We present a complete classification of the distributed computational complexity of local optimization problems in directed cycles for both the deterministic and the randomized LOCAL model. We show that for any local optimization problem Π (that can be of the form min-sum, max-sum, min-max, or max-min, for any local cost or utility function over some finite alphabet), and for any constant approximation ratio α, the task of finding an α-approximation of Π in directed cycles has one of the following complexities: 1) O(1) rounds in deterministic LOCAL, O(1) rounds in randomized LOCAL, 2) Θ(log^* n) rounds in deterministic LOCAL, O(1) rounds in randomized LOCAL, 3) Θ(log^* n) rounds in deterministic LOCAL, Θ(log^* n) rounds in randomized LOCAL, 4) Θ(n) rounds in deterministic LOCAL, Θ(n) rounds in randomized LOCAL. Moreover, for any given Π and α, we can determine the complexity class automatically, with an efficient (centralized, sequential) meta-algorithm, and we can also efficiently synthesize an asymptotically optimal distributed algorithm. Before this work, similar results were only known for local search problems (e.g., locally checkable labeling problems). The family of local optimization problems is a strict generalization of local search problems, and it contains numerous commonly studied distributed tasks, such as the problems of finding approximations of the maximum independent set, minimum vertex cover, minimum dominating set, and minimum vertex coloring.

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We prove that given any α-approximation LOCAL algorithm for Minimum Dominating Set (MDS) on planar graphs, we can construct an f(g)-round (3α + 1)-approximation LOCAL algorithm for MDS on graphs embeddable in a given Euler genus-g surface. Heydt et al. [European Journal of Combinatorics (2025)] gave an algorithm with α = 11 + ϵ, from which we derive a (34 + ϵ)-approximation algorithm for graphs of genus g, therefore improving upon the current state of the art of 24g + O(1) due to Amiri et al. [ACM Transactions on Algorithms (2019)]. It also improves the approximation ratio of 91 + ϵ due to Czygrinow et al. [Theoretical Computer Science (2019)] in the particular case of orientable surfaces.

Common definitions of the “standard” LOCAL model tend to be sloppy and even self-contradictory on one point: do the nodes update their state using an arbitrary function or a computable function? So far, this distinction has been safe to neglect, since problems where it matters seem contrived and quite different from e.g. typical local graph problems studied in this context.

Locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer, are the standard formalism for studying local distributed graph problems. They capture many natural problems, such as coloring, maximal independent set, and sinkless orientation, while still being restrictive enough to enable general complexity-theoretic results. However, recent work has also revealed artificial LCLs with counterintuitive behavior, including quantum and shared-randomness advantages, exotic round complexities, dependence on computability assumptions, and undecidability phenomena. This raises a natural question: are these phenomena artifacts of the particular Naor-Stockmeyer definition, or are they inherent to local checkability?

We show that there is a one-round randomized distributed algorithm that can 2-color cycles such that the expected fraction of monochromatic edges is less than 0.24118. We also show that a one-round algorithm cannot achieve a fraction less than 0.23879. Before this work, the best upper and lower bounds were 0.25 and 0.2. Our proof was largely discovered and developed by large language models, and both the upper and lower bounds have been formalized in Lean 4.

Recent work on distributed graph algorithms [e.g. STOC 2022, ITCS 2022, PODC 2020] has drawn attention to the following open question: are round elimination fixed points a universal technique for proving lower bounds? That is, given a locally checkable problem \(\Pi\) that requires at least \(\Omega(\log n)\) rounds in the deterministic LOCAL model, can we always find a relaxation \(\Pi'\) of \(\Pi\) that is a nontrivial fixed point for the round elimination technique [see STOC 2016, PODC 2019]? If yes, then a key part of distributed computational complexity would be also decidable.

By prior work, we have many results related to distributed graph algorithms for problems that can be defined with local constraints; the formal framework used in prior work is locally checkable labeling problems (LCLs), introduced by Naor and Stockmeyer in the 1990s. It is known, for example, that if we have a deterministic algorithm that solves an LCL in $o(\log n)$ rounds, we can speed it up to $O(\log^*n)$ rounds, and if we have a randomized $O(\log^*n)$ rounds algorithm, we can derandomize it for free. It is also known that randomness helps with some LCL problems: there are LCL problems with randomized complexity $Θ(\log\log n)$ and deterministic complexity $Θ(\log n)$. However, so far there have not been any LCL problems in which the use of shared randomness has been necessary; in all prior algorithms it has been enough that the nodes have access to their own private sources of randomness. Could it be the case that shared randomness never helps with LCLs? Could we have a general technique that takes any distributed graph algorithm for any LCL that uses shared randomness, and turns it into an equally fast algorithm where private randomness is enough? In this work we show that the answer is no. We present an LCL problem $Π$ such that the round complexity of $Π$ is $Ω(\sqrt n)$ in the usual randomized \local model with private randomness, but if the nodes have access to a source of shared randomness, then the complexity drops to $O(\log n)$. As corollaries, we also resolve several other open questions related to the landscape of distributed computing in the context of LCL problems. In particular, problem $Π$ demonstrates that distributed quantum algorithms for LCL problems strictly benefit from a shared quantum state. Problem $Π$ also gives a separation between finitely dependent distributions and non-signaling distributions.

We connect three distinct lines of research that have recently explored extensions of the classical LOCAL model of distributed computing: A. distributed quantum computing and non-signaling distributions [e.g. STOC 2024], B. finitely-dependent processes [e.g. Forum Math. Pi 2016], and C. locality in online graph algorithms and dynamic graph algorithms [e.g. ICALP 2023]. We prove new results on the capabilities and limitations of all of these models of computing, for locally checkable labeling problems (LCLs). We show that all these settings can be sandwiched between the classical LOCAL model and what we call the randomized online-LOCAL model. Our work implies limitations on the quantum advantage in the distributed setting, and we also exhibit a new barrier for proving tighter bounds. Our main technical results are these: 1. All LCL problems solvable with locality $O(\log^\star n)$ in the classical deterministic LOCAL model admit a finitely-dependent distribution with locality $O(1)$. This answers an open question by Holroyd [2024], and also presents a new barrier for proving bounds on distributed quantum advantage using causality-based arguments. 2. In rooted trees, if we can solve an LCL problem with locality $o(\log \log \log n)$ in the randomized online-LOCAL model (or any of the weaker models, such as quantum-LOCAL), we can solve it with locality $O(\log^\star n)$ in the classical deterministic LOCAL model. One of many implications is that in rooted trees, $O(\log^\star n)$ locality in quantum-LOCAL is not stronger than $O(\log^\star n)$ locality in classical LOCAL.

