1 Introduction

Although the maximum cut problem (MaxCut) is fundamental in combinatorial optimization, it has not been intensively studied from the perspective of distributed algorithms. The folklore algorithm consisting in choosing uniformly at random one side of the cut for each vertex of a graph G can however be seen as a distributed randomized algorithm with no rounds of communication. By the linearity of expectation, this algorithm gives a cut (a bipartition of the vertex set) of size at least m/2 in average, where m is the number of edges of G. Here, by the size of the cut, we mean the number of edges connecting the two parts of the bipartition. Since every cut in G contains at most m edges, this algorithm has approximation ratio at least \(\tfrac{1}{2}\) in average, which means that the size of the cut given by the algorithm is at least \(\tfrac{1}{2}\) of the size of the maximum cut in average.

A natural question is whether a better approximation ratio can be obtained if more rounds of communications are allowed. This question was answered positively by Shearer [26] in the case of triangle-free d-regular graphs. A d-regular graph is a graph in which every vertex has degree d. In the case of triangle-free d-regular graphs, Shearer gave a simple randomized algorithm finding a cut of size at least \( m\cdot (\frac{1}{2}+\frac{0.177}{\sqrt{d}})\) in average, and thus achieving an approximation ratio of \(\frac{1}{2}+\frac{0.177}{\sqrt{d}}\) in average. Shearer’s algorithm uses a single round of communication, messages consisting of a single bit, and at most 3 random bits per vertex. This was recently improved by Hirvonen, Rybicki, Schmid and Suomela [15], who obtained a simpler algorithm finding a cut of size at least \(m\cdot \left( \frac{1}{2}+\frac{0.28125}{\sqrt{d}} \right) \) in average. Their algorithm uses a single round of communication, messages consisting of a single bit, and a single random bit per vertex.

The case where d is small and the girth (length of a shortest cycle) is large has also been considered: for 3-regular graphs, Kardoš, Král’ and Volec [16] showed that when the girth is at least 637789, there exists a randomized distributed algorithm that outputs a cut of average size at least 0.88672 m in at most 318894 rounds (the important value here is the size of the cut). This was improved by Lyons [19], who proved a lower bound of 0.89 m for cubic graphs of girth at least 655. The best known lower bound for cubic graphs of large girth, 0.90 m, was proved by Gamarnik and Li [9], using a result of Csóka, Gerencsér, Harangi, and Virág [3]. The bound of Lyons [19] holds for any d-regular graphs of large enough (but constant) girth: such graphs have a cut of size at least \( m\cdot (\frac{1}{2}+\frac{2}{\pi \sqrt{d}})\approx m\cdot (\frac{1}{2}+\frac{0.637}{\sqrt{d}})\). On the other hand, Dembo, Montanari and Sen [6] showed that in random d-regular graphs, the maximum cut has size \(m\cdot (\frac{1}{2}+\frac{0.763+o(1)}{\sqrt{d}})+o(m)\) with high probability, proving a conjecture of [9]. The existence of this constant \({\approx }0.763\) is also connected to a conjecture of Hatami, Lovász and Szegedy [14] on limits of sparse graphs (see also the conclusion of [24] where the conjecture is strongly disproved for maximum independent sets, improving on an earlier result of [10]).

All the results mentioned above (except the result of Gamarnik and Li [9]) can be translated into efficient algorithms working in the CONGEST model. In this model, each node of the graph corresponds to a processor with infinite computational power and has a unique ID (each ID is an integer between 1 and \(\text {poly}(n)\), where n denotes the number of vertices in the graph). Nodes can communicate with their neighbors in the graph in synchronous rounds until each node outputs 0 or 1, corresponding to its side in the cut. In the CONGEST model, each message sent by a node to a neighbor has size \(O(\log n)\), while in some of the algorithms above, the messages have size at most 1. Let us call \(\textsf {CONGEST}(B)\) the variant of the CONGEST model in which messages are restricted to have size at most B (instead of \(O(\log n)\)), and let us say that an algorithm is local in a model if it runs in a constant number of rounds in this model. In particular the results of [15, 16, 26] mentioned above can be translated into local algorithms in the CONGEST(O (1)) model, while the results of [3, 19] can be translated into local algorithms in the CONGEST model.

Note that some of our lower bounds are also valid in the less restricted LOCAL model where the size of each message is not limited. In the following, we will make it clear if this applies. On the other hand, all our algorithms can be implemented in the PO model (anonymous network with port numbering and orientations), which is significantly stronger than the CONGEST model (see [13] for some results on local algorithms in PO and CONGEST).

