Vertex Nim played on graphs

Duchêne, Éric; Renault, Gabriel

doi:10.1016/j.tcs.2013.11.025

Cited by 10 publications

(10 citation statements)

References 19 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In [5], they found a polynomial algorithm to compute the normal outcome of the game NimG-RM and asked if there is one in misère convention. We will show that the misère version of NimG-RM is pspace-Hard on general graphs.…”

Section: Complexity Results For Nimg In Misère Conventionmentioning

confidence: 99%

“…Because loops may sometimes make a difference, we note +L when there is a loop on all the vertices, and +NL when loops are not permitted. As said in the introduction, the results for the polynomial complexity of NimG-RM in normal play are due to Renault and Duchene [5], whereas the pspace-Hardness results for NimG-MR are due to Burke and George [4].…”

Section: Complexity Results For Nimg In Misère Conventionmentioning

confidence: 99%

“…Burke and George [4] proved that the game NimG-MR is pspace-Hard in normal convention, whereas Duchêne and Renault [5] found that NimG-RM is solvable in polynomial time. Lichtenstein and Sipser [9] proved that finding the normal outcome of a Vertex Geography position on a directed graph is pspace-complete.…”

Section: Edge Geography Vertex Geographymentioning

confidence: 99%

See 2 more Smart Citations

On the complexity of the misère version of three games played on graphs

Renault

Schmidt

2015

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

We investigate the complexity of finding a winning strategy for the misère version of three games played on graphs: two variants of the game NimG, introduced by Stockmann in 2004 and the game Vertex Geography on both directed and undirected graphs. We show that on general graphs those three games are pspace-Hard or Complete. For one pspace-Hard variant of NimG, we find an algorithm to compute an effective winning strategy in time

show abstract

Section: Complexity Results For Nimg In Misère Conventionmentioning

confidence: 99%

Section: Complexity Results For Nimg In Misère Conventionmentioning

confidence: 99%

See 1 more Smart Citation

On the complexity of the misère version of three games played on graphs

Renault

Schmidt

2015

Theoretical Computer Science

Self Cite

View full text Add to dashboard Cite

show abstract

“…This fundamental problem-commonly involving deep alternation-has been the subject of intense mathematical and computational studies [3,12,21,30,34,15,13,31,18,9,23,2]. A ruleset R defines a natural game tree, capturing this alternation for each of its positions, G, by recursively branching with feasible options.…”

Section: The Value Of Games Beyond Winningmentioning

confidence: 99%

Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

Burke¹,

Ferland²,

Teng³

2021

Preprint

View full text Add to dashboard Cite

We settle two long-standing complexity-theoretical questions-open since 1981 and 1993-in combinatorial game theory (CGT).We prove that the Grundy value (a.k.a. nim-value, or nimber) of Undirected Geography is PSPACE-complete to compute. This exhibits a stark contrast with a result from 1993 that Undirected Geography is polynomial-time solvable. By distilling to a simple reduction, our proof further establishes a dichotomy theorem, providing a "phase transition to intractability" in Grundy-value computation, sharply characterized by a maximum degree of four: The Grundy value of Undirected Geography over any degree-three graph is polynomial-time computable, but over degree-four graphs-even when planar & bipartite-is PSPACE-hard. Additionally, we show, for the first time, how to construct Undirected Geography instances with Grundy value * n and size polynomial in n.We strengthen a result from 1981 showing that sums of tractable partisan games are PSPACEcomplete in two fundamental ways. First, since Undirected Geography is an impartial ruleset, we extend the hardness of sums to impartial games, a strict subset of partisan. Second, the 1981 construction is not built from a natural ruleset, instead using a long sum of tailored short-depth game positions. We use the sum of two Undirected Geography positions to create our hard instances. Our result also has computational implications to Sprague-Grundy Theory (1930s) which shows that the Grundy value of the disjunctive sum of any two impartial games can be computed-in polynomial time-from their Grundy values. In contrast, we prove that assuming PSPACE = P, there is no general polynomial-time method to summarize two polynomial-time solvable impartial games to efficiently solve their disjunctive sum.

show abstract

“…Choosing an edge in Nim on graphs is the same as choosing a heap in Nim; reducing a weight is the same as removing counters from a heap, and switching the playing piece between vertices is just the act of taking turns between players. This is just one example of a graph game from the literature; for other examples, see [13], [21].…”

Section: Games On Graphsmentioning

confidence: 99%

Impartial Intersection Restriction Games

Huggan¹

View full text Add to dashboard Cite

Intersection restriction games are games played on hypergraphs in which options for a player are restricted based on previous play via some intersection property. This paper focuses on two games within this class: Arc-Kayles and a Triple Packing game. Arc-Kayles is a game where, on their turn, players remove an edge and all adjacent edges from a graph. Together, players are forming a maximal matching. The Triple Packing game is a combinatorial design game where players are choosing triples such that no two triples chosen share a pair. Both games are played under normal play. We give new results for Arc-Kayles played on a special star graph and the wheel graph as well as partial results for the Triple Packing game played on the complete graph. ii Acknowledgements I would like to thank my supervisor Dr. Brett Stevens. Brett's knowledge and insight into mathematical connections has been very inspirational to my academic curiosity. Brett has provided me with invaluable national and international academic opportunities which expanded my mathematical thinking to new creative levels.I would also like to thank my examination committee members for their suggestions and thought-provoking questions throughout my thesis defence.

show abstract

Vertex Nim played on graphs

Cited by 10 publications

References 19 publications

On the complexity of the misère version of three games played on graphs

On the complexity of the misère version of three games played on graphs

Winning the War by (Strategically) Losing Battles: Settling the Complexity of Grundy-Values in Undirected Geography

Impartial Intersection Restriction Games

Contact Info

Product

Resources

About