We present the first local problem that shows a super-constant separation between the classical randomized LOCAL model of distributed computing and its quantum counterpart. By prior work, such a separation was known only for an artificial graph problem with an inherently global definition [Le Gall et al. 2019]. We present a problem that we call iterated GHZ, which is defined using only local constraints. Formally, it is a family of locally checkable labeling problems [Naor and Stockmeyer 1995]; in particular, solutions can be verified with a constant-round distributed algorithm. We show that in graphs of maximum degree $Δ$, any classical (deterministic or randomized) LOCAL model algorithm will require $Ω(Δ)$ rounds to solve the iterated GHZ problem, while the problem can be solved in $1$ round in quantum-LOCAL. We use the round elimination technique to prove that the iterated GHZ problem requires $Ω(Δ)$ rounds for classical algorithms. This is the first work that shows that round elimination is indeed able to separate the two models, and this also demonstrates that round elimination cannot be used to prove lower bounds for quantum-LOCAL. To apply round elimination, we introduce a new technique that allows us to discover appropriate problem relaxations in a mechanical way; it turns out that this new technique extends beyond the scope of the iterated GHZ problem and can be used to e.g. reproduce prior results on maximal matchings [FOCS 2019, PODC 2020] in a systematic manner.

Algorithms with advice have received ample attention in the distributed and online settings, and they have recently proven useful also in dynamic settings. In this work we study local computation with advice: the goal is to solve a graph problem Π with a distributed algorithm in T(Δ) communication rounds, for some function T that only depends on the maximum degree Δ of the graph, and the key question is how many bits of advice per node are needed. Some of our results regard Locally Checkable Labeling problems (LCLs), which is an important family of problems that includes various coloring and orientation problems on finite-degree graphs. These are constraint-satisfaction graph problems that can be defined with a finite set of valid input/output-labeled neighborhoods. Our main results are: 1) Any locally checkable labeling problem can be solved with only 1 bit of advice per node in graphs with sub-exponential growth (the number of nodes within radius r is sub-exponential in r; for example, grids are such graphs). Moreover, we can make the set of nodes that carry advice bits arbitrarily sparse. As a corollary, any locally checkable labeling problem admits a locally checkable proof with 1 bit per node in graphs with sub-exponential growth. 2) The assumption of sub-exponential growth is complemented by a conditional lower bound: assuming the Exponential-Time Hypothesis, there are locally checkable labeling problems that cannot be solved in general with any constant number of bits per node. 3) In any graph we can find an almost-balanced orientation (indegrees and outdegrees differ by at most one) with 1 bit of advice per node, and again we can make the advice arbitrarily sparse. As a corollary, we can also compress an arbitrary subset of edges so that a node of degree d stores only d/2 + 2 bits, and we can decompress it locally, in T(Δ) rounds. 4) In any graph of maximum degree Δ, we can find a Δ-coloring (if it exists) with 1 bit of advice per node, and again, we can make the advice arbitrarily sparse. 5) In any 3-colorable graph, we can find a 3-coloring with 1 bit of advice per node. As a corollary, in bounded-degree graphs there is a locally checkable proof that certifies 3-colorability with 1 bit of advice per node, while prior work shows that this is not possible with a proof labeling scheme (PLS), which is a more restricted setting where the verifier can only see up to distance 1. Our work shows that for many problems the key threshold is not whether we can achieve 1 bit of advice per node, but whether we can make the advice arbitrarily sparse. To formalize this idea, we develop a general framework of composable schemas that enables us to build algorithms for local computation with advice in a modular fashion: once we have (1) a schema for solving Π₁ and (2) a schema for solving Π₂ assuming an oracle for Π₁, we can also compose them and obtain (3) a schema that solves Π₂ without the oracle. It turns out that many natural problems admit composable schemas, all of them can be solved with only 1 bit of advice, and we can make the advice arbitrarily sparse.

In this work, we give two results that put new limits on distributed quantum advantage in the context of the LOCAL model of distributed computing. First, we show that there is no distributed quantum advantage for any linear program. Put otherwise, if there is a quantum-LOCAL algorithm $\mathcal{A}$ that finds an $α$-approximation of some linear optimization problem $Π$ in $T$ communication rounds, we can construct a classical, deterministic LOCAL algorithm $\mathcal{A}'$ that finds an $α$-approximation of $Π$ in $T$ rounds. As a corollary, all classical lower bounds for linear programs, including the KMW bound, hold verbatim in quantum-LOCAL. Second, using the above result, we show that there exists a locally checkable labeling problem (LCL) for which quantum-LOCAL is strictly weaker than the classical deterministic SLOCAL model. Our results extend from quantum-LOCAL also to finitely dependent and non-signaling distributions, and one of the corollaries of our work is that the non-signaling model and the SLOCAL model are incomparable in the context of LCL problems: By prior work, there exists an LCL problem for which SLOCAL is strictly weaker than the non-signaling model, and our work provides a separation in the opposite direction.

The randomized online-LOCAL model captures a number of models of computing; it is at least as strong as all of these models: - the classical LOCAL model of distributed graph algorithms, - the quantum version of the LOCAL model, - finitely dependent distributions [e.g. Holroyd 2016], - any model that does not violate physical causality [Gavoille, Kosowski, Markiewicz, DISC 2009], - the SLOCAL model [Ghaffari, Kuhn, Maus, STOC 2017], and - the dynamic-LOCAL and online-LOCAL models [Akbari et al., ICALP 2023]. In general, the online-LOCAL model can be much stronger than the LOCAL model. For example, there are locally checkable labeling problems (LCLs) that can be solved with logarithmic locality in the online-LOCAL model but that require polynomial locality in the LOCAL model. However, in this work we show that in trees, many classes of LCL problems have the same locality in deterministic LOCAL and randomized online-LOCAL (and as a corollary across all the above-mentioned models). In particular, these classes of problems do not admit any distributed quantum advantage. We present a near-complete classification for the case of rooted regular trees. We also fully classify the super-logarithmic region in unrooted regular trees. Finally, we show that in general trees (rooted or unrooted, possibly irregular, possibly with input labels) problems that are global in deterministic LOCAL remain global also in the randomized online-LOCAL model.

