of...

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“…It is now known that L(p, 1)-Labeling on trees can be solved in linear time [27]. For more algorithmic results, see [28].…”

Section: Introductionmentioning

confidence: 99%

Computing $L(p,1)$-Labeling with Combined Parameters

Hanaka¹,

Kawai²,

Ono³

2020

Preprint

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Given a graph, an L(p, 1)-labeling of the graph is an assignment f from the vertex set to the set of nonnegative integers such that for any pair of vertices (u, v), |f (u) − f (v)| ≥ p if u and v are adjacent, and f (u) = f (v) if u and v are at distance 2. The L(p, 1)-labeling problem is to minimize the span of f (i.e.,maxu∈V (f (u)) − minu∈V (f (u)) + 1). It is known to be NP-hard even for graphs of maximum degree 3 or graphs with tree-width 2, whereas it is fixedparameter tractable with respect to vertex cover number. Since vertex cover number is a kind of the strongest parameter, there is a large gap between tractability and intractability from the viewpoint of parameterization. To fill up the gap, in this paper, we propose new fixed-parameter algorithms for L(p, 1)-Labeling by the twin cover number plus the maximum clique size and by the tree-width plus the maximum degree. These algorithms reduce the gap in terms of several combinations of parameters.

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Algorithmic aspects of distance constrained labeling: a survey

Cited by 7 publications

References 65 publications

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

Solving Distance-constrained Labeling Problems for Small Diameter Graphs via TSP

Price of anarchy for graph coloring games with concave payoff

Computing $L(p,1)$-Labeling with Combined Parameters

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