Complexity of inheritance of ℱ-convexity for restricted games induced by minimum partitions

2020 ◽  
Vol 54 (1) ◽  
pp. 143-161
Author(s):  
A. Skoda
Keyword(s):  
Cooperative Game ◽  
Polynomial Time ◽  
Minimum Weight ◽  
Connected Components ◽  
Communication Graph ◽  
Convex Game ◽  
Restricted Game ◽  
Restricted Games ◽  
Unweighted Graph

Let G = (N, E, w) be a weighted communication graph. For any subset A ⊆ N, we delete all minimum-weight edges in the subgraph induced by A. The connected components of the resultant subgraph constitute the partition 𝒫min(A) of A. Then, for every cooperative game (N, v), the 𝒫min-restricted game (N, v̅) is defined by v̅(A)=∑F∈𝒫min(A)v(F) for all A ⊆ N. We prove that we can decide in polynomial time if there is inheritance of ℱ-convexity, i.e., if for every ℱ-convex game the 𝒫min-restricted game is ℱ-convex, where ℱ-convexity is obtained by restricting convexity to connected subsets. This implies that we can also decide in polynomial time for any unweighted graph if there is inheritance of convexity for Myerson’s graph-restricted game.

scholarly journals Convexity of graph-restricted games induced by minimum partitions

2019 ◽  
Vol 53 (3) ◽  
pp. 841-866 ◽  
Author(s):  
Alexandre Skoda
Keyword(s):  
Necessary Conditions ◽  
Minimum Weight ◽  
Weighted Graph ◽  
Connected Components ◽  
Weighted Graphs ◽  
Classical Definition ◽  
Restricted Game ◽  
Definition Of ◽  
Adjacent Cycles ◽  
Restricted Games

We consider restricted games on weighted graphs associated with minimum partitions. We replace in the classical definition of Myerson restricted game the connected components of any subgraph by the sub-components corresponding to a minimum partition. This minimum partition 𝒫min is i nduced by the deletion of the minimum weight edges. We provide five necessary conditions on the graph edge-weights to have inheritance of convexity from the underlying game to the restricted game associated with 𝒫min. Then, we establish that these conditions are also sufficient for a weaker condition, called ℱ-convexity, obtained by restriction of convexity to connected subsets. Moreover, we prove that inheritance of convexity for Myerson restricted game associated with a given graph G is equivalent to inheritance of ℱ-convexity for the 𝒫min-restricted game associated with a particular weighted graph G′ built from G by adding a dominating vertex, and with only two different edge-weights. Then, we prove that G is cycle-complete if and only if a specific condition on adjacent cycles is satisfied on G′.

scholarly journals A Symmetric Banzhaf Cooperation Value for Games with a Proximity Relation among the Agents

Symmetry ◽  
2020 ◽  
Vol 12 (7) ◽  
pp. 1196
Author(s):  
Inés Gallego ◽  
Julio R. Fernández ◽  
Andrés Jiménez-Losada ◽  
Manuel Ordóñez
Keyword(s):  
Common Good ◽  
A Priori ◽  
Group Symmetry ◽  
Connected Components ◽  
Banzhaf Value ◽  
Connected Component ◽  
Communication Graph ◽  
Political Situation ◽  
Proximity Relation ◽  
Natural Way

A cooperative game represents a situation in which a set of agents form coalitions in order to achieve a common good. To allocate the benefits of the result of this cooperation there exist several values such as the Shapley value or the Banzhaf value. Sometimes it is considered that not all communications between players are feasible and a graph is introduced to represent them. Myerson (1977) introduced a Shapley-type value for these situations. Another model for cooperative games is the Owen model, Owen (1977), in which players that have similar interests form a priori unions that bargain as a block in order to get a fair payoff. The model of cooperation introduced in this paper combines these two models following Casajus (2007). The situation consists of a communication graph where a two-step value is defined. In the first step a negotiation among the connected components is made and in the second one players inside each connected component bargain. This model can be extended to fuzzy contexts such as proximity relations that consider leveled closeness between agents as we proposed in 2016. There are two extensions of the Banzhaf value to the Owen model, because the natural way loses the group symmetry property. In this paper we construct an appropriate value to extend the symmetric option for situations with a proximity relation and provide it with an axiomatization. Then we apply this value to a political situation.

