邻域猜想及其对SAT的推论的反证
Disproof of the Neighborhood Conjecture with Implications to SAT
摘要：
我们研究了贝克描述的制造者/破坏者博弈。其结果是，我们反驳了贝克猜想的位置游戏，建立这个游戏和SAT之间的连接，并构造一个不可满足的k-CNF公式，每个变量的出现次数很少，从而改善了以前的结果由Hoory和Szeider和显示，从Lovasz局部引理得到的界限是紧到一个常数因子。我们研究的Maker/Breaker游戏如下。Maker和Breaker轮流从给定的n-一致超图F中选择顶点，Maker先去。Maker的目标是完全占据一个超边，而Breaker试图避免这一点。Beck指出，如果F的最大邻域大小至多为2^（n-1），则Breaker有获胜策略。我们通过建立一个最大邻域大小为3*2^（n - 3）的n-一致超图来反驳这个猜想，其中Maker有一个获胜策略。此外，我们给出了如何构造一个最大度为（2^（n-1））/n的n-一致超图，其中maker有一个获胜策略.最后，我们建立了SAT和我们研究的制造商/断路器游戏之间的联系。我们可以利用这种联系来推导SAT中的新结果。Kratochvil，Savicky和Tuza证明了对于任意k >= 3，存在整数f（k）使得每个（k，f（k））-公式是可满足的，但是（k，f（k）+ 1）-SAT已经是NP-完全的（不知道f（k）是否可计算）。Kratochvil、Savicky和Tuza也给出了最著名的下界f（k）= Omega（2^k/k），这是Lovasz局部引理的一个推论。我们证明了f（k）= Theta（2^k/k），改进了Hoory和Szeider给出的最好的上界O（（log k）* 2^k/k）.
英文摘要：
We study a Maker/Breaker game described by Beck. As a result we disprove a conjecture of Beck on positional games, establish a connection between this game and SAT and construct an unsatisfiable k-CNF formula with few occurrences per variable, thereby improving a previous result by Hoory and Szeider and showing that the bound obtained from the Lovasz Local Lemma is tight up to a constant factor. The Maker/Breaker game we study is as follows. Maker and Breaker take turns in choosing vertices from a given n-uniform hypergraph F, with Maker going first. Maker's goal is to completely occupy a hyperedge and Breaker tries to avoid this. Beck conjectures that if the maximum neighborhood size of F is at most 2^(n-1) then Breaker has a winning strategy. We disprove this conjecture by establishing an n-uniform hypergraph with maximum neighborhood size 3*2^(n - 3) where Maker has a winning strategy. Moreover, we show how to construct an n-uniform hypergraph with maximum degree (2^(n-1))/n where maker has a winning strategy. Finally, we establish a connection between SAT and the Maker/Breaker game we study. We can use this connection to derive new results in SAT. Kratochvil, Savicky and Tuza showed that for every k >= 3 there is an integer f(k) such that every (k,f(k))-formula is satisfiable, but (k,f(k) + 1)-SAT is already NP-complete (it is not known whether f(k) is computable). Kratochvil, Savicky and Tuza also gave the best known lower bound f(k) = Omega(2^k/k), which is a consequence of the Lovasz Local Lemma. We prove that, in fact, f(k) = Theta(2^k/k), improving upon the best known upper bound O((log k) * 2^k/k) by Hoory and Szeider.
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