We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round "barrier" achieved by Luby's algorithm, but these yield o(log n)-round complexity only when the maximum degree Delta is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby's algorithm) even for moderately small Delta (i.e., for Delta = Omega(log n) and Delta = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby's algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the Theta(log n) time complexity barrier and the Theta(m) message complexity barrier in the Congest model for MIS or closely-related symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A beta-ruling set is an independent set such that every node in the graph is at most beta hops from a node in the independent set. We present the following results: - Time Complexity: We show that we can break the O(log n) "barrier" for 2- and 3-ruling sets. We compute 3-ruling sets in O(log n/log log n) rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O(log Delta (log n)^(1/2 + epsilon) + log n/log log n) rounds for any epsilon > 0, which is o(log n) for a wide range of Delta values (e.g., Delta = 2^(log n)^(1/2-epsilon)). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby's algorithm in the Congest model. - Message Complexity: We show an Omega(n^2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log^2 n) messages and runs in O(Delta log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).
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Tight Bounds for Online Edge Coloring
Vizing’s celebrated theorem asserts that any graph of maximum degree ∆ admits an edge coloring using at most ∆ + 1 colors. In contrast, Bar-Noy, Motwani and Naor showed over a quarter century ago that the trivial greedy algorithm, which uses 2∆−1 colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to lowdegree graphs, with ∆ = O(log n), and they conjectured the existence of online algorithms using ∆(1 + o(1)) colors for ∆ = ω(log n). Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS’03 and Bahmani et al., SODA’10). We resolve the above conjecture for adversarial vertex arrivals in bipartite graphs, for which we present a (1+o(1))∆-edge-coloring algorithm for ∆ = ω(log n) known a priori. Surprisingly, if ∆ is not known ahead of time, we show that no (e/(e−1)−Ω(1))∆-edge-coloring algorithm exists.We then provide an optimal, (e/(e−1) +o(1))∆-edge-coloring algorithm for unknown ∆ = ω(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.
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- NSF-PAR ID:
- 10121531
- Date Published:
- Journal Name:
- IEEE Symposium on Foundations of Computer Science
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We study local symmetry breaking problems in the Congest model, focusing on ruling set problems, which generalize the fundamental Maximal Independent Set (MIS) problem. The time (round) complexity of MIS (and ruling sets) have attracted much attention in the Local model. Indeed, recent results (Barenboim et al., FOCS 2012, Ghaffari SODA 2016) for the MIS problem have tried to break the long-standing O(log n)-round “barrier” achieved by Luby’s algorithm, but these yield o(log n)-round complexity only when the maximum degree is somewhat small relative to n. More importantly, these results apply only in the Local model. In fact, the best known time bound in the Congest model is still O(log n) (via Luby’s algorithm) even for moderately small (i.e., for = (log n) and = o(n)). Furthermore, message complexity has been largely ignored in the context of local symmetry breaking. Luby’s algorithm takes O(m) messages on m-edge graphs and this is the best known bound with respect to messages. Our work is motivated by the following central question: can we break the (log n) time complexity barrier and the (m) message complexity barrier in the Congest model for MIS or closelyrelated symmetry breaking problems? This paper presents progress towards this question for the distributed ruling set problem in the Congest model. A -ruling set is an independent set such that every node in the graph is at most hops from a node in the independent set. We present the following results: Time Complexity: We show that we can break the O(log n) “barrier” for 2- and 3-ruling sets. We compute 3-ruling sets in O log n log log n rounds with high probability (whp). More generally we show that 2-ruling sets can be computed in O log · (log n)1/2+" + log n log log n rounds for any " > 0, which is o(log n) for a wide range of values (e.g., = 2(log n)1/2−" ). These are the first 2- and 3-ruling set algorithms to improve over the O(log n)-round complexity of Luby’s algorithm in the Congest model. Message Complexity: We show an (n2) lower bound on the message complexity of computing an MIS (i.e., 1-ruling set) which holds also for randomized algorithms and present a contrast to this by showing a randomized algorithm for 2-ruling sets that, whp, uses only O(n log2 n) messages and runs in O( log n) rounds. This is the first message-efficient algorithm known for ruling sets, which has message complexity nearly linear in n (which is optimal up to a polylogarithmic factor).more » « less
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Abstract In a Merlin–Arthur proof system, the proof verifier (Arthur) accepts valid proofs (from Merlin) with probability 1, and rejects invalid proofs with probability arbitrarily close to 1. The running time of such a system is defined to be the length of Merlin’s proof plus the running time of Arthur. We provide new Merlin–Arthur proof systems for some key problems in fine-grained complexity. In several cases our proof systems have optimal running time. Our main results include:
Certifying that a list of
n integers has no 3-SUM solution can be done in Merlin–Arthur time$$\tilde{O}(n)$$ $$\tilde{O}(n^{1.5})$$ $$\tilde{O}(n^{1.5})$$ $$\tilde{O}(n^{1.5})$$ Counting the number of
k -cliques with total edge weight equal to zero in ann -node graph can be done in Merlin–Arthur time$${\tilde{O}}(n^{\lceil k/2\rceil })$$ $$k\ge 3$$ k , this bound can be further improved for sparse graphs: for example, counting the number of zero-weight triangles in anm -edge graph can be done in Merlin–Arthur time$${\tilde{O}}(m)$$ k -cliques in unweighted graphs, and had worse running times for smallk .Computing the All-Pairs Shortest Distances matrix for an
n -node graph can be done in Merlin–Arthur time$$\tilde{O}(n^2)$$ $$\Omega (n^2)$$ $$\tilde{O}(n^{2.94})$$ Certifying that an
n -variablek -CNF is unsatisfiable can be done in Merlin–Arthur time$$2^{n/2 - n/O(k)}$$ $$2^{n/2}\cdot \textrm{poly}(n)$$ $$\#$$ k -UNSAT running in$$2^{n/2}/n^{\omega (1)}$$ Certifying a Quantified Boolean Formula is true can be done in Merlin–Arthur time
$$2^{4n/5}\cdot \textrm{poly}(n)$$ $$2^{2n/3}\cdot \textrm{poly}(n)$$ n integers can be done in Merlin–Arthur time$$2^{n/3}\cdot \textrm{poly}(n)$$ $$2^{0.49991n}\cdot \textrm{poly}(n)$$ -
A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an n-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most Δ and aim for colorings using a small number f(Δ) of colors. A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, non-robust, streaming setting, (Δ+1)-colorings can be obtained while using only Õ(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(Δ²) colors and that robust O(Δ)-coloring requires a linear amount of space, namely Ω(nΔ). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust. We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(Δ²)-coloring using Õ(n √Δ) space and an O(Δ³)-coloring using Õ(n) space.more » « less
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