We now review recent results on distributed approximation of MaxCut. On the deterministic side, Censor-Hillel, Levy, and Shachnai [2] designed a deterministic \(\frac{1}{2}\)-approximation that runs in \(\tilde{O}\left( \varDelta +\log ^* n \right) \) rounds in the CONGEST model on any graph of maximum degree at most \(\varDelta \). More recently, Kawarabayashi and Schwartzman [17] improved the complexity for constant factor approximation by providing a deterministic \(\left( \frac{1}{2}-\epsilon \right) \)-approximation that runs in \(O(\log ^* n)\) rounds (for any \(\epsilon >0\)), in the CONGEST model. However, no deterministic local approximation for MaxCut (i.e. running in a constant number of rounds) in the CONGEST model is known.

There is a similar gap between randomized and deterministic approximations for the maximum directed cut problem. Censor-Hillel, Levy, and Shachnai [2] provided a deterministic algorithm running in \(O(\varDelta +\log ^* n)\) rounds that guarantees a \(\frac{1}{3}\)-approximation as well as a randomized \(\frac{1}{2}\)-approximation with the same round complexity. The round complexities were improved by Kawarabayashi and Schwartzman [17] who provided a deterministic \(\left( \frac{1}{3}-\epsilon \right) \)-approximation running in \(O(\log ^* n)\) rounds as well as a randomized \(\left( \frac{1}{2}-\epsilon \right) \)-approximation in \(O(\epsilon ^{-1})\) rounds. All these results are stated in the CONGEST model. Similarly, no deterministic local algorithm is known to achieve a constant factor approximation for this problem.

1.1 Our Results

Our work focuses on bridging the gap between extremely efficient randomized local algorithms and slower deterministic algorithms for MaxCut. It should be noted that there are generic tools to derandomize distributed algorithms (see [4, 11] for recent results in this direction) but existing techniques mainly apply to locally checkable problems (problem for which a solution can be checked locally), which is not the case of (approximations of) MaxCut.

In Sect. 2 we show that any deterministic algorithm that guarantees a constant factor approximation for MaxCut on the class of bipartite d-regular graphs when d is a (fixed) even integer requires \(\varOmega (\log ^*n)\) rounds, which matches the complexity of the algorithm of Kawarabayashi and Schwartzman [17] mentioned above. When d is odd, we show that one cannot achieve a approximation ratio better than \(\frac{1}{d}\) in a constant number of rounds. Our proofs use an elementary graph construction and then apply Ramsey’s theorem [25]. Both these arguments are not new in distributed algorithms: our construction is inspired from Linial’s seminal paper [18] that provides a lower bound on the round complexity of coloring cycles and from a more recent paper by Åstrand, Polishchuk, Rybicki, Suomela, and Uitto [1] which applies Ramsey’s theorem in a similar setting to prove that there is no deterministic and local constant factor approximation for the maximum matching problem. It was pointed out to us that similar arguments were also used by Czygrinow, Hanckowiak, and Wawrzyniak [5] to prove lower bounds for the approximation of maximum independent sets in cycles. Here, our results hold for any d-regular graph (d is not necessarily equal to 2), so some additional work needs to be done compared to the simple case of cycles.

In Sect. 3, we show that this barrier of \(\frac{1}{d}\) when d is odd is sharp: we first remark that a result of Naor and Stockmeyer [22] on weak 2-coloring of graphs directly gives a deterministic local algorithm that guarantees a \(\frac{1}{d}\)-approximation. We then provide a much simpler and faster deterministic local algorithm achieving the same approximation ratio. It runs in a single round with messages of size \(O(\log n)\) and we also argue that this cannot be improved.

For the Maximum Directed Cut problem in d-regular graphs, we prove that a similar situation occurs. If d is even, a constant factor approximation cannot be achieved in \(o(\log ^* n)\) rounds, and if d is odd, no \((\tfrac{2}{d}+\epsilon )\)-approximation can be achieved in \(o(\log ^* n)\) rounds (for any \(\epsilon >0\)). On the other hand, if d is odd, a \(\tfrac{2}{d+1/d}\) factor approximation can be achieved in 0 round, and a \(\tfrac{2}{d+1/d-\varOmega (1/d^2)}\) factor approximation can be achieved in 2 rounds. Note that there is a small gap between the lower bounds and the upper bound of \(\tfrac{2}{d}\), and we explain some obstacles towards closing the gap.

