Improved distributed algorithms for coloring interval graphs with application to multicoloring trees

Post-print (lokagerð höfundar)We give a distributed (1+eps)-approximation algorithm for the minimum vertex coloring problem on interval graphs, which runs in the LOCAL model and operates in O((1/eps) log* n) rounds. If nodes are aware of their interval representations, then the algorithm can be adapted to the CONGEST model using the same number of rounds. Prior to this work, only constant factor approximations using O(log* n) rounds were known. Linial's ring coloring lower bound implies that the dependency on log* n cannot be improved. We further prove that the dependency on 1/eps is also optimal. To obtain our CONGEST model algorithm, we develop a color rotation technique that may be of independent interest. We demonstrate that color rotations can also be applied to obtain a (1+eps)-approximate multicoloring of directed trees in O((1/eps)log* n) rounds.Magnus M. Halldorsson is supported by grants 152679-05 and 174484-05 from the Icelandic Research Fund. Christian Konrad is supported by the Centre for Discrete Mathematics and its Applications (DIMAP) at Warwick University and by EPSRC award EP/N011163/1."Peer Reviewed

Local Conflict Coloring

Locally finding a solution to symmetry-breaking tasks such as vertex-coloring, edge-coloring, maximal matching, maximal independent set, etc., is a long-standing challenge in distributed network computing. More recently, it has also become a challenge in the framework of centralized local computation. We introduce conflict coloring as a general symmetry-breaking task that includes all the aforementioned tasks as specific instantiations --- conflict coloring includes all locally checkable labeling tasks from [Naor\&Stockmeyer, STOC 1993]. Conflict coloring is characterized by two parameters $l$ and $d$, where the former measures the amount of freedom given to the nodes for selecting their colors, and the latter measures the number of constraints which colors of adjacent nodes are subject to.We show that, in the standard LOCAL model for distributed network computing, if l/d \textgreater{} \Delta, then conflict coloring can be solved in $\tilde O(\sqrt{\Delta})+\log^*n$ rounds in $n$-node graphs with maximum degree $\Delta$, where $\tilde O$ ignores the polylog factors in $\Delta$. The dependency in~$n$ is optimal, as a consequence of the $\Omega(\log^*n)$ lower bound by [Linial, SIAM J. Comp. 1992] for $(\Delta+1)$-coloring. An important special case of our result is a significant improvement over the best known algorithm for distributed $(\Delta+1)$-coloring due to [Barenboim, PODC 2015], which required $\tilde O(\Delta^{3/4})+\log^*n$ rounds. Improvements for other variants of coloring, including $(\Delta+1)$-list-coloring, $(2\Delta-1)$-edge-coloring, $T$-coloring, etc., also follow from our general result on conflict coloring. Likewise, in the framework of centralized local computation algorithms (LCAs), our general result yields an LCA which requires a smaller number of probes than the previously best known algorithm for vertex-coloring, and works for a wide range of coloring problems

Performance of distributed mechanisms for flow admission in wireless adhoc networks

Given a wireless network where some pairs of communication links interfere with each other, we study sufficient conditions for determining whether a given set of minimum bandwidth quality-of-service (QoS) requirements can be satisfied. We are especially interested in algorithms which have low communication overhead and low processing complexity. The interference in the network is modeled using a conflict graph whose vertices correspond to the communication links in the network. Two links are adjacent in this graph if and only if they interfere with each other due to being in the same vicinity and hence cannot be simultaneously active. The problem of scheduling the transmission of the various links is then essentially a fractional, weighted vertex coloring problem, for which upper bounds on the fractional chromatic number are sought using only localized information. We recall some distributed algorithms for this problem, and then assess their worst-case performance. Our results on this fundamental problem imply that for some well known classes of networks and interference models, the performance of these distributed algorithms is within a bounded factor away from that of an optimal, centralized algorithm. The performance bounds are simple expressions in terms of graph invariants. It is seen that the induced star number of a network plays an important role in the design and performance of such networks.Comment: 21 pages, submitted. Journal version of arXiv:0906.378