11 research outputs found
Optimal strategies for a game on amenable semigroups
The semigroup game is a two-person zero-sum game defined on a semigroup S as
follows: Players 1 and 2 choose elements x and y in S, respectively, and player
1 receives a payoff f(xy) defined by a function f from S to [-1,1]. If the
semigroup is amenable in the sense of Day and von Neumann, one can extend the
set of classical strategies, namely countably additive probability measures on
S, to include some finitely additive measures in a natural way. This extended
game has a value and the players have optimal strategies. This theorem extends
previous results for the multiplication game on a compact group or on the
positive integers with a specific payoff. We also prove that the procedure of
extending the set of allowed strategies preserves classical solutions: if a
semigroup game has a classical solution, this solution solves also the extended
game.Comment: 17 pages. To appear in International Journal of Game Theor
The Effect of Planarization on Width
We study the effects of planarization (the construction of a planar diagram
from a non-planar graph by replacing each crossing by a new vertex) on
graph width parameters. We show that for treewidth, pathwidth, branchwidth,
clique-width, and tree-depth there exists a family of -vertex graphs with
bounded parameter value, all of whose planarizations have parameter value
. However, for bandwidth, cutwidth, and carving width, every graph
with bounded parameter value has a planarization of linear size whose parameter
value remains bounded. The same is true for the treewidth, pathwidth, and
branchwidth of graphs of bounded degree.Comment: 15 pages, 6 figures. To appear at the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Achieving Good Angular Resolution in 3D Arc Diagrams
We study a three-dimensional analogue to the well-known graph visualization
approach known as arc diagrams. We provide several algorithms that achieve good
angular resolution for 3D arc diagrams, even for cases when the arcs must
project to a given 2D straight-line drawing of the input graph. Our methods
make use of various graph coloring algorithms, including an algorithm for a new
coloring problem, which we call localized edge coloring.Comment: 12 pages, 5 figures; to appear at the 21st International Symposium on
Graph Drawing (GD 2013
Fixed Linear Crossing Minimization by Reduction to the Maximum Cut Problem
Many real-life scheduling, routing and locating problems can be formulated as combinatorial optimization problems whose goal is to find a linear layout of an input graph in such a way that the number of edge crossings is minimized. In this paper, we study a restricted version of the linear layout problem where the order of vertices on the line is fixed, the so-called fixed linear crossing number problem (FLCNP). We show that this NP-hard problem can be reduced to the well-known maximum cut problem. The latter problem was intensively studied in the literature; practically efficient exact algorithms based on the branch-and-cut technique have been developed. By an experimental evaluation on a variety of graphs, we prove that using this reduction for solving FLCNP compares favorably to earlier branch-and-bound algorithms
Planar crossing numbers of genus g graphs
Pach and Tóth [15] proved that any n-vertex graph of genus g and maximum degree d has a planar crossing number at most c g dn, for a constant c>1. We improve on this results by decreasing the bound to O(dgn), if g = o(n), and to O(g²), otherwise, and also prove that our result is tight within a constant factor