In this work we study local computation with advice: the goal is to solve a graph problem Π with a distributed algorithm in f (Δ) communication rounds, for some function f that only depends on the maximum degree Δ of the graph, and the key question is how many bits of advice per node are needed. Our main results are:

In prior work, Gupta et al. (SPAA 2022) presented a distributed algorithm for multiplying sparse n x n matrices, using n computers. They assumed that the input matrices are uniformly sparse---there are at most d non-zeros in each row and column---and the task is to compute a uniformly sparse part of the product matrix. Initially each computer knows one row of each input matrix, and eventually each computer needs to know one row of the product matrix. In each communication round each computer can send and receive one O(łog n)-bit message. Their algorithm solves this task in O(d^1.907 ) rounds, while the trivial bound is O(d^2).

We give an almost complete characterization of the hardness of c-coloring χ-chromatic graphs with distributed algorithms, for a wide range of models of distributed computing. In particular, we show that these problems do not admit any distributed quantum advantage. To do that:

By prior work, it is known that any distributed graph algorithm that finds a maximal matching requires Ω(log⁎⁡n) communication rounds, while it is possible to find a maximal fractional matching in O(1) rounds in bounded-degree graphs. However, all prior O(1)-round algorithms for maximal fractional matching use arbitrarily fine-grained fractional values. In particular, none of them is able to find a half-integral solution, using only values from {0,12,1}. We show that the use of fine-grained fractional values is necessary, and moreover we give a complete characterization on exactly how small values are needed: if we consider maximal fractional matching in graphs of maximum degree Δ=2d, and any distributed graph algorithm with round complexity T(Δ) that only depends on Δ and is independent of n, we show that the algorithm has to use fractional values with a denominator at least 2d. We give a new algorithm that shows that this is also sufficient.

In this work, we give a unifying view of locality in four settings: distributed algorithms, sequential greedy algorithms, dynamic algorithms, and online algorithms. We introduce a new model of computing, called the online-LOCAL 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 we get to see its radius-T neighborhood before choosing the output. We compare the online-LOCAL model with three other models: the LOCAL model of distributed computing, where each node produces its output based on its radius-T neighborhood, its sequential counterpart SLOCAL, and the dynamic-LOCAL model, where changes in the dynamic input graph only influence the radius-T neighborhood of the point of change. The SLOCAL and dynamic-LOCAL models are sandwiched between the LOCAL and online-LOCAL models, with LOCAL being the weakest and online-LOCAL the strongest model. In general, all models are distinct, but we study in particular locally checkable labeling problems (LCLs), which is a family of graph problems studied in the context of distributed graph algorithms. We prove that for LCL problems in paths, cycles, and rooted trees, all models are roughly equivalent: the locality of any LCL problem falls in the same broad class - $O(\log^* n)$, $Θ(\log n)$, or $n^{Θ(1)}$ - in all four models. In particular, this result enables one to generalize prior lower-bound results from the LOCAL model to all four models, and it also allows one to simulate e.g. dynamic-LOCAL algorithms efficiently in the LOCAL model. We also show that this equivalence does not hold in general bipartite graphs. We provide an online-LOCAL algorithm with locality $O(\log n)$ for the $3$-coloring problem in bipartite graphs - this is a problem with locality $Ω(n^{1/2})$ in the LOCAL model and $Ω(n^{1/10})$ in the SLOCAL model.

We present a deterministic algorithm for solving a wide range of dynamic programming problems in trees in O(log D) rounds in the massively parallel computation model (MPC), with O(nδ) words of local memory per machine, for any given constant 0 < δ < 1. Here D is the diameter of the tree and n is the number of nodes---we emphasize that our running time is independent of n.

One of the cornerstones of the distributed complexity theory is the derandomization result by Chang, Kopelowitz, and Pettie [FOCS 2016]: any randomized LOCAL algorithm that solves a locally checkable labeling problem (LCL) can be derandomized with at most exponential overhead. The original proof assumes that the number of random bits is bounded by some function of the input size. We give a new, simple proof that does not make any such assumptions-it holds even if the randomized algorithm uses infinitely many bits. While at it, we also broaden the scope of the result so that it is directly applicable far beyond LCL problems.

Abstract Consider any locally checkable labeling problem $$\Pi $$ Π in rooted regular trees: there is a finite set of labels $$\Sigma $$ Σ , and for each label $$x \in \Sigma $$ x ∈ Σ we specify what are permitted label combinations of the children for an internal node of label x (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set. We show that the distributed computational complexity of any such problem $$\Pi $$ Π falls in one of the following classes: it is O(1), $$\Theta (\log ^* n)$$ Θ ( log ∗ n ) , $$\Theta (\log n)$$ Θ ( log n ) , or $$n^{\Theta (1)}$$ n Θ ( 1 ) rounds in trees with n nodes (and all of these classes are nonempty). We show that the complexity of any given problem is the same in all four standard models of distributed graph algorithms: deterministic $$\mathsf {LOCAL}$$ LOCAL , randomized $$\mathsf {LOCAL}$$ LOCAL , deterministic $$\mathsf {CONGEST}$$ CONGEST , and randomized $$\mathsf {CONGEST}$$ CONGEST model. In particular, we show that randomness does not help in this setting, and the complexity class $$\Theta (\log \log n)$$ Θ ( log log n ) does not exist (while it does exist in the broader setting of general trees). We also show how to systematically determine the complexity class of any such problem $$\Pi $$ Π , i.e., whether $$\Pi $$ Π takes O(1), $$\Theta (\log ^* n)$$ Θ ( log ∗ n ) , $$\Theta (\log n)$$ Θ ( log n ) , or $$n^{\Theta (1)}$$ n Θ ( 1 ) rounds. While the algorithm may take exponential time in the size of the description of $$\Pi $$ Π , it is nevertheless practical: we provide a freely available implementation of the classifier algorithm, and it is fast enough to classify many problems of interest.