Our results imply that while finding a constant factor approximation for the (directed) maximum cut problem in regular graphs of even degree does not require any communication for randomized distributed algorithm (i.e. it can be solved in 0 round), for deterministic algorithms an unbounded number of rounds are needed in this case. Note that this separation is not possible for locally checkable problems (see Theorem 3 in [4]). The (perhaps) surprising aspect is that in the case of regular graphs of odd degree, the problem can be solved by a deterministic algorithm without communication (if some orientation is given).

Note that another example of non locally checkable problem with such a separation between the randomized and deterministic complexities was given in [11]. Their problem consists in marking \((1+o(1))\sqrt{n}\) vertices of an n-cycle; the randomized version can also be solved in 0 round, while the deterministic version needs \(\varOmega (\sqrt{n})\) rounds.

1.2 Definitions

A cut in a graph G is a bipartition (A, B) of its vertex set V(G). We usually refer to A and B as the left side and the right side of the cut, respectively. The size of a cut (A, B) is the number of edges with one end in A and the other in B. The MaxCut problem in a graph G consists in finding a cut in G whose size is maximum.

Given an oriented graph G, a directed cut is again a bipartition (A, B) of the vertex set of G, and the size of the directed cut (A, B) is the number of arcs with their tail in A and their head in B. The MaxDiCut problem in an oriented graph G consists in finding a directed cut in G whose size is maximum.

Our results in this paper mainly concern d-regular graph, i.e. graphs in which each vertex has degree d. When we refer to an oriented d-regular graph G, we mean that the underlying unoriented graph is d-regular (the out-degrees can be arbitrary).

For an integer \(k\geqslant 1\), the tower function \(\mathrm {twr}_k\) is the function defined as \(\mathrm {twr}_1(x)=x\) and \(\mathrm {twr}_{k}(x)=2^{\mathrm {twr}_{k-1}(x)}\) for \(k\geqslant 2\). The iterated logarithm of an integer n, denoted by \(\log ^*n\) is defined as 0 if \(n\leqslant 1\), and as \(1+\log ^*(\log n)\) otherwise (here and everywhere else in the paper, \(\log \) denotes the logarithm base 2). The following can be easily derived by induction on k.

Claim 1

For any \(k,n\geqslant 1\):

$$\log ^* (\mathrm {twr}_{k}(n))= k-1+\log ^*(n)$$

2 Many Rounds for Deterministic Constant Factor Approximation in Regular Graphs

As mentioned in the introduction of this paper, Kawarabayashi and Schwartzman [17] provided a deterministic approximation running in \(O(\log ^* n)\) rounds for both problems studied here. In this section, we show with simple arguments based on bounds on Ramsey numbers that their bound is best possible.

In this section, we set \([n]=\left\{ 1,\ldots , n \right\} \). The q-color Ramsey number \(r_k(n;q)\) is the minimum N such that in any q-coloring of the k-element subsets of [N], there is an n-element subset S of [N] such that all k-element subsets of S have the same color (see [21] for a recent survey on Ramsey numbers).

Theorem 2

([7, 8]). There exists \(c>0\) such that for any positive integers q, k, and n, we have \(r_k(n;q) \leqslant \mathrm {twr}_k(c\cdot n \cdot q \log q)\).

We will also need two simple constructions of d-regular bipartite graphs.

We first assume that d is even. We consider a cycle C of size n, with n even, and then add an edge between each pair of vertices that are at distance exactly i in C for every \(i\in \left\{ 3,5,7, \ldots , d-1\right\} \). This graph, which we denote by \(C_n^d\), is certainly bipartite (the bipartition corresponds to the vertices at even distance from some arbitrary vertex in C, and the vertices at odd distance from this vertex). See Fig. 1 for an example of this graph. By a slight abuse of language, we say that two (or more) vertices of \(C_n^d\) are consecutive if they are consecutive in C. Similarly, when we refer to the clockwise order around \(C_n^d\), we indeed refer to the clockwise order around C.

Assume now d is odd. We take two disjoint copies of \(C_n^{d-1}\) and assume that the vertices of the cycle C in the first copy are \(u_1,u_2,\ldots ,u_n\), in clockwise order, and the vertices of the cycle C in the second copy are \(v_1,v_2,\ldots ,v_n\) in clockwise order. We then connect \(u_i\) and \(v_i\) by an edge, for any \(1\leqslant i \leqslant n\). This graph, which we denote by \(D_{2n}^d\), is clearly bipartite and d-regular, see Fig. 2 for an example.