ON THE COMPLEXITY OF COMPUTING VALUES OF RESTRICTED GAMES

2002 ◽  
Vol 13 (05) ◽  
pp. 633-651 ◽  
Author(s):  
J. M. BILBAO ◽  
J. R. FERNÁNDEZ ◽  
J. J. LÓPEZ
Keyword(s):  
Cooperative Games ◽  
Partially Ordered Set ◽  
Polynomial Algorithm ◽  
Ordered Set ◽  
Combinatorial Structure ◽  
Distributive Lattices ◽  
Communication Graph ◽  
Partially Ordered ◽  
Connected Subgraphs ◽  
Restricted Games

The aim of this paper is to compute Shapley's and Banzhaf's values of cooperative games restricted by a combinatorial structure. There have been previous models developed to study the problem of games with partial cooperation. Games restricted by a communication graph were introduced by Myerson and Owen. Another type of combinatorial structure introduced by Gilles, Owen and van den Brink is equivalent to a subclass of antimatroids. Cooperative games in which the set of players is a partially ordered set, that is, games on distributive lattices was investigated by Faigle and Kern. We introduce a new combinatorial structure called augmenting system which is a generaligation of the antimatroid structure and the system of connected subgraphs of graph. We present new algorithmic procedures for computing values of games under augmenting systems restrictions and we show that there exist problems with polynomial algorithm complexity.

DIAMONDS ARE NOT A MINIMUM WEIGHT TRIANGULATION'S BEST FRIEND

2002 ◽  
Vol 12 (06) ◽  
pp. 445-453 ◽  
Author(s):  
PROSENJIT BOSE ◽  
LUC DEVROYE ◽  
WILLIAM EVANS
Keyword(s):  
Polynomial Time ◽  
Minimum Weight ◽  
Connected Subgraph ◽  
Constant Number ◽  
Point Sets ◽  
Expected Number ◽  
Number Of Components ◽  
Best Friend ◽  
Point Set ◽  
Minimum Weight Triangulation

Two recent methods have increased hopes of finding a polynomial time solution to the problem of computing the minimum weight triangulation of a set S of n points in the plane. Both involve computing what was believed to be a connected or nearly connected subgraph of the minimum weight triangulation, and then completing the triangulation optimally. The first method uses the light graph of S as its initial subgraph. The second method uses the LMT-skeleton of S. Both methods rely, for their polynomial time bound, on the initial subgraphs having only a constant number of components. Experiments performed by the authors of these methods seemed to confirm that randomly chosen point sets displayed this desired property. We show that there exist point sets where the number of components is linear in n. In fact, the expected number of components in either graph on a randomly chosen point set is linear in n, and the probability of the number of components exceeding some constant times n tends to one.

EXACT AND APPROXIMATION ALGORITHMS FOR FINDING AN OPTIMAL BRIDGE CONNECTING TWO SIMPLE POLYGONS

2005 ◽  
Vol 15 (06) ◽  
pp. 609-630
Author(s):  
AMIT M. BHOSLE ◽  
TEOFILO F. GONZALEZ
Keyword(s):  
Polynomial Time ◽  
Approximation Scheme ◽  
Geodesic Distance ◽  
Minimum Weight ◽  
Additional Restriction ◽  
Polynomial Time Approximation Scheme ◽  
Time Approximation ◽  
Simple Polygons ◽  
Compact Region ◽  
Polynomial Time Approximation