The locality of a graph problem is the smallest distance T such that each node can choose its own part of the solution based on its radius-T neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality. In this work we seek to automate the study of solvability and locality: given the description of a graph problem Π, we would like to determine if Π is solvable and what is the asymptotic locality of Π as a function of the size of the graph. Put otherwise, we seek to automatically synthesize efficient distributed and parallel algorithms for solving Π. We focus on locally checkable graph problems; these are problems in which a solution is globally feasible if it looks feasible in all constant-radius neighborhoods. Prior work on such problems has brought primarily bad news: questions related to locality are undecidable in general, and even if we focus on the case of labeled paths and cycles, determining locality is PSPACE-hard (Balliu et al., PODC 2019). We complement prior negative results with efficient algorithms for the cases of unlabeled paths and cycles and, as an extension, for rooted trees. We study locally checkable graph problems from an automata-theoretic perspective by representing a locally checkable problem Π as a nondeterministic finite automaton M over a unary alphabet. We identify polynomial-time-computable properties of the automaton M that near-completely capture the solvability and locality of Π in cycles and paths, with the exception of one specific case that is co-NP-complete.

In this paper, we study the notion of mending: given a partial solution to a graph problem, how much effort is needed to take one step towards a proper solution? For example, if we have a partial coloring of a graph, how hard is it to properly color one more node? In prior work (SIROCCO 2022), this question was formalized and studied from the perspective of mending radius: if there is a hole that we need to patch, how far do we need to modify the solution? In this work, we investigate a complementary notion of mending volume: how many nodes need to be modified to patch a hole? We focus on the case of locally checkable labeling problems (LCLs) in trees, and show that already in this setting there are two infinite hierarchies of problems: for infinitely many values 0 < α ≤ 1, there is an LCL problem with mending volume Θ(n^α), and for infinitely many values k ≥ 1, there is an LCL problem with mending volume Θ(log^k n). Hence the mendability of LCL problems on trees is a much more fine-grained question than what one would expect based on the mending radius alone.

We study matrix multiplication in the low-bandwidth model: There are n computers, and we need to compute the product of two n × n matrices. Initially computer i knows row i of each input matrix. In one communication round each computer can send and receive one O(logn)-bit message. Eventually computer i has to output row i of the product matrix.

We give practical, efficient algorithms that automatically determine the asymptotic distributed round complexity of a given locally checkable graph problem in the $[Θ(\log n), Θ(n)]$ region, in two settings. We present one algorithm for unrooted regular trees and another algorithm for rooted regular trees. The algorithms take the description of a locally checkable labeling problem as input, and the running time is polynomial in the size of the problem description. The algorithms decide if the problem is solvable in $O(\log n)$ rounds. If not, it is known that the complexity has to be $Θ(n^{1/k})$ for some $k = 1, 2, \dotsc$, and in this case the algorithms also output the right value of the exponent $k$. In rooted trees in the $O(\log n)$ case we can then further determine the exact complexity class by using algorithms from prior work; for unrooted trees the more fine-grained classification in the $O(\log n)$ region remains an open question.

Online algorithms make decisions based on past inputs, with the goal of being competitive against an algorithm that sees also future inputs. In this work, we introduce time-local online algorithms; these are online algorithms in which the output at any given time is a function of only T latest inputs. Our main observation is that time-local online algorithms are closely connected to local distributed graph algorithms: distributed algorithms make decisions based on the local information in the spatial dimension, while time-local online algorithms make decisions based on the local information in the temporal dimension. We formalize this connection, and show how we can directly use the tools developed to study distributed approximability of graph optimization problems to prove upper and lower bounds on the competitive ratio achieved with time-local online algorithms. Moreover, we show how to use computational techniques to synthesize optimal time-local algorithms.

Consider any locally checkable labeling problem Π in rooted regular trees: there is a finite set of labels Σ, and for each label χ x Σ we specify what are permitted label combinations of the children for an internal node of label x (the leaf nodes are unconstrained). This formalism is expressive enough to capture many classic problems studied in distributed computing, including vertex coloring, edge coloring, and maximal independent set.

We introduce a new graph problem, the token dropping game, and we show how to solve it efficiently in a distributed setting. We use the token dropping game as a tool to design an efficient distributed algorithm for stable orientations and more generally for locally optimal semi-matchings. The prior work by Czygrinow et al. (DISC 2012) finds a stable orientation in O(Δ^5) rounds in graphs of maximum degree Δ, while we improve it to O(Δ^4) and also prove a lower bound of Ω(Δ). For the more general problem of locally optimal semi-matchings, the prior upper bound is O(S^5) and our new algorithm runs in O(C · S^4) rounds, which is an improvement for C = o(S); here C and S are the maximum degrees of customers and servers, respectively.

A rich line of work has been addressing the computational complexity of locally checkable labelings (LCLs), illustrating the landscape of possible complexities. In this paper, we study the landscape of LCL complexities under bandwidth restrictions. Our main results are twofold. First, we show that on trees, the CONGEST complexity of an LCL problem is asymptotically equal to its complexity in the LOCAL model. An analog statement for non-LCL problems is known to be false. Second, we show that for general graphs this equivalence does not hold, by providing an LCL problem for which we show that it can be solved in O(log n) rounds in the LOCAL model, but requires Ω̃(n^{1/2}) rounds in the CONGEST model.

We show that any algorithm that solves the sinkless orientation problem in the supported LOCAL model requires Ω(log n) rounds, and this is tight. The supported LOCAL is at least as strong as the usual LOCAL model, and as a corollary this also gives a new, short and elementary proof that shows that the round complexity of the sinkless orientation problem in the deterministic LOCAL model is Ω(log n).

Abstract The landscape of the distributed time complexity is nowadays well-understood for subpolynomial complexities. When we look at deterministic algorithms in the $$\mathsf {LOCAL}$$ LOCAL model and locally checkable problems ( $$\mathsf {LCL}$$ LCL s) in bounded-degree graphs, the following picture emerges: There are lots of problems with time complexities of $$\varTheta (\log ^* n)$$ Θ ( log ∗ n ) or $$\varTheta (\log n)$$ Θ ( log n ) . It is not possible to have a problem with complexity between $$\omega (\log ^* n)$$ ω ( log ∗ n ) and $$o(\log n)$$ o ( log n ) . In general graphs, we can construct $$\mathsf {LCL}$$ LCL problems with infinitely many complexities between $$\omega (\log n)$$ ω ( log n ) and $$n^{o(1)}$$ n o ( 1 ) . In trees, problems with such complexities do not exist. However, the high end of the complexity spectrum was left open by prior work. In general graphs there are $$\mathsf {LCL}$$ LCL problems with complexities of the form $$\varTheta (n^\alpha )$$ Θ ( n α ) for any rational $$0 < \alpha \le 1/2$$ 0 < α ≤ 1 / 2 , while for trees only complexities of the form $$\varTheta (n^{1/k})$$ Θ ( n 1 / k ) are known. No $$\mathsf {LCL}$$ LCL problem with complexity between $$\omega (\sqrt{n})$$ ω ( n ) and o(n) is known, and neither are there results that would show that such problems do not exist. We show that: In general graphs, we can construct $$\mathsf {LCL}$$ LCL problems with infinitely many complexities between $$\omega (\sqrt{n})$$ ω ( n ) and o(n). In trees, problems with such complexities do not exist. Put otherwise, we show that any $$\mathsf {LCL}$$ LCL

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O ( Δ + log * n ) communication rounds; here, n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: These problems cannot be solved in o (log * n ) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that any algorithm that finds a maximal matching or maximal independent set with probability at least 1-1/ n requires Ω (min { Δ , log log n / log log log n }) rounds in the LOCAL model of distributed computing. As a corollary, it follows that any deterministic algorithm that finds a maximal matching or maximal independent set requires Ω (min { Δ , log n / log log n }) rounds; this is an improvement over prior lower bounds also as a function of n .

Computer scientists seek to understand what can be automated, but what do we know about automating our own work? Can we outsource our own research questions to computers? In this talk I will discuss this question from the perspective of the theory of distributed computing. I will present not only recent examples of human-computer-collaborations that have resulted in major breakthroughs in our understanding of distributed computing, but I will also explore the fundamental limits of such approaches.

We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees in the usual LOCAL model of distributed computing. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2-vertex coloring, and perfect matching. We show that the complexity of any such problem is in one of the following classes: O(1), Θ(log n), Θ(n), or unsolvable. Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it.

Locally checkable labeling problems (LCLs) are distributed graph problems in which a solution is globally feasible if it is locally feasible in all constant-radius neighborhoods. Vertex colorings, maximal independent sets, and maximal matchings are examples of LCLs.

Assume we have a graph problem that is locally checkable but not locally solvable---given a solution we can check that it is feasible by verifying all constant-radius neighborhoods, but to find a feasible solution each node needs to explore the input graph at least up to distance Ω (log n) in order to produce its own part of the solution.

We present a complete classification of the deterministic distributed time complexity for a family of graph problems: binary labeling problems in trees. These are locally checkable problems that can be encoded with an alphabet of size two in the edge labeling formalism. Examples of binary labeling problems include sinkless orientation, sinkless and sourceless orientation, 2-vertex coloring, perfect matching, and the task of coloring edges red and blue such that all nodes are incident to at least one red and at least one blue edge. More generally, we can encode e.g. any cardinality constraints on indegrees and outdegrees. We study the deterministic time complexity of solving a given binary labeling problem in trees, in the usual LOCAL model of distributed computing. We show that the complexity of any such problem is in one of the following classes: $O(1)$, $Θ(\log n)$, $Θ(n)$, or unsolvable. In particular, a problem that can be represented in the binary labeling formalism cannot have time complexity $Θ(\log^* n)$, and hence we know that e.g. any encoding of maximal matchings has to use at least three labels (which is tight). Furthermore, given the description of any binary labeling problem, we can easily determine in which of the four classes it is and what is an asymptotically optimal algorithm for solving it. Hence the distributed time complexity of binary labeling problems is decidable, not only in principle, but also in practice: there is a simple and efficient algorithm that takes the description of a binary labeling problem and outputs its distributed time complexity.

The locality of a graph problem is the smallest distance $T$ such that each node can choose its own part of the solution based on its radius-$T$ neighborhood. In many settings, a graph problem can be solved efficiently with a distributed or parallel algorithm if and only if it has a small locality. In this work we seek to automate the study of solvability and locality: given the description of a graph problem $Π$, we would like to determine if $Π$ is solvable and what is the asymptotic locality of $Π$ as a function of the size of the graph. Put otherwise, we seek to automatically synthesize efficient distributed and parallel algorithms for solving $Π$. We focus on locally checkable graph problems; these are problems in which a solution is globally feasible if it looks feasible in all constant-radius neighborhoods. Prior work on such problems has brought primarily bad news: questions related to locality are undecidable in general, and even if we focus on the case of labeled paths and cycles, determining locality is $\mathsf{PSPACE}$-hard (Balliu et al., PODC 2019). We complement prior negative results with efficient algorithms for the cases of unlabeled paths and cycles and, as an extension, for rooted trees. We introduce a new automata-theoretic perspective for studying locally checkable graph problems. We represent a locally checkable problem $Π$ as a nondeterministic finite automaton $\mathcal{M}$ over a unary alphabet. We identify polynomial-time-computable properties of the automaton $\mathcal{M}$ that near-completely capture the solvability and locality of $Π$ in cycles and paths, with the exception of one specific case that is $\mbox{co-$\mathsf{NP}$}$-complete.

There are distributed graph algorithms for finding maximal matchings and maximal independent sets in O(Δ + log^* n) communication rounds; here n is the number of nodes and Δ is the maximum degree. The lower bound by Linial (1987, 1992) shows that the dependency on n is optimal: these problems cannot be solved in o(log^* n) rounds even if Δ = 2. However, the dependency on Δ is a long-standing open question, and there is currently an exponential gap between the upper and lower bounds. We prove that the upper bounds are tight. We show that maximal matchings and maximal independent sets cannot be found in o(Δ + log log n / log log log n) rounds with any randomized algorithm in the LOCAL model of distributed computing. As a corollary, it follows that there is no deterministic algorithm for maximal matchings or maximal independent sets that runs in o(Δ + log n / log log n) rounds; this is an improvement over prior lower bounds also as a function of n.

