19 research outputs found
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
Query complexity of Boolean functions on the middle slice of the cube
We study the query complexity of slices of Boolean functions. Among other
results we show that there exists a Boolean function for which we need to query
all but 7 input bits to compute its value, even if we know beforehand that the
number of 0's and 1's in the input are the same, i.e. when our input is from
the middle slice. This answers a question of Byramji. Our proof is
non-constructive, but we also propose a concrete candidate function that might
have the above property. Our results are related to certain natural discrepancy
type questions that -- somewhat surprisingly -- have not been studied before.Comment: 10 page
Size Ramsey Number of Bounded Degree Graphs for Games
We study Maker/Breaker games on the edges of sparse graphs. Maker and Breaker take turns at claiming previously unclaimed edges of a given graph H. Maker aims to occupy a given target graph G and Breaker tries to prevent Maker from achieving his goal. We show that for every d there is a constant c = c(d) with the property that for every graph G on n vertices of maximum degree d there is a graph H on at most cn edges such that Maker has a strategy to occupy a copy of G in the game on H. This is a result about a game-theoretic variant of the size Ramsey number. For a given graph G, is defined as the smallest number M for which there exists a graph H with M edges such that Maker has a strategy to occupy a copy of G in the game on H. In this language, our result yields that for every connected graph G of constant maximum degree, . Moreover, we can also use our method to settle the corresponding extremal number for universal graphs: for a constant d and for the class of n-vertex graphs of maximum degree d, denotes the minimum number such that there exists a graph H with M edges where, for every G ∈ , Maker has a strategy to build a copy of G in the game on H. We obtain that $s({\cal G}_{n}) = \Theta(n^{2 - \frac{2}{d}})
Avoider-Enforcer star games
In this paper, we study Avoider-Enforcer games played on the edge
set of the complete graph on vertices. For every constant we
analyse the -star game, where Avoider tries to avoid claiming edges
incident to the same vertex. We analyse both versions of Avoider-Enforcer games
-- the strict and the monotone -- and for each provide explicit winning
strategies for both players. We determine the order of magnitude of the
threshold biases , and
, where is the hypergraph of the game
Efficient Splitting of Necklaces
We provide efficient approximation algorithms for the Necklace Splitting problem. The input consists of a sequence of beads of n types and an integer k. The objective is to split the necklace, with a small number of cuts made between consecutive beads, and distribute the resulting intervals into k collections so that the discrepancy between the shares of any two collections, according to each type, is at most 1. We also consider an approximate version where each collection should contain at least a (1-?)/k and at most a (1+?)/k fraction of the beads of each type. It is known that there is always a solution making at most n(k-1) cuts, and this number of cuts is optimal in general. The proof is topological and provides no efficient procedure for finding these cuts. It is also known that for k = 2, and some fixed positive ?, finding a solution with n cuts is PPAD-hard.
We describe an efficient algorithm that produces an ?-approximate solution for k = 2 making n (2+log (1/?)) cuts. This is an exponential improvement of a (1/?)^O(n) bound of Bhatt and Leighton from the 80s. We also present an online algorithm for the problem (in its natural online model), in which the number of cuts made to produce discrepancy at most 1 on each type is O?(m^{2/3} n), where m is the maximum number of beads of any type. Lastly, we establish a lower bound showing that for the online setup this is tight up to logarithmic factors. Similar results are obtained for k > 2
Pozicione igre na grafovima
\section*{Abstract} We study Maker-Breaker games played on the edges of the complete graph on vertices, , whose family of winning sets \cF consists of all edge sets of subgraphs which possess a predetermined monotone increasing property. Two players, Maker and Breaker, take turns in claiming , respectively , unclaimed edges per move. We are interested in finding the threshold bias b_{\cF}(a) for all values of , so that for every , b\leq b_{\cF}(a), Maker wins the game and for all values of , such that b>b_{\cF}(a), Breaker wins the game. We are particularly interested in cases where both and can be greater than . We focus on the \textit{Connectivity game}, where the winning sets are the edge sets of all spanning trees of and on the \textit{Hamiltonicity game}, where the winning sets are the edge sets of all Hamilton cycles on . Next, we consider biased Avoider-Enforcer games, also played on the edges of . For every constant we analyse the -star game, where Avoider tries to avoid claiming edges incident to the same vertex. We analyse both versions of Avoider-Enforcer games, the strict and the monotone, and for each provide explicit winning strategies for both players. Consequentially, we establish bounds on the threshold biases f^{mon}_\cF, f^-_\cF and f^+_\cF, where \cF is the hypergraph of the game (the family of target sets). We also study the monotone version of -game, where Avoider wants to avoid claiming all the edges of some graph isomorphic to in . Finally, we search for the fast winning strategies for Maker in Perfect matching game and Hamiltonicity game, again played on the edge set of . Here, we look at the biased games, where Maker's bias is 1, and Breaker's bias is .\section*{Izvod} Prou\v{c}avamo takozvane Mejker-Brejker (Maker-Breaker) igre koje se igraju na granama kompletnog grafa sa \v{c}vorova, , \v{c}ija familija pobedni\v{c}kih skupova \cF obuhvata sve skupove grana grafa koji imaju neku monotono rastu\'{c}u osobinu. Dva igra\v{c}a, \textit{Mejker} (\textit{Pravi\v{s}a}) i \textit{Brejker} (\textit{Kva\-ri\-\v{s}a}) se smenjuju u odabiru , odnosno , slobodnih grana po potezu. Interesuje nas da prona\dj emo grani\v{c}ni bias b_{\cF}(a) za sve vrednosti pa\-ra\-me\-tra , tako da za svako , b\le b_{\cF}(a), Mejker pobe\dj uje u igri, a za svako , takvo da je b>b_{\cF}(a), Brejker pobe\dj uje. Posebno nas interesuju slu\v{c}ajevi u kojima oba parametra i mogu imati vrednost ve\'cu od 1. Na\v{s}a pa\v{z}nja je posve\'{c}ena igri povezanosti, gde su pobedni\v{c}ki skupovi grane svih pokrivaju\'cih stabala grafa , kao i igri Hamiltonove konture, gde su pobedni\v{c}ki skupovi grane svih Hamiltonovih kontura grafa . Zatim posmatramo igre tipa Avojder-Enforser (Avoider-Enforcer), sa biasom , koje se tako\dj e igraju na granama kompletnog grafa sa \v{c}vorova, . Za svaku konstantu , analiziramo igru -zvezde (zvezde sa krakova), u kojoj \textit{Avojder} poku\v{s}va da izbegne da ima svojih grana incidentnih sa istim \v{c}vorom. Posmatramo obe verzije ove igre, striktnu i monotonu, i za svaku dajemo eksplicitnu pobedni\v{c}ku strategiju za oba igra\v{c}a. Kao rezultat, dobijamo gornje i donje ograni\v{c}enje za grani\v{c}ne biase f^{mon}_\cF, f^-_\cF i f^+_\cF, gde \cF predstavlja hipergraf igre (familija ciljnih skupova). %, and . Tako\dj e, posmatramo i monotonu verziju -igre, gde Avojder \v{z}eli da izbegne da graf koji \v{c}ine njegove grane sadr\v{z}i graf izomorfan sa . Kona\v{c}no, \v{z}elimo da prona\dj emo strategije za brzu pobedu Mejkera u igrama savr\v{s}enog me\v{c}inga i Hamiltonove konture, koje se tako\dj e igraju na granama kompletnog grafa . Ovde posmatramo asimetri\v{c}ne igre gde je bias Mejkera 1, a bias Brejkera ,
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Combinatorics
This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics