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Register a new userDistributed recoloring of interval and chordal graphs
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Added by
Laurent Feuilloley
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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One of the fundamental and most-studied algorithmic problems in distributed computing on networks is graph coloring, both in bounded-degree and in general graphs. Recently, the study of this problem has been extended in two directions. First, the problem of recoloring, that is computing an efficient transformation between two given colorings (instead of computing a new coloring), has been considered, both to model radio network updates, and as a useful subroutine for coloring. Second, as it appears that general graphs and bounded-degree graphs do not model real networks very well (with, respectively, pathological worst-case topologies and too strong assumptions), coloring has been studied in more specific graph classes. In this paper, we study the intersection of these two directions: distributed recoloring in two relevant graph classes, interval and chordal graphs.
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Two (proper) colorings of a graph are adjacent if they differ on exactly one vertex. Jerrum proved that any $(d + 2)$-coloring of any d-degenerate graph can be transformed into any other via a sequence of adjacent colorings. A result of Bonamy et al. ensures that a shortest transformation can have a quadratic length even for $d = 1$. Bousquet and Perarnau proved that a linear transformation exists for between $(2d + 2)$-colorings. It is open to determine if this bound can be reduced. In this note, we prove that it can be reduced for graphs of treewidth 2, which are 2-degenerate. There exists a linear transformation between 5-colorings. It completes the picture for graphs of treewidth 2 since there exist graphs of treewidth 2 such a linear transformation between 4-colorings does not exist.
Naor, Parter, and Yogev (SODA 2020) have recently demonstrated the existence of a emph{distributed interactive proof} for planarity (i.e., for certifying that a network is planar), using a sophisticated generic technique for constructing distributed IP protocols based on sequential IP protocols. The interactive proof for planarity is based on a distributed certification of the correct execution of any given sequential linear-time algorithm for planarity testing. It involves three interactions between the prover and the randomized distributed verifier (i.e., it is a dMAM/ protocol), and uses small certificates, on $O(log n)$ bits in $n$-node networks. We show that a single interaction from the prover suffices, and randomization is unecessary, by providing an explicit description of a emph{proof-labeling scheme} for planarity, still using certificates on just $O(log n)$ bits. We also show that there are no proof-labeling schemes -- in fact, even no emph{locally checkable proofs} -- for planarity using certificates on $o(log n)$ bits.
Coloring unit-disk graphs efficiently is an important problem in the global and distributed setting, with applications in radio channel assignment problems when the communication relies on omni-directional antennas of the same power. In this context it is important to bound not only the complexity of the coloring algorithms, but also the number of colors used. In this paper, we consider two natural distributed settings. In the location-aware setting (when nodes know their coordinates in the plane), we give a constant time distributed algorithm coloring any unit-disk graph $G$ with at most $(3+epsilon)omega(G)+6$ colors, for any constant $epsilon>0$, where $omega(G)$ is the clique number of $G$. This improves upon a classical 3-approximation algorithm for this problem, for all unit-disk graphs whose chromatic number significantly exceeds their clique number. When nodes do not know their coordinates in the plane, we give a distributed algorithm in the LOCAL model that colors every unit-disk graph $G$ with at most $5.68omega(G)$ colors in $O(2^{sqrt{log log n}})$ rounds. Moreover, when $omega(G)=O(1)$, the algorithm runs in $O(log^* n)$ rounds. This algorithm is based on a study of the local structure of unit-disk graphs, which is of independent interest. We conjecture that every unit-disk graph $G$ has average degree at most $4omega(G)$, which would imply the existence of a $O(log n)$ round algorithm coloring any unit-disk graph $G$ with (approximatively) $4omega(G)$ colors.
Suppose that D is an acyclic orientation of a graph G. An arc of D is called dependent if its reversal creates a directed cycle. Let m and M denote the minimum and the maximum of the number of dependent arcs over all acyclic orientations of G. We call G fully orientable if G has an acyclic orientation with exactly d dependent arcs for every d satisfying m <= d <= M. A graph G is called chordal if every cycle in G of length at least four has a chord. We show that all chordal graphs are fully orientable.
We consider the problem of aggregating data in a dynamic graph, that is, aggregating the data that originates from all nodes in the graph to a specific node, the sink. We are interested in giving lower bounds for this problem, under different kinds of adversaries. In our model, nodes are endowed with unlimited memory and unlimited computational power. Yet, we assume that communications between nodes are carried out with pairwise interactions, where nodes can exchange control information before deciding whether they transmit their data or not, given that each node is allowed to transmit its data at most once. When a node receives a data from a neighbor, the node may aggregate it with its own data. We consider three possible adversaries: the online adaptive adversary, the oblivious adversary , and the randomized adversary that chooses the pairwise interactions uniformly at random. For the online adaptive and the oblivious adversary, we give impossibility results when nodes have no knowledge about the graph and are not aware of the future. Also, we give several tight bounds depending on the knowledge (be it topology related or time related) of the nodes. For the randomized adversary, we show that the Gathering algorithm, which always commands a node to transmit, is optimal if nodes have no knowledge at all. Also, we propose an algorithm called Waiting Greedy, where a node either waits or transmits depending on some parameter, that is optimal when each node knows its future pairwise interactions with the sink.
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