As communication networks are growing at a fast pace, the need for more scalable approaches to operate such networks is pressing. Decentralization and locality are key concepts to provide scalability. Existing models for which local algorithms are designed fail to model an important aspect of many modern communication networks such as software-defined networks: the possibility to precompute distributed network state. We take this as an opportunity to study the fundamental question of how and to what extent local algorithms can benefit from preprocessing. In particular, we show that preprocessing allows for significant speedups of various networking problems. A main benefit is the precomputation of structural primitives, where purely distributed algorithms have to start from scratch. Maybe surprisingly, we also show that there are strict limitations on how much preprocessing can help in different scenarios. To this end, we provide approximation bounds for the maximum independent set problem-which however show that our obtained speedups are asymptotically optimal. Even though we show that physical link failures in general hinder the power of preprocessing, we can still facilitate the precomputation of symmetry breaking processes to bypass various runtime barriers. We believe that our model and results are of interest beyond the scope of this paper and apply to other dynamic networks as well.

The committee decided to award the 2019 Edsger W. Dijkstra Prize in Distributed Computing to Alessandro Panconesi and Aravind Srinivasan for their paper Randomized Distributed Edge Coloring via an Extension of the Chernoff-Hoeffding Bounds, SIAM Journal on Computing, volume 26, number 2, 1997, pages 350-368. A preliminary version of this paper appeared as Fast Randomized Algorithms for Distributed Edge Coloring, Proceedings of the Eleventh Annual ACM Symposium Principles of Distributed Computing (PODC), 1992, pages 251-262.

Consider a computer network that consists of a path with n nodes. The nodes are labeled with inputs from a constant-sized set, and the task is to find output labels from a constant-sized set subject to some local constraints---more formally, we have an LCL (locally checkable labeling) problem. How many communication rounds are needed (in the standard LOCAL model of computing) to solve this problem?

A graph is weakly 2-colored if the nodes are labeled with colors black and white such that each black node is adjacent to at least one white node and vice versa. In this work we study the distributed computational complexity of weak 2-coloring in the standard łocal model of distributed computing, and how it is related to the distributed computational complexity of other graph problems.

We study a combinatorial optimization problem that is motivated by the scenario of autonomous cars driving on a multi-lane highway: some cars need to change lanes before the next intersection, and if there is congestion, cars need to slow down to make space for those who are changing lanes. There are two natural objective functions to minimize: (1) how long does it take for all traffic to clear the road, and (2) the total number of maneuvers. In this work, we present an approximation algorithm for solving these problems in the two-lane case and a hardness result for the multi-lane case.

The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now known that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use various more traditional complexity theory tools to build a clearer picture of the complexity landscape of the congested clique: \beginitemize ıtem Nondeterminism and beyond: We introduce the nondeterministic congested clique model (analogous to NP) and show that there is a natural canonical problem family that captures all problems solvable in constant time with nondeterministic algorithms. We further generalise these notions by introducing the constant-round decision hierarchy (analogous to the polynomial hierarchy). ıtem Non-constructive lower bounds: We lift the prior non-uniform counting arguments to a general technique for proving non-constructive uniform lower bounds for the congested clique. In particular, we prove a time hierarchy theorem for the congested clique, showing that there are decision problems of essentially all complexities, both in the deterministic and nondeterministic settings. ıtem Fine-grained complexity: We map out relationships between various natural problems in the congested clique model, arguing that a reduction-based complexity theory currently gives us a fairly good picture of the complexity landscape of the congested clique. \enditemize

A number of recent papers – e.g. Brandt et al. (STOC 2016), Chang et al. (FOCS 2016), Ghaffari & Su (SODA 2017), Brandt et al. (PODC 2017), and Chang & Pettie (FOCS 2017) – have advanced our understanding of one of the most fundamental questions in theory of distributed computing: what are the possible time complexity classes of LCL problems in the LOCAL model? In essence, we have a graph problem Π in which a solution can be verified by checking all radius-O(1) neighbourhoods, and the question is what is the smallest T such that a solution can be computed so that each node chooses its own output based on its radius-T neighbourhood. Here T is the distributed time complexity of Π. The time complexity classes for deterministic algorithms in bounded-degree graphs that are known to exist by prior work are Θ(1), Θ(log* n), Θ(logn), Θ(n1/k), and Θ(n). It is also known that there are two gaps: one between ω(1) and o(loglog* n), and another between ω(log* n) and o(logn). It has been conjectured that many more gaps exist, and that the overall time hierarchy is relatively simple – indeed, this is known to be the case in restricted graph families such as cycles and grids. We show that the picture is much more diverse than previously expected. We present a general technique for engineering LCL problems with numerous different deterministic time complexities, including Θ(logα n) for any α ≥ 1, 2Θ(logα n) for any α ≤ 1, and Θ(nα) for any α < 1/2 in the high end of the complexity spectrum, and Θ(logα log* n) for any α ≥ 1, 2Θ(logα log* n) for any α ≤ 1, and Θ((log* n)α) for any α ≤ 1 in the low end of the complexity spectrum; here α is a positive rational number.

Given two colorings of a graph, we consider the following problem: can we recolor the graph from one coloring to the other through a series of elementary changes, such that the graph is properly colored after each step? We introduce the notion of distributed recoloring: The input graph represents a network of computers that needs to be recolored. Initially, each node is aware of its own input color and target color. The nodes can exchange messages with each other, and eventually each node has to stop and output its own recoloring schedule, indicating when and how the node changes its color. The recoloring schedules have to be globally consistent so that the graph remains properly colored at each point, and we require that adjacent nodes do not change their colors simultaneously. We are interested in the following questions: How many communication rounds are needed (in the deterministic LOCAL model of distributed computing) to find a recoloring schedule? What is the length of the recoloring schedule? And how does the picture change if we can use extra colors to make recoloring easier? The main contributions of this work are related to distributed recoloring with one extra color in the following graph classes: trees, 3-regular graphs, and toroidal grids.

