Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userOnline Edge Coloring Algorithms via the Nibble Method
65
0
0.0
(
0
)
Added by
David Wajc
Publication date
2020
fields
Informatics Engineering
and research's language is
English
Ask ChatGPT about the research
No Arabic abstract
Nearly thirty years ago, Bar-Noy, Motwani and Naor [IPL92] conjectured that an online $(1+o(1))Delta$-edge-coloring algorithm exists for $n$-node graphs of maximum degree $Delta=omega(log n)$. This conjecture remains open in general, though it was recently proven for bipartite graphs under emph{one-sided vertex arrivals} by Cohen et al.~[FOCS19]. In a similar vein, we study edge coloring under widely-studied relaxations of the online model. Our main result is in the emph{random-order} online model. For this model, known results fall short of the Bar-Noy et al.~conjecture, either in the degree bound [Aggarwal et al.~FOCS03], or number of colors used [Bahmani et al.~SODA10]. We achieve the best of both worlds, thus resolving the Bar-Noy et al.~conjecture in the affirmative for this model. Our second result is in the adversarial online (and dynamic) model with emph{recourse}. A recent algorithm of Duan et al.~[SODA19] yields a $(1+epsilon)Delta$-edge-coloring with poly$(log n/epsilon)$ recourse. We achieve the same with poly$(1/epsilon)$ recourse, thus removing all dependence on $n$. Underlying our results is one common offline algorithm, which we show how to implement in these two online models. Our algorithm, based on the Rodl Nibble Method, is an adaptation of the distributed algorithm of Dubhashi et al.~[TCS98]. The Nibble Method has proven successful for distributed edge coloring. We display its usefulness in the context of online algorithms.
rate research
Read More
Vizings celebrated theorem asserts that any graph of maximum degree $Delta$ admits an edge coloring using at most $Delta+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses $2Delta-1$ colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with $Delta=O(log n)$, and they conjectured the existence of online algorithms using $Delta(1+o(1))$ colors for $Delta=omega(log n)$. Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS03 and Bahmani et al., SODA10). We resolve the above conjecture for emph{adversarial} vertex arrivals in bipartite graphs, for which we present a $(1+o(1))Delta$-edge-coloring algorithm for $Delta=omega(log n)$ known a priori. Surprisingly, if $Delta$ is not known ahead of time, we show that no $big(frac{e}{e-1} - Omega(1) big) Delta$-edge-coloring algorithm exists. We then provide an optimal, $big(frac{e}{e-1}+o(1)big)Delta$-edge-coloring algorithm for unknown $Delta=omega(log n)$. Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.
We investigate the parameterized complexity of the following edge coloring problem motivated by the problem of channel assignment in wireless networks. For an integer q>1 and a graph G, the goal is to find a coloring of the edges of G with the maximum number of colors such that every vertex of the graph sees at most q colors. This problem is NP-hard for q>1, and has been well-studied from the point of view of approximation. Our main focus is the case when q=2, which is already theoretically intricate and practically relevant. We show fixed-parameter tractable algorithms for both the standard and the dual parameter, and for the latter problem, the result is based on a linear vertex kernel.
We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. We demonstrate the power of our new approach by obtaining better algorithms for sequential, distributed, and parallel models of computation. Our end results include the following randomized algorithms for computing edge connectivity with high probability: -- Two sequential algorithms with complexities $O(m log n)$ and $O(m+n log^3 n)$. These improve on a long line of developments including a celebrated $O(m log^3 n)$ algorithm of Karger [STOC96] and the state of the art $O(m log^2 n (loglog n)^2)$ algorithm of Henzinger et al. [SODA17]. Moreover, our $O(m+n log^3 n)$ algorithm is optimal whenever $m = Omega(n log^3 n)$. Within our new time bounds, whp, we can also construct the cactus representation of all minimal cuts. -- An $~O(n^{0.8} D^{0.2} + n^{0.9})$ round distributed algorithm, where D denotes the graph diameter. This improves substantially on a recent breakthrough of Daga et al. [STOC19], which achieved a round complexity of $~O(n^{1-1/353}D^{1/353} + n^{1-1/706})$, hence providing the first sublinear distributed algorithm for exactly computing the edge connectivity. -- The first $O(1)$ round algorithm for the massively parallel computation setting with linear memory per machine.
Online algorithms make decisions based on past inputs. In general, the decision may depend on the entire history of inputs. If many computers run the same online algorithm with the same input stream but are started at different times, they do not necessarily make consistent decisions. In this work we introduce time-local online algorithms. These are online algorithms where the output at a given time only depends on $T = O(1)$ latest inputs. The use of (deterministic) time-local algorithms in a distributed setting automatically leads to globally consistent decisions. Our key observation is that time-local online algorithms (in which the output at a given time only depends on local inputs in the temporal dimension) are closely connected to local distributed graph algorithms (in which the output of a given node only depends on local inputs in the spatial dimension). This makes it possible to interpret prior work on distributed graph algorithms from the perspective of online algorithms. We describe an algorithm synthesis method that one can use to design optimal time-local online algorithms for small values of $T$. We demonstrate the power of the technique in the context of a variant of the online file migration problem, and show that e.g. for two nodes and unit migration costs there exists a $3$-competitive time-local algorithm with horizon $T=4$, while no deterministic online algorithm (in the classic sense) can do better. We also derive upper and lower bounds for a more general version of the problem; we show that there is a $6$-competitive deterministic time-local algorithm and a $2.62$-competitive randomized time-local algorithm for any migration cost $alpha ge 1$.
We introduce a new model of computation: the online LOCAL model (OLOCAL). In this model, the adversary reveals the nodes of the input graph one by one, in the same way as in classical online algorithms, but for each new node the algorithm can also inspect its radius-$T$ neighborhood before choosing the output; instead of looking ahead in time, we have the power of looking around in space. It is natural to compare OLOCAL with the LOCAL model of distributed computing, in which all nodes make decisions simultaneously in parallel based on their radius-$T$ neighborhoods.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