Fig. 1.
figure 1

\(C_{12}^4\)

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Fig. 2.
figure 2

\(D_{24}^{5}\)

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We are now ready to state the main result of this section.

Theorem 3

Let \(d\geqslant 2\) be a fixed integer.

  • If d is even, then any deterministic algorithm in the LOCAL model that guarantees a constant factor approximation for MaxCut on the class of bipartite d-regular n-vertex graphs runs in \(\varOmega (\log ^*n)\) rounds.

  • If d is odd, then for any \(\epsilon >0\), any deterministic \(\left( \frac{1}{d}+\epsilon \right) \)-approximation algorithm in the LOCAL model for MaxCut on the class of bipartite d-regular n-vertex graphs runs in \(\varOmega (\log ^*n)\) rounds.

Note that since the LOCAL model is less restrictive than the CONGEST model, this theorem is also valid in the CONGEST model.

Due to space limitation, the full proof of Theorem 3 is given in the full version of the paper. The idea is to use the 2 color version of Theorem 2 to find a large number of blocks of ID’s that behave similarly with respect to a given algorithm, and to place these blocks of ID’s consecutively around the graphs \(C^d_n\) and \(D^d_{2n}\). In both cases a significant portion of the edges of a maximum cut will be missed. In the second case the barrier of \(\tfrac{1}{d}\) comes from the perfect matching between the outer cycle and the inner cycle, on which little can be said.

A direct consequence of our theorem is the following corollary that matches the round complexity obtained by Kawarabayashi and Schwartzman [17]:

Corollary 4

Deterministic constant factor approximation on general graphs for MaxCut in the LOCAL model requires \(\varOmega (\log ^* n)\) rounds.

2.1 Directed Cut

In this section, we consider the similar problem MaxDiCut where edges are oriented and we only count the edges going from the left side of the cut to the right side. We can prove similar bounds on the quality of the solution one can hope to achieve by simply orienting our lower bound graphs \(C^d_n\) and \(D^d_{2n}\): we will define \(\overrightarrow{C^d_n}\) as the same graph as \(C^d_n\) where we orient all the edges in clockwise order. Similarly, \(\overrightarrow{D^d_{2n}}\) is obtained from \(D^d_{2n}\) by orienting all the edges in clockwise order on both the inner and outer cycle, and all the edges in the remaining perfect matching from the outer cycle to the inner cycle. We can again apply Ramsey’s theorem as in the proof of Theorem 3 to obtain the following result:

Theorem 5

Let \(d>0\) be a fixed integer.

  • If d is even, any deterministic algorithm that guarantees a constant factor approximation for MaxDiCut on the class of d-regular bipartite n-vertex oriented graphs requires \(\varOmega (\log ^*n)\) rounds in the LOCAL model.

  • If d is odd, then for any \(\epsilon >0\), any deterministic \(\left( \frac{2}{d}+\epsilon \right) \)-approximation of MaxDiCut on the class of d-regular bipartite n-vertex oriented graphs requires \(\varOmega (\log ^*n)\) rounds in the LOCAL model.

We note a slight difference with Theorem 3 in the case where d is odd. In Theorem 5 the approximation ratio is only bounded by \(\frac{2}{d}\), instead \(\frac{1}{d}\). This happens because with our definition of \(\overrightarrow{D^d_{2n}}\), one can check that the optimal directed cut is of size \(\frac{nd}{4}=\frac{m}{2}\) instead of m in the undirected case.

3 Matching the Approximation Ratio When d Is Odd

3.1 Weak-Coloring

In a landmark paper, Naor and Stockmeyer [22] addressed the issue of what can or cannot be computed locally. In particular, they proved one result that turns out to be relevant in our case.

A weak coloring of a graph is a coloring of its vertices such that each vertex has at least one neighbor with a different color. Observe that a weak coloring using only 2 colors is a \(\frac{1}{d}\)-approximation of the MaxCut problem when the graph is d-regular. Let \(O_d\) be the class of graphs of maximum degree d where the degree of every vertex is odd. Naor and Stockmeyer proved the following theorem.

Theorem 6

([22]). There is a constant b such that, for every d, there is a deterministic algorithm with round complexity \(\log ^* d + b\) in the CONGEST model that solves the weak 2-coloring problem in the class \(O_d\).

As discussed above, this result directly implies that one can produce a local deterministic \(\frac{1}{d}\)-approximation of the MaxCut problem on d-regular graphs. However, the result given here is much stronger than what we are looking for as in this case every vertex has at least one incident edge in the cut. A natural question is whether a faster algorithm (of round complexity that does not depend on d) exists for the MaxCut problem on d-regular graphs with d odd. In the next section, we prove that such an algorithm exists.

