19,108 research outputs found
Impartial coloring games
Coloring games are combinatorial games where the players alternate painting
uncolored vertices of a graph one of colors. Each different ruleset
specifies that game's coloring constraints. This paper investigates six
impartial rulesets (five new), derived from previously-studied graph coloring
schemes, including proper map coloring, oriented coloring, 2-distance coloring,
weak coloring, and sequential coloring. For each, we study the outcome classes
for special cases and general computational complexity. In some cases we pay
special attention to the Grundy function
Positional Games
Positional games are a branch of combinatorics, researching a variety of
two-player games, ranging from popular recreational games such as Tic-Tac-Toe
and Hex, to purely abstract games played on graphs and hypergraphs. It is
closely connected to many other combinatorial disciplines such as Ramsey
theory, extremal graph and set theory, probabilistic combinatorics, and to
computer science. We survey the basic notions of the field, its approaches and
tools, as well as numerous recent advances, standing open problems and
promising research directions.Comment: Submitted to Proceedings of the ICM 201
Core Stability in Chain-Component Additive Games
Chain-component additive games are graph-restricted superadditive games, where an exogenously given line-graph determines the cooperative possibilities of the players.These games can model various multi-agent decision situations, such as strictly hierarchical organisations or sequencing / scheduling related problems, where an order of the agents is fixed by some external factor, and with respect to this order only consecutive coalitions can generate added value. In this paper we characterise core stability of chain-component additive games in terms of polynomial many linear inequalities and equalities that arise from the combinatorial structure of the game.Furthermore we show that core stability is equivalent to essential extendibility.We also obtain that largeness of the core as well as extendibility and exactness of the game are equivalent properties which are all sufficient for core stability.Moreover, we also characterise these properties in terms of linear inequalities.Core stability;graph-restricted games;large core;exact game
Hardness of Graph Pricing through Generalized Max-Dicut
The Graph Pricing problem is among the fundamental problems whose
approximability is not well-understood. While there is a simple combinatorial
1/4-approximation algorithm, the best hardness result remains at 1/2 assuming
the Unique Games Conjecture (UGC). We show that it is NP-hard to approximate
within a factor better than 1/4 under the UGC, so that the simple combinatorial
algorithm might be the best possible. We also prove that for any , there exists such that the integrality gap of
-rounds of the Sherali-Adams hierarchy of linear programming for
Graph Pricing is at most 1/2 + .
This work is based on the effort to view the Graph Pricing problem as a
Constraint Satisfaction Problem (CSP) simpler than the standard and complicated
formulation. We propose the problem called Generalized Max-Dicut(), which
has a domain size for every . Generalized Max-Dicut(1) is
well-known Max-Dicut. There is an approximation-preserving reduction from
Generalized Max-Dicut on directed acyclic graphs (DAGs) to Graph Pricing, and
both our results are achieved through this reduction. Besides its connection to
Graph Pricing, the hardness of Generalized Max-Dicut is interesting in its own
right since in most arity two CSPs studied in the literature, SDP-based
algorithms perform better than LP-based or combinatorial algorithms --- for
this arity two CSP, a simple combinatorial algorithm does the best.Comment: 28 page
Arkhipov's theorem, graph minors, and linear system nonlocal games
The perfect quantum strategies of a linear system game correspond to certain
representations of its solution group. We study the solution groups of graph
incidence games, which are linear system games in which the underlying linear
system is the incidence system of a (non-properly) two-coloured graph. While it
is undecidable to determine whether a general linear system game has a perfect
quantum strategy, for graph incidence games this problem is solved by
Arkhipov's theorem, which states that the graph incidence game of a connected
graph has a perfect quantum strategy if and only if it either has a perfect
classical strategy, or the graph is nonplanar. Arkhipov's criterion can be
rephrased as a forbidden minor condition on connected two-coloured graphs. We
extend Arkhipov's theorem by showing that, for graph incidence games of
connected two-coloured graphs, every quotient closed property of the solution
group has a forbidden minor characterization. We rederive Arkhipov's theorem
from the group theoretic point of view, and then find the forbidden minors for
two new properties: finiteness and abelianness. Our methods are entirely
combinatorial, and finding the forbidden minors for other quotient closed
properties seems to be an interesting combinatorial problem.Comment: Minor updates. Also see video abstract at
https://youtu.be/uTudADhT1p
On the Concavity of Delivery Games
Delivery games, introduced by Hamers, Borm, van de Leensel and Tijs (1994), are combinatorial optimization games that arise from delivery problems closely related to the Chinese postman problem (CPP). They showed that delivery games are not necessarily balanced. For delivery problems corresponding to the class of bridge-connected Euler graphs they showed that the related games are balanced. This paper focuses on the concavity property for delivery games. A delivery game arising from a delivery model corresponding to a bridge-connected Euler graph needs not to be concave. The main result will be that for delivery problems corresponding to the class of bridge-connected cyclic graphs, which is a subclass of the class of bridge-connected Euler graphs, the related delivery games are concave.
Grim Under a Compensation Variant
Games on graphs are a well studied subset of combinatorial games. Balance and strategies for winning are often looked at in these games. One such combinatorial graph game is Grim. Many of the winning strategies of Grim are already known. We note that many of these winning strategies are only available to the first player. Hoping to develop a fairer Grim, we look at Grim played under a slighlty different rule set. We develop winning strategies and known outcomes for this altered Grim. Throughout, we discuss whether our altered Grim is a fairer game then the original
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