While the relationship of time and space is an established topic in traditional centralised com- plexity theory, this is not the case in distributed computing. We aim to remedy this by studying the time and space complexity of algorithms in a weak message-passing model of distributed com- puting. While a constant number of communication rounds implies a constant number of states visited during the execution, the other direction is not clear at all. We show that indeed, there exist non-trivial graph problems that are solvable by constant-space algorithms but that require a non-constant running time. Somewhat surprisingly, this holds even when restricted to the class of only cycle and path graphs. Our work provides us with a new complexity class for distributed computing and raises interesting questions about the existence of further combinations of time and space complexity.

LCLs or locally checkable labelling problems (e.g. maximal independent set, maximal matching, and vertex colouring) in the LOCAL model of computation are very well-understood in cycles (toroidal 1-dimensional grids): every problem has a complexity of O(1), Θ(log* n), or Θ(n), and the design of optimal algorithms can be fully automated. This work develops the complexity theory of LCL problems for toroidal 2-dimensional grids. The complexity classes are the same as in the 1-dimensional case: O(1), Θ(log* n), and Θ(n). However, given an LCL problem it is undecidable whether its complexity is Θ(log* n) or Θ(n) in 2-dimensional grids.

The degree splitting problem requires coloring the edges of a graph red or blue such that each node has almost the same number of edges in each color, up to a small additive discrepancy. The directed variant of the problem requires orienting the edges such that each node has almost the same number of incoming and outgoing edges, again up to a small additive discrepancy. We present deterministic distributed algorithms for both variants, which improve on their counterparts presented by Ghaffari and Su [SODA'17]: our algorithms are significantly simpler and faster, and have a much smaller discrepancy. This also leads to a faster and simpler deterministic algorithm for (2+o(1))Delta-edge-coloring, improving on that of Ghaffari and Su.

The congested clique model of distributed computing has been receiving attention as a model for densely connected distributed systems. While there has been significant progress on the side of upper bounds, we have very little in terms of lower bounds for the congested clique; indeed, it is now know that proving explicit congested clique lower bounds is as difficult as proving circuit lower bounds. In this work, we use traditional complexity-theoretic tools to build a clearer picture of the complexity landscape of the congested clique, proving non-constructive lower bounds and studying the relationships between natural problems.

Consider a complete communication network of $n$ nodes, where the nodes receive a common clock pulse. We study the synchronous $c$-counting problem: given any starting state and up to $f$ faulty nodes with arbitrary behavior, the task is to eventually have all correct nodes labeling the pulses with increasing values modulo $c$ in agreement. Thus, we are considering algorithms that are self-stabilizing despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have optimal resilience, (3) have a linear stabilization time in $f$ (asymptotically optimal), (4) use a small number of states, and, consequently, (5) communicate a small number of bits per round. Prior algorithms either resort to randomization, use a large number of states and need high communication bandwidth, or have suboptimal resilience. In particular, we achieve an exponential improvement in both state complexity and message size for deterministic algorithms. Moreover, we present two complementary approaches for reducing the number of bits communicated during and after stabilization.

We show that any randomised Monte Carlo distributed algorithm for the Lovász local lemma requires Omega(log log n) communication rounds, assuming that it finds a correct assignment with high probability. Our result holds even in the special case of d = O(1), where d is the maximum degree of the dependency graph. By prior work, there are distributed algorithms for the Lovász local lemma with a running time of O(log n) rounds in bounded-degree graphs, and the best lower bound before our work was Omega(log* n) rounds [Chung et al. 2014].

In the relay placement problem, the input is a set of sensors and a number r ⩾ 1, the communication range of a relay. In the one-tier version of the problem, the objective is to place a minimum number of relays so that between every pair of sensors there is a path through sensors and/or relays such that the consecutive vertices of the path are within distance r if both vertices are relays and within distance 1 otherwise. The two-tier version adds the restrictions that the path must go through relays, and not through sensors . We present a 3.11-approximation algorithm for the one-tier version and a polynomial-time approximation scheme (PTAS) for the two-tier version. We also show that the one-tier version admits no PTAS, assuming P ≠ NP.

In this work, we use algebraic methods for studying distance computation and subgraph detection tasks in the congested clique model. Specifically, we adapt parallel matrix multiplication implementations to the congested clique, obtaining an O(n1-2/ω) round matrix multiplication algorithm, where ω < 2.3728639 is the exponent of matrix multiplication. In conjunction with known techniques from centralised algorithmics, this gives significant improvements over previous best upper bounds in the congested clique model. The highlight results include: triangle and 4-cycle counting in O(n0.158) rounds, improving upon the O(n1/3) triangle counting algorithm of Dolev et al. [DISC 2012], a (1 + o(1))-approximation of all-pairs shortest paths in O(n0.158) rounds, improving upon the ~O (n1/2)-round (2 + o(1))-approximation algorithm of Nanongkai [STOC 2014], and computing the girth in O(n0.158) rounds, which is the first non-trivial solution in this model. In addition, we present a novel constant-round combinatorial algorithm for detecting 4-cycles.

Consider a complete communication network of n nodes, in which the nodes receive a common clock pulse. We study the synchronous c-counting problem: given any starting state and up to f faulty nodes with arbitrary behaviour, the task is to eventually have all correct nodes count modulo c in agreement. Thus, we are considering algorithms that are self-stabilising despite Byzantine failures. In this work, we give new algorithms for the synchronous counting problem that (1) are deterministic, (2) have linear stabilisation time in f, (3) use a small number of states, and (4) achieve almost-optimal resilience. Prior algorithms either resort to randomisation, use a large number of states, or have poor resilience. In particular, we achieve an exponential improvement in the state complexity of deterministic algorithms, while still achieving linear stabilisation time and almost-linear resilience.

By prior work, there is a distributed graph algorithm that finds a maximal fractional matching (maximal edge packing) in O(Δ) rounds, independently of n; here Δ is the maximum degree of the graph and n is the number of nodes in the graph. We show that this is optimal: there is no distributed algorithm that finds a maximal fractional matching in o(Δ) rounds, independently of n. Our work gives the first linear-in-Δ lower bound for a natural graph problem in the standard LOCAL model of distributed computing---prior lower bounds for a wide range of graph problems have been at best logarithmic in Δ.