3.2 A Simpler and Faster Algorithm

Consider the following algorithm: every vertex v collects the list of IDs of its neighbors, then v chooses its side of the cut depending on whether the median value of this list is higher or lower than its own ID. We call this algorithm the median algorithm. It runs in a single round and we prove the following theorem:

Theorem 7

When the input is a d-regular graph on n vertices, with d odd, the median algorithm finds in 1 round a \(\frac{1}{d}\)-approximation for the MaxCut problem in the CONGEST model.

We will actually give two different proofs of this result, the first proof shows a slightly better result in term of size of the cut while the second proof holds even for graphs in \(O_d\). In the first one, we prove the following slightly stronger statement.

Theorem 8

When the input is a d-regular graph on n vertices, with d odd, the median algorithm outputs in 1 round (in the CONGEST model) a cut of size at least \(\tfrac{n}{2}+\tfrac{(d-1)(d+1)}{4}\).

The proof of Theorem 8 is given in the full version of the paper. The idea is to orient each edge from the lower ID to the higher ID. The obtained orientation is acyclic, and the median algorithm places the vertices with out-degree higher than in-degree on one side of the cut and the other vertices on the other side of the cut. In the analysis the fact that the orientation is acyclic is important to obtain the additive term \(\tfrac{(d-1)(d+1)}{4}\).

An interesting aspect of Theorem 8 is that it shows that in Theorem 3, it is crucial that d is a fixed constant (independent of n). Indeed, if \(d=\varOmega (\sqrt{n})\), then \(\tfrac{n}{2}+\tfrac{(d-1)(d+1)}{4}\geqslant (1+\varOmega (1))\tfrac{n}{2}\) and thus the median algorithm achieves a \(\tfrac{1+\epsilon }{d}\)-approximation, for some \(\epsilon >0\). This is impossible when d is a constant, as shown by Theorem 3.

The median algorithm is based on finding an (acyclic) orientation of the input graph. Here, we do it by simply orienting the edges from the end with lower ID to the end with higher ID. This costs a single round of communication, with messages of size \(\log n\) (since vertices have to send their ID to their neighbors). It follows that in the more restricted \(\textsf {CONGEST}(b)\) model, where messages have size at most b, our algorithm takes \(\tfrac{\log n}{b}\) rounds. In particular, if only messages of size 1 are allowed, our algorithm takes \(\log n\) rounds. It turns out that this is close to best possible.

Theorem 9

Let \(D^d=\left\{ D_{2n}^d, n>0 \right\} \) for d odd. Any deterministic constant factor approximation of MaxCut on the class \(D^d\) requires at least \((1-o(1))\log n\) rounds in the \(\textsf {CONGEST}(1)\) model.

The proof of Theorem 9 is given in the full version of the paper. It is based on a simple symmetry argument.

3.3 Directed Cuts

Given a bipartition \((V_1,V_2)\) of an oriented graph G, the set of arcs oriented from \(V_1\) to \(V_2\) (the directed cut from \(V_1\) to \(V_2\)) is denoted by \(\overrightarrow{E}(V_1,V_2)\). The maximum cardinality of a directed cut in G is denoted by \(\mathrm {maxdicut}(G)\).

Let G be an oriented graph. For each vertex v, we define the deficit of v as \(\delta (v)=d^+(v)-d^-(v)\), where \(d^+(v)\) and \(d^-(v)\) denote the out-degree and in-degree of v, respectively. We define the sign of a vertex v as the sign of \(\delta (v)\), and we say that that a vertex is positive or negative accordingly. The set of positive vertices is denoted by \(V^+\) and the set of negative vertices is denoted by \(V^-\). Note that if all the vertices of G have odd degree (in particular if G is d-regular with d odd), then every vertex is positive or negative and this case \(V^+,V^-\) form a bipartition of the vertex set V of G.

Note that the median algorithm described in the previous subsection can be rephrased as: find an acyclic orientation of G and then choose the cut \((V^-,V^+)\) with respect to this orientation. Our second proof of Theorem 7 will be a direct consequence of the following general result (which proves that not only the cut, but also the directed cut between \(V^+\) and \(V^-\) has size at least n/2, and that the original orientation does not need to be acyclic).

