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Directed Switching Games on graphs and matroids

Yahya Ould Hamidoune, Michel Las Vergnas
1986 Journal of combinatorial theory. Series B (Print)  
EXTENDED AND UNROOTED DIRECTED SWITCHING GAMES We derive in this section the classifications of the Extended Directed Shannon Switching Game and the graphic and cographic Unrooted Directed Switching Game  ...  Unlike the undirected case the Extended Directed Shannon Switching Game and the cographic Unrooted Directed Switching Game are not dual games.  ...  Analogously a natural context for directed switching games is provided by oriented matroids. We introduce in this section several directed switching games on oriented matroids.  ... 
doi:10.1016/0095-8956(86)90083-3 fatcat:dtkegvvi6vcd3ios3xtdidscge

The directed switching game on Lawrence oriented matroids

David Forge, Adrien Vieilleribière
2009 European journal of combinatorics (Print)  
The game has been generalized and solved for matroids by A. Lehman [1] . A directed switching game on graphs and oriented matroids was introduced by Y. O. Hamidoune and M. Las Vergnas in 1986 [2].  ...  The main content of the note is a proof of the conjecture of Hamidoune and Las Vergnas on the directed switching game in the case of Lawrence oriented matroids. introduced the switching game for graphs  ...  The directed switching game on a Lawrence matroid of rank r and of order n is winning for Maker playing first if and only if n ≥ 2r. Proof.  ... 
doi:10.1016/j.ejc.2008.12.002 fatcat:2bwzmrfu2fbpvlh2nplk7twbi4

Page 1810 of Mathematical Reviews Vol. , Issue 85e [page]

1985 Mathematical Reviews  
Hamidoune, Yahya Ould (F-PARIS6); Las Vergnas, Michel (F-PARIS6) Jeux de commutation orientés sur les graphes et les matroides. (English summary) [Directed switching games on graphs and matroids] C.  ...  We give complete clas- sifications of directed switching games on graphic and cographic oriented matroids.” Li, Wei Xuan On matroids of the greatest W-connectivity. J. Combin. Theory Ser.  ... 

The switching game on unions of oriented matroids

Vanessa Chatelain, Jorge Luis Ramírez Alfonsín
2012 European journal of combinatorics (Print)  
They conjectured that the classification of the directed switching game on oriented matroids is identical to the classification of the nonoriented version.  ...  In 1986, Hamidoune and Las Vergnas [3] introduced an oriented matroid version of the so-called Shannon's switching game.  ...  The directed switching game for oriented matroids. Let M be an oriented matroid and e one of its elements.  ... 
doi:10.1016/j.ejc.2011.08.006 fatcat:cp7yxt27ongadinhy3vcqxlfam

Page 6045 of Mathematical Reviews Vol. , Issue 87j [page]

1987 Mathematical Reviews  
Hamidoune, Yahya Ould (F-PARIS6); Las Vergnas, Michel (F-PARIS6) Directed switching games on graphs and matroids. J. Combin. Theory Ser. B 40 (1986), no. 3, 237-269.  ...  In the Directed Shannon Switching Game, the graph G is again undirected with distinguished vertices zp and z;.  ... 

An extension of elementary bipartite graphs to matroid pairs and its characterization

Masataka Nakamura
1994 Discrete Applied Mathematics  
In this paper we shall present a generalization of elementary bipartite graphs to a certain pair of matroids, which we shall call an "elementary pair", and show that a matroid pair is elementary if and  ...  only if it is irreducible with respect to a decomposition method associated with matroid intersection problems.  ...  And when applied to a graph, this decomposition provides a constructive solution to the Shannon switching game [l].  ... 
doi:10.1016/0166-218x(92)00181-k fatcat:ctaozxcx45hezinkbwx4srhmx4

Matroids are Immune to Braess Paradox [article]

Satoru Fujishige, Michel X. Goemans, Tobias Harks, Britta Peis and Rico Zenklusen
2017 arXiv   pre-print
In this paper we consider general nonatomic congestion games and give a characterization of the maximal combinatorial property of strategy spaces for which Braess paradox does not occur.  ...  In a nutshell, bases of matroids are exactly this maximal structure.  ...  Perhaps the most famous example of a nonatomic congestion game appears in the traffic model of Wardrop where the resources form a (directed) graph and a population corresponds to a continuum of players  ... 
arXiv:1504.07545v2 fatcat:urckk3k5lnfzzfqchdx74m7t5m

Convergence Dynamics of Graphical Congestion Games [chapter]

