7,934 research outputs found

    A tight analysis of Kierstead-Trotter algorithm for online unit interval coloring

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    Kierstead and Trotter (Congressus Numerantium 33, 1981) proved that their algorithm is an optimal online algorithm for the online interval coloring problem. In this paper, for online unit interval coloring, we show that the number of colors used by the Kierstead-Trotter algorithm is at most 3ω(G)33 \omega(G) - 3, where ω(G)\omega(G) is the size of the maximum clique in a given graph GG, and it is the best possible.Comment: 4 page

    An easy subexponential bound for online chain partitioning

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    Bosek and Krawczyk exhibited an online algorithm for partitioning an online poset of width ww into w14lgww^{14\lg w} chains. We improve this to w6.5lgw+7w^{6.5 \lg w + 7} with a simpler and shorter proof by combining the work of Bosek & Krawczyk with work of Kierstead & Smith on First-Fit chain partitioning of ladder-free posets. We also provide examples illustrating the limits of our approach.Comment: 23 pages, 11 figure

    An Improved Bound for First-Fit on Posets Without Two Long Incomparable Chains

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    It is known that the First-Fit algorithm for partitioning a poset P into chains uses relatively few chains when P does not have two incomparable chains each of size k. In particular, if P has width w then Bosek, Krawczyk, and Szczypka (SIAM J. Discrete Math., 23(4):1992--1999, 2010) proved an upper bound of ckw^{2} on the number of chains used by First-Fit for some constant c, while Joret and Milans (Order, 28(3):455--464, 2011) gave one of ck^{2}w. In this paper we prove an upper bound of the form ckw. This is best possible up to the value of c.Comment: v3: referees' comments incorporate

    An on-line competitive algorithm for coloring bipartite graphs without long induced paths

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    The existence of an on-line competitive algorithm for coloring bipartite graphs remains a tantalizing open problem. So far there are only partial positive results for bipartite graphs with certain small forbidden graphs as induced subgraphs. We propose a new on-line competitive coloring algorithm for P9P_9-free bipartite graphs

    Lower bounds for on-line graph colorings

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    We propose two strategies for Presenter in on-line graph coloring games. The first one constructs bipartite graphs and forces any on-line coloring algorithm to use 2log2n102\log_2 n - 10 colors, where nn is the number of vertices in the constructed graph. This is best possible up to an additive constant. The second strategy constructs graphs that contain neither C3C_3 nor C5C_5 as a subgraph and forces Ω(nlogn13)\Omega(\frac{n}{\log n}^\frac{1}{3}) colors. The best known on-line coloring algorithm for these graphs uses O(n12)O(n^{\frac{1}{2}}) colors

    Optimal Online Edge Coloring of Planar Graphs with Advice

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    Using the framework of advice complexity, we study the amount of knowledge about the future that an online algorithm needs to color the edges of a graph optimally, i.e., using as few colors as possible. For graphs of maximum degree Δ\Delta, it follows from Vizing's Theorem that O(mlogΔ)O(m\log \Delta) bits of advice suffice to achieve optimality, where mm is the number of edges. We show that for graphs of bounded degeneracy (a class of graphs including e.g. trees and planar graphs), only O(m)O(m) bits of advice are needed to compute an optimal solution online, independently of how large Δ\Delta is. On the other hand, we show that Ω(m)\Omega (m) bits of advice are necessary just to achieve a competitive ratio better than that of the best deterministic online algorithm without advice. Furthermore, we consider algorithms which use a fixed number of advice bits per edge (our algorithm for graphs of bounded degeneracy belongs to this class of algorithms). We show that for bipartite graphs, any such algorithm must use at least Ω(mlogΔ)\Omega(m\log \Delta) bits of advice to achieve optimality.Comment: CIAC 201
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