Online Dual Edge Coloring of Paths and Trees

We study a dual version of online edge coloring, where the goal is to color as many edges as possible using only a given number, \(k\), of available colors. All of our results are with regard to competitive analysis. For paths, we consider \(k=2\), and for trees, we consider any \(k \ge 2\). We prove that a natural greedy algorithm called First-Fit is optimal among deterministic algorithms on paths as well as trees. This is the first time that an optimal algorithm for online dual edge coloring has been identified for a class of graphs. For paths, we give a randomized algorithm, which is optimal and better than the best possible deterministic algorithm. Again, it is the first time that this has been done for a class of graphs. For trees, we also show that even randomized algorithms cannot be much better than First-Fit.

[1]  Joan Boyar,et al.  The relative worst order ratio for online algorithms , 2007, TALG.

[2]  Joan Boyar,et al.  The Seat Reservation Problem , 1999, Algorithmica.

[3]  Zhi-Zhong Chen,et al.  Approximating Maximum Edge 2-Coloring in Simple Graphs , 2010, AAIM.

[4]  Marcin Kaminski,et al.  Approximating the Maximum 3- and 4-Edge-Colorable Subgraph , 2010, SWAT.

[5]  Allan Borodin,et al.  Online computation and competitive analysis , 1998 .

[6]  Joan Boyar,et al.  The relative worst order ratio applied to paging , 2005, SODA '05.

[7]  Lene M. Favrholdt,et al.  Comparing First-Fit and Next-Fit for online edge coloring , 2010, Theor. Comput. Sci..

[8]  Anna R. Karlin,et al.  Competitive snoopy caching , 1986, 27th Annual Symposium on Foundations of Computer Science (sfcs 1986).

[9]  Robert E. Tarjan,et al.  Amortized efficiency of list update and paging rules , 1985, CACM.

[10]  Andrew Chi-Chih Yao,et al.  Probabilistic computations: Toward a unified measure of complexity , 1977, 18th Annual Symposium on Foundations of Computer Science (sfcs 1977).

[11]  Romeo Rizzi.  Approximating the maximum 3-edge-colorable subgraph problem , 2009, Discret. Math..

[12]  Uriel Feige,et al.  Approximating Maximum Edge Coloring in Multigraphs , 2002, APPROX.

[13]  Adrian Kosowski.  Approximating the maximum 2- and 3-edge-colorable subgraph problems , 2009, Discret. Appl. Math..

[14]  Rajeev Motwani,et al.  The Greedy Algorithm is Optimal for On-Line Edge Coloring , 1992, Inf. Process. Lett..

[15]  Hal A. Kierstead,et al.  Coloring Graphs On-line , 1996, Online Algorithms.

[16]  Tao Jiang,et al.  Tight Bounds on the Competitive Ratio on Accommodating Sequences for the Seat Reservation Problem , 2003, J. Sched..

[17]  Morten N. Nielsen,et al.  On-Line Edge-Coloring with a Fixed Number of Colors , 2000, Algorithmica.

[18]  Lukasz Kowalik,et al.  Beyond the Vizing's Bound for at Most Seven Colors , 2014, SIAM J. Discret. Math..