Richard Southwell, Yanjiao Chen, Jianwei Huang, Qian Zhang
2012 Lecture Notes of the Institute for Computer Sciences, Social Informatics and Telecommunications Engineering  
When the collection of sets of resources available to each player is a matroid, we show that pure Nash equilibria may not exist in the most general case.  ...  However, if the resources are homogenous, the game can converge to a Nash equilibrium in polynomial time.  ...  Such games are proved to have the finite improvement property when the graph is undirected or acyclic. But the authors illustrated a game on a directed graph with no pure Nash equilibria.  ... 
doi:10.1007/978-3-642-35582-0_3 fatcat:nfoxs3ojvnbopizlyemodvdbhm

Computing Pure Nash and Strong Equilibria in Bottleneck Congestion Games [chapter]

Tobias Harks, Martin Hoefer, Max Klimm, Alexander Skopalik
2010 Lecture Notes in Computer Science  
In contrast, computing even a single coalitional improvement move in matroid and single-commodity games is strongly NP-hard.  ...  In this paper, we study the computational complexity of pure Nash and strong equilibria in these games.  ...  Let us now consider strategy spaces defined as a-arborescences, which are in general not matroids. Let D = (V, R) be a directed graph with |R| = m.  ... 
doi:10.1007/978-3-642-15781-3_3 fatcat:5qpyjudfvzc6pppyesaqob5yya

Page 38 of Mathematical Reviews Vol. , Issue 87a [page]

1987 Mathematical Reviews  
Contents: Mikio Kano, Regular-component factors of graphs (pp. 1-7); Yohei Yamasaki, Skannon switching games without termi- nals (Japanese) (pp. 8-12); Akira Nakayama and Hikoe Enomoto, Linear arboricity  ...  Székely, On two concepts of discrepancy in a class of combinatorial games (pp. 679-683); V. D. Tonchev, Quasiresidual designs, codes and graphs (pp. 685-695). D. T.  ... 

Resource Buying Games [article]

Tobias Harks, Britta Peis
2012 arXiv   pre-print
We show that for marginally non-increasing cost functions, matroids are exactly the right structure to consider, and that resource buying games with marginally non-decreasing cost functions always admit  ...  In contrast to classical congestion games for which equilibria are guaranteed to exist, the existence of equilibria in resource buying games strongly depends on the underlying structure of the S_i's and  ...  Matroids have a rich combinatorial structure and include, for instance, the setting where each player wants to build a spanning tree in a graph. In Section 4, we study weighted resource buying games.  ... 
arXiv:1204.4111v1 fatcat:6h57gr7p6nbwjisvb4d3xz2tma

Resource Buying Games [chapter]

Tobias Harks, Britta Peis
2012 Lecture Notes in Computer Science  
We show that for marginally non-increasing cost functions, matroids are exactly the right structure to consider, and that resource buying games with marginally non-decreasing cost functions always admit  ...  The cost of a resource e depends on the number (or load) of players using e, and has to be paid completely by the players before it becomes available.  ...  Matroids have a rich combinatorial structure and include, for instance, the setting where each player wants to build a spanning tree in a graph. In Section 4, we study weighted resource buying games.  ... 
doi:10.1007/978-3-642-33090-2_49 fatcat:7psv5holufek3gsddmih5lfklq

Matroids Are Immune to Braess' Paradox

Satoru Fujishige, Michel X. Goemans, Tobias Harks, Britta Peis, Rico Zenklusen
2017 Mathematics of Operations Research  
In this paper, we consider general nonatomic congestion games and give a characterization of the combinatorial property of strategy spaces for which the Braess paradox does not occur.  ...  In short, matroid bases are precisely the required structure.  ...  In particular, we are grateful to one reviewer who pointed out the special role of clutters in our characterization.  ... 
doi:10.1287/moor.2016.0825 fatcat:qqmsftbd5revtjmtdvmut2u274

Totally Unimodular Congestion Games [article]

Alberto Del Pia, Michael Ferris, Carla Michini
2016 arXiv   pre-print
We also show how this technique can be extended to matroid congestion games.  ...  We introduce some combinatorial TU congestion games, where the players' strategies are matchings, vertex covers, edge covers, and stable sets of a given bipartite graph.  ...  Examples of TU matrices that are not incidence matrices of directed graphs include incidence matrices of bipartite graphs, matrices with the consecutive-ones property, and network matrices.  ... 
arXiv:1511.02784v2 fatcat:huq3ovo52vhvtjsq36cmi33xoi

On the Structure of the Graph of Unique Symmetric Base Exchanges of Bispanning Graphs [article]

Timo Bingmann
2016 arXiv   pre-print
We are interested in a restriction of these graphs to only unique symmetric base exchanges, which are edge exchanges wherein selecting one edge leaves only one choice for selecting the other.  ...  exchange graphs and forwarding edges at the join seam.  ...  Directed Graphs When considering exchange graphs on game, we require directed graphs in intermediate steps.  ... 
arXiv:1601.03526v3 fatcat:42buu5ux7jdilf2zaycnpe5vdu
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