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Parametric and Kinetic Minimum Spanning Trees
"... We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * r ..."
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Cited by 29 (7 self)
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We consider the parametric minimum spanning treeproblem, in which we are given a graph with edge weights that are linear functions of a parameter * and wish tocompute the sequence of minimum spanning trees generated as * varies. We also consider the kinetic minimumspanning tree problem, in which * represents time and the graph is subject in addition to changes such as edge insertions, deletions, and modifications of the weight functions as time progresses. We solve both problems in time O(n2=3 log4=3 n) per combinatorial change in the tree (or randomized O(n2=3 log n) per change). Our time bounds reduce to O(n1=2 log3=2 n) per change (O(n1=2 log n) randomized) for planar graphs or other minorclosed families of graphs, and O(n1=4 log3=2 n) per change (O(n1=4 log n) randomized) for planar graphs with weight changes but no insertions or deletions.
Geometric Lower Bounds for Parametric Matroid Optimization
, 1998
"... We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: ksets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in suc ..."
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Cited by 16 (2 self)
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We relate the sequence of minimum bases of a matroid with linearly varying weights to three problems from combinatorial geometry: ksets, lower envelopes of line segments, and convex polygons in line arrangements. Using these relations we show new lower bounds on the number of base changes in such sequences: #(nr 1/3 ) for a general nelement matroid with rank r , and #(m#(n)) for the special case of parametric graph minimum spanning trees. The only previous lower bound was #(n log r) for uniform matroids; upper bounds of O(mn 1/2 ) for arbitrary matroids and O(mn 1/2 / log # n) for uniform matroids were also known. 1 Introduction In this paper we study connections between combinatorial geometry and matroid optimization theory, as represented by the following problem. Parametric matroid optimization. Given a matroid for which the elements have weights that vary as a linear function of a parameter t , what is the sequence of minimum weight bases over the range of values o...
The Stackelberg Minimum Spanning Tree Game
 In Proc. of 10th WADS
, 2007
"... Abstract. We consider a oneround twoplayer 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’ ..."
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Cited by 10 (1 self)
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Abstract. We consider a oneround twoplayer 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 wellstudied Stackelberg shortestpath game. We analyze the complexity and approximability of the first player’s best strategy in StackMST. In particular, we prove that the problem is APXhard 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. 1