Using sparsification for parametric minimum spanning tree problems

Fernández-Baca, David; Slutzki, Giora; Eppstein, David

doi:10.1007/3-540-61422-2_128

Cited by 21 publications

(28 citation statements)

References 34 publications

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“…Finally, our problem should not be confused with other spanning tree games found in cooperative games and mechanism design theory [8,11], with parametric spanning tree problems [7,6], or with two-stage stochastic minimum spanning tree problems [5].…”

Section: Related Workmentioning

confidence: 99%

The Stackelberg Minimum Spanning Tree Game

Cardinal

Demaine²,

Fiorini

et al. 2009

Algorithmica

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Abstract. We consider a one-round two-player network pricing game, the Stackelberg Minimum Spanning Tree game or StackMST. The game is played on a graph (representing a network), whose edges are colored either red or blue, and where the red edges have a given fixed cost (representing the competitor's prices). The first player chooses an assignment of prices to the blue edges, and the second player then buys the cheapest possible minimum spanning tree, using any combination of red and blue edges. The goal of the first player is to maximize the total price of purchased blue edges. This game is the minimum spanning tree analog of the well-studied Stackelberg shortest-path game.We analyze the complexity and approximability of the first player's best strategy in StackMST. In particular, we prove that the problem is APX-hard even if there are only two different red costs, and give an approximation algorithm whose approximation ratio is at most min{k, 3 + 2 ln b, 1 + ln W }, where k is the number of distinct red costs, b is the number of blue edges, and W is the maximum ratio between red costs. We also give a natural integer linear programming formulation of the problem, and show that the integrality gap of the fractional relaxation asymptotically matches the approximation guarantee of our algorithm.

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Section: Related Workmentioning

confidence: 99%

The Stackelberg Minimum Spanning Tree Game

Cardinal

Demaine²,

Fiorini

et al. 2009

Algorithmica

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“…The parametric minimum spanning tree problem (with linear edge weights) has polynomially many solutions that can be constructed in polynomial time [1,15,18]. In contrast, the parametric shortest path problem is not polynomial, at least if the output must be represented as an explicit list of paths: it has a number of solutions and running time that are exponential in log 2 n on n-vertex graphs [10].…”

Section: Related Workmentioning

confidence: 99%

The Parametric Closure Problem

Eppstein

2015

Lecture Notes in Computer Science

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Abstract. We define the parametric closure problem, in which the input is a partially ordered set whose elements have linearly varying weights and the goal is to compute the sequence of minimum-weight downsets of the partial order as the weights vary. We give polynomial time solutions to many important special cases of this problem including semiorders, reachability orders of bounded-treewidth graphs, partial orders of bounded width, and series-parallel partial orders. Our result for series-parallel orders provides a significant generalization of a previous result of Carlson and Eppstein on bicriterion subtree problems.

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“…where Π(δ X = v) stands for the possibility degree that δ X = v. Since the statement "X is optimal under w w w" is equivalent to the condition δ X (w w w) = 0, we get the following relationships between optimality degrees (6), (7) and deviation (10) Using (3) we can express µ∆ X in the following way: N(X is optimal) = 1 − inf{λ : δ λ X = 0} (13) and N(X is optimal) = 0 if δ 1 X > 0. Exactly the same reasoning can be applied to elements.…”

Section: The Optimality Evaluation and Fuzzy Deviation Intervalmentioning

confidence: 99%

“…Exactly the same reasoning can be applied to elements. It is enough to replace X with f in formulae (10)- (13).…”

Section: The Optimality Evaluation and Fuzzy Deviation Intervalmentioning

confidence: 99%

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Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

Kasperski¹,

Zieliński²

2010

European Journal of Operational Research

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This paper deals with a general combinatorial optimization problem in which closed intervals and fuzzy intervals model uncertain element weights. The notion of a deviation interval is introduced, which allows us to characterize the optimality and the robustness of solutions and elements. The problem of computing deviation intervals is addressed and some new complexity results in this field are provided. Possibility theory is then applied to generalize a deviation interval and a solution concept to fuzzy ones.

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Using sparsification for parametric minimum spanning tree problems

Cited by 21 publications

References 34 publications

The Stackelberg Minimum Spanning Tree Game

The Stackelberg Minimum Spanning Tree Game

The Parametric Closure Problem

Minmax regret approach and optimality evaluation in combinatorial optimization problems with interval and fuzzy weights

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