Given two simple polygons P and Q we define the weight of a bridge (p,q), with p ∈ ρ(P) and q ∈ ρ(Q), where ρ() denotes the compact region enclosed by the boundary of the polygon, between the two polygons as gd(p,P) + d(p,q) + gd(q,Q), where d(p,q) is the Euclidean distance between the points p and q, and gd(x,X) is the geodesic distance between x and its geodesic furthest neighbor on X. Our problem differs from another version of the problem where the additional restriction of requiring the endpoints of the bridge to be mutually visible was imposed. We show that an optimal bridge always exists such that the endpoints of the bridge lie on the boundaries of the two polygons. Using this critical property, we present an algorithm to find an optimal bridge (of minimum weight) in O(n2 log n) time. We present a polynomial time approximation scheme that for any ∊ > 0 generates a bridge with objective function within a factor of 1 + ∊ of the optimal value in O(kn log kn) time, where [Formula: see text]. An improved polynomial time approximation scheme and algorithms for generalized versions of our problems are also discussed.

scholarly journals Independent sets in (P₆, diamond)-free graphs

2009 ◽  
Vol Vol. 11 no. 1 (Graph and Algorithms) ◽  
Author(s):  
Raffaele Mosca
Keyword(s):  
Polynomial Time ◽  
Dominating Set ◽  
Minimum Weight ◽  
Independent Set ◽  
Independent Sets ◽  
Maximum Weight ◽  
Independent Set Problem ◽  
International Audience ◽  
Dominating Set Problem ◽  
Independent Dominating Set

Graphs and Algorithms International audience We prove that on the class of (P6,diamond)-free graphs the Maximum-Weight Independent Set problem and the Minimum-Weight Independent Dominating Set problem can be solved in polynomial time.

scholarly journals Parameterized Algorithms for Power-Efficiently Connecting Wireless Sensor Networks: Theory and Experiments

2021 ◽  
Author(s):  
Matthias Bentert ◽  
René van Bevern ◽  
André Nichterlein ◽  
Rolf Niedermeier ◽  
Pavel V. Smirnov
Keyword(s):  
Communication Network ◽  
Wireless Sensor Networks ◽  
Sensor Networks ◽  
Sensor Network ◽  
Polynomial Time ◽  
Exact Algorithms ◽  
Wireless Sensor ◽  
Connected Components ◽  
Np Hard ◽  
Transmission Power

We study a problem of energy-efficiently connecting a symmetric wireless communication network: given an n-vertex graph with edge weights, find a connected spanning subgraph of minimum cost, where the cost is determined by each vertex paying the heaviest edge incident to it in the subgraph. The problem is known to be NP-hard. Strengthening this hardness result, we show that even o(log n)-approximating the difference d between the optimal solution cost and a natural lower bound is NP-hard. Moreover, we show that under the exponential time hypothesis, there are no exact algorithms running in 2o(n) time or in [Formula: see text] time for any computable function f. We also show that the special case of connecting c network components with minimum additional cost generally cannot be polynomial-time reduced to instances of size cO(1) unless the polynomial-time hierarchy collapses. On the positive side, we provide an algorithm that reconnects O(log n)-connected components with minimum additional cost in polynomial time. These algorithms are motivated by application scenarios of monitoring areas or where an existing sensor network may fall apart into several connected components because of sensor faults. In experiments, the algorithm outperforms CPLEX with known integer linear programming (ILP) formulations when n is sufficiently large compared with c. Summary of Contribution: Wireless sensor networks are used to monitor air pollution, water pollution, and machine health; in forest fire and landslide detection; and in natural disaster prevention. Sensors in wireless sensor networks are often battery-powered and disposable, so one may be interested in lowering the energy consumption of the sensors in order to achieve a long lifetime of the network. We study the min-power symmetric connectivity problem, which models the task of assigning transmission powers to sensors so as to achieve a connected communication network with minimum total power consumption. The problem is NP-hard. We provide perhaps the first parameterized complexity study of optimal and approximate solutions for the problem. Our algorithms work in polynomial time in the scenario where one has to reconnect a sensor network with n sensors and O(log n)-connected components by means of a minimum transmission power increase or if one can find transmission power lower bounds that already yield a network with O(log n)-connected components. In experiments, we show that, in this scenario, our algorithms outperform previously known exact algorithms based on ILP formulations.

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