We show that there is no deterministic local algorithm (constant-time distributed graph algorithm) that finds a (7-ε)-approximation of a minimum dominating set on planar graphs, for any positive constant ε. In prior work, the best lower bound on the approximation ratio has been 5-ε; there is also an upper bound of 52.

Linial's seminal result shows that any deterministic distributed algorithm that finds a 3-colouring of an $n$-cycle requires at least log*(n)/2 - 1 communication rounds. We give a new simpler proof of this theorem.

Do unique node identifiers help in deciding whether a network G has a prescribed property P? We study this question in the context of distributed local decision, where the objective is to decide whether G has property P by having each node run a constant-time distributed decision algorithm. In a yes-instance all nodes should output yes, while in a no-instance at least one node should output no.

In the study of deterministic distributed algorithms, it is commonly assumed that each node has a unique O (log n )-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient. Our result holds for so-called simple PO- checkable graph optimisation problems ; this includes many classical packing and covering problems such as vertex covers, edge covers, matchings, independent sets, dominating sets, and edge dominating sets. We focus on the case of bounded-degree graphs and show that if a local algorithm finds a constant-factor approximation of a simple PO-checkable graph problem with the help of unique identifiers, then the same approximation ratio can be achieved on anonymous networks. As a corollary of our result, we derive a tight lower bound on the local approximability of the minimum edge dominating set problem . By prior work, there is a deterministic local algorithm that achieves the approximation factor of 4--1/⌊Δ/2⌋ in graphs of maximum degree Δ. This approximation ratio is known to be optimal in the port-numbering model—our main theorem implies that it is optimal also in the standard model in which each node has a unique identifier. Our main technical tool is an algebraic construction of homogeneously ordered graphs : We say that a graph is (α, r )-homogeneous if its nodes are linearly ordered so that an α fraction of nodes have pairwise isomorphic radius- r neighbourhoods. We show that there exists a finite (α, r )-homogeneous 2 k -regular graph of girth at least g for any α < 1 and any r , k , and g .

In the study of deterministic distributed algorithms it is commonly assumed that each node has a unique O(log n)-bit identifier. We prove that for a general class of graph problems, local algorithms (constant-time distributed algorithms) do not need such identifiers: a port numbering and orientation is sufficient.

This work presents a classification of weak models of distributed computing. We focus on deterministic distributed algorithms, and we study models of computing that are weaker versions of the widely-studied port-numbering model. In the port-numbering model, a node of degree d receives messages through d input ports and it sends messages through d output ports, both numbered with 1,2,...,d. In this work, VVc is the class of all graph problems that can be solved in the standard port-numbering model. We study the following subclasses of VVc:

We study distributed algorithms that find a maximal matching in an anonymous, edge-coloured graph. If the edges are properly coloured with k colours, there is a trivial greedy algorithm that finds a maximal matching in k-1 synchronous communication rounds. The present work shows that the greedy algorithm is optimal in the general case: if A is a deterministic distributed algorithm that finds a maximal matching in anonymous, k-edge-coloured graphs, then there is a worst-case input in which the running time of A is at least k1 rounds.

This work studies decision problems from the perspective of nondeterministic distributed algorithms. For a yes instance there must exist a proof that can be verified with a distributed algorithm: all nodes must accept a valid proof, and at least one node must reject an invalid proof. We focus on locally checkable proofs that can be verified with a constant-time distributed algorithm.

We study the stable marriage problem in a distributed setting. The communication network is a bipartite graph, with men on one side and women on the other. Acceptable partners are connected by edges, and each participant has chosen a linear order on the adjacent nodes, indicating the matching preferences.

An edge dominating set for a graph G is a set D of edges such that each edge of G is in D or adjacent to at least one edge in D. This work studies deterministic distributed approximation algorithms for finding minimum-size edge dominating sets. The focus is on anonymous port-numbered networks: there are no unique identifiers, but a node of degree d can refer to its neighbours by integers 1, 2, ..., d. The present work shows that in the port-numbering model, edge dominating sets can be approximated as follows: in d-regular graphs, to within 4-6/(d+1) for an odd d and to within 4-2/d for an even d; and in graphs with maximum degree Δ, to within 4-2/(Δ-1) for an odd Δ and to within 4-2/Δ for an even Δ. These approximation ratios are tight for all values of d and Δ: there are matching lower bounds.

We present a distributed algorithm that finds a maximal edge packing in O(Δ + log* W) synchronous communication rounds in a weighted graph, independent of the number of nodes in the network; here Δ is the maximum degree of the graph and W is the maximum weight. As a direct application, we have a distributed 2-approximation algorithm for minimum-weight vertex cover, with the same running time. We also show how to find an $f$-approximation of minimum-weight set cover in O(f2k2 + fk log* W) rounds; here k is the maximum size of a subset in the set cover instance, f is the maximum frequency of an element, and W is the maximum weight of a subset. The algorithms are deterministic, and they can be applied in anonymous networks.

In a max-min LP, the objective is to maximise ω subject to Ax ≤ 1, Cx ≥ ω1, and x ≥ 0 for nonnegative matrices A and C. We present a local algorithm (constant-time distributed algorithm) for approximating max-min LPs. The approximation ratio of our algorithm is the best possible for any local algorithm; there is a matching unconditional lower bound.

A local algorithm is a distributed algorithm where each node must operate solely based on the information that was available at system startup within a constant-size neighbourhood of the node. We study the applicability of local algorithms to max-min LPs where the objective is to maximise minkSigmav CkvXv subject to SigmavalphaivXv les 1 far each i and Xv ges 0 far each v. Here ckvges 0, and the support sets Vi= {v : alphaiv> 0}, Vk= {v : ckv> 0}, Iv= {i: alphaiv> 0} and Kv= {k : Ckv> 0} have bounded size. In the distributed setting, each agent v is responsible for choosing the value of Xv, and the communication network is a hypergraph H where the sets Vkand Viconstitute the hyperedges. We present inapproximability results for a wide range of structural assumptions; for example, even if |Vi| and |Vk| are bounded by some constants larger than 2, there is no local approximation scheme. To contrast the negative results, we present a local approximation algorithm which achieves good approximation ratios if we can bound the relative growth of the vertex neighbourhoods in H.