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Conflict-free coloring.

機譯：無沖突的著色。

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Graph and hypergraph colorings constitute an important subject in combinatorics and algorithm theory. In this work, we study conflict-free coloring for hypergraphs. Conflict-free coloring is one possible generalization of traditional graph coloring. Conflict-free coloring hypergraphs induced by geometric shapes, like intervals on the line, or disks on the plane, has applications in frequency assignment in cellular networks. Colors model frequencies and since the frequency spectrum is limited and expensive, the goal of an algorithm is to minimize the number of assigned frequencies, that is, reuse frequencies as much as possible.;We concentrate on an online variation of the problem, especially in the case where the hypergraph is induced by intervals. For deterministic algorithms, we introduce a hierarchy of models ranging from static to online and we compute lower and upper bounds on the numbers of colors used.;In the randomized oblivious adversary model, we introduce a framework for conflict-free coloring a specific class of hypergraphs with a logarithmic number of colors. This specific class includes many hypergraphs arising in geometry and gives online randomized algorithm that use fewer colors and fewer random bits than other algorithms in the literature. Based on the same framework, we initiate the study of online deterministic algorithms that recolor few points.;For the problem of conlict-free coloring points with respect to a given set of intervals, we describe an efficient algorithm that computes a coloring with at most twice the number of colors of an optimal coloring. We also show that there is a family of inputs that force our algorithm to use two times the number of colors of an optimal solution.;Then, we study conflict-free coloring problems in graphs. We compare conflict-free coloring with respect to paths of graphs to a closely related problem, called vertex ranking, or ordered coloring. For conflict-free coloring with respect to neighborhoods of vertices of graphs, we prove that number of colors in the order of the square root of the number of vertices is sufficient and sometimes necessary.;Finally, we initiate the study of Ramsey-type problems for conflict-free colorings and compute a van der Waerden-like number.
機譯：圖和超圖著色是組合和算法理論中的重要主題。在這項工作中，我們研究了超圖的無沖突著色。無沖突著色是傳統(tǒng)圖形著色的一種可能概括。由幾何形狀（如直線上的間隔或平面上的磁盤）引起的無沖突著色超圖，已在蜂窩網(wǎng)絡的頻率分配中得到應用。對模型頻率進行著色，并且由于頻譜有限且昂貴，因此算法的目標是最大程度地減少分配的頻率，即盡可能多地重復使用頻率。；我們專注于問題的在線變化，尤其是在間隔誘發(fā)超圖的情況。對于確定性算法，我們引入了從靜態(tài)到在線的模型層次結(jié)構(gòu)，并計算了所使用顏色的數(shù)量上限和下限;在隨機遺忘的對手模型中，我們引入了為特定類別的對象進行無沖突著色的框架顏色數(shù)量為對數(shù)的超圖。該特定類包括許多出現(xiàn)在幾何圖形中的超圖，并提供了在線隨機算法，該算法比文獻中的其他算法使用更少的顏色和更少的隨機位?；谙嗤目蚣?，我們開始研究對少數(shù)點進行重新著色的在線確定性算法。針對針對給定間隔的無沖突著色點的問題，我們描述了一種高效的算法，該算法最多可以計算著色次數(shù)最佳著色的顏色數(shù)量的兩倍。我們還表明，有一系列輸入迫使我們的算法使用最佳解決方案的兩倍顏色;然后，我們研究了圖形中的無沖突著色問題。我們將與圖路徑有關的無沖突著色與一個緊密相關的問題（稱為頂點排序或有序著色）進行比較。對于關于圖的頂點鄰域的無沖突著色，我們證明了按頂點數(shù)的平方根順序排列的顏色數(shù)是足夠的，有時是必要的。最后，我們開始研究Ramsey型問題用于無沖突的著色并計算類似范德瓦爾登的數(shù)。

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