Theorem 10

Let G be an n-vertex oriented d-regular graph with d odd, and let \(V^+\) and \(V^-\) be defined as above. Then the directed cut \(\overrightarrow{E}(V^+,V^-)\) contains at least \(\max \{\tfrac{n}{2},\tfrac{2}{d+1/d} \cdot \mathrm {maxdicut}(G)\}\) arcs.

The proof of Theorem 10 is given in the full version of the paper. From now on, we call the 0-round algorithm resulting from Theorem 10 the oriented median algorithm. The factor \(\frac{2}{d+1/d}\) might seem a little surprising, but it turns out to be sharp, in the following sense: there are infinite families of d-regular oriented graphs G for which the oriented median algorithm outputs a cut of size precisely \(\frac{2}{d+1/d}\mathrm {maxdicut}(G)\) (an example is given in the full version of the paper). So the problem does not come from the analysis of the algorithm, but rather from the algorithm itself.

To overcome this issue and close the gap with the \(\frac{2}{d}\) bound, one might be tempted to consider local improvements. In the following, a vertex will be stable if it has at least one neighbor on the other side of the cut. Otherwise it will be unstable. We now consider the following simple algorithm: at every round, every unstable vertex changes side. When we perform one round of this algorithm, we say we perform a flip (as this algorithm can be seen as a variant of the well known FLIP algorithm that is further discussed in the conclusion).

Theorem 11

Assume that \(d\geqslant 3\) is odd. Then the 2-round algorithm consisting of the oriented median algorithm followed by two flips provides a \(\frac{2}{d+1/d-c/d^2}\)-approximation (for some \(c>0)\) for the MaxDiCut problem in d-regular graphs.

The proof of Theorem 11 is given in the full version of the paper. Theorem 11 proves that after 2 flips, we can slightly improve on the approximation ratio of Theorem 10. A natural question is whether the same can be achieved after a single flip (in the full version of the paper, we give an example showing that the answer is negative).

4 Conclusion

4.1 FLIP

In Sect. 3, we have designed a very simple one-round algorithm approximating MaxCut in regular graphs (with odd degrees). Once a solution has been obtained, it might be tempting to run a few more rounds of computation to see if the solution can be improved locally.

We have already seen a simple way to improve the quality of a solution (by moving the so-called unstable vertices to the other side of the cut), but the notion of stability we used was specifically designed to improve the approximation ratio in a small number of rounds. Another simple way to locally improve a cut (in the sequential setting this time) is to take a vertex with more neighbors in its own part than in the other part, and change its side. If this is done until no such vertex exists, the resulting cut is maximal, and in this case is a \(\tfrac{1}{2}\)-approximation of the maximum cut. This operation, called FLIP, has been studied for a long time. When the edges are weighted, it was proved by Poljak [23] that any sequence of FLIPs takes only polynomially many steps before reaching a maximal cut in cubic graphs, while Monien and Tscheuschner [20] proved that there are graphs of maximum degree 4 for which a sequence of FLIPs can take exponentially many steps to reach a maximal cut. In the unweighted case however, since each flip improves the cut by at least one, the maximum number of flips before reaching a maximal cut is bounded by the number edges (which is linear in n in bounded degree graphs). In the distributed framework, it might be tempting to consider running some rounds of the distributed FLIP dynamics: at each round, each vertex with more neighbors in its own part than in the other part changes side. The graph \(D^d_{2n}\) constructed in the previous sections shows that it might not be helpful at all: if all the vertices of the outer cycle are in one side of the cut, and all the vertices of the inner cycle are on the other side of the cut, then at each round, all the vertices of the graph would change side, not improving the solution.

It might be worth noting that in our application of the median algorithm, not all vertices of the outer cycle of \(D^d_{2n}\) are on the same side of the cut (given some bad labelling): due to some side-effects, roughly d vertices in the outer cycle are not on the same side of the cut as the others, and similarly for the inner cycle. It can then be checked that if we run the distributed FLIP dynamics in this instance, the solution does improve over time, but improving the approximation ratio from \(\tfrac{1}{d}\) to \(\tfrac{1}{d}+\epsilon \) requires \(\varOmega (\epsilon n)\) rounds, which is extremely unpractical. This has to be compared with the lower bound of Theorem 3, which says that in order to achieve an approximation ratio of \(\tfrac{1}{d}+\epsilon \) in general, one needs a number of rounds of the order of \(\varOmega (\log ^*n)\).

4.2 SLOCAL vs LOCAL Model in the Deterministic Setting

Recently introduced by Ghaffari, Kuhn and Maus in [12], the SLOCAL model is designed to study the influence of the two major issues in LOCAL algorithms separately: in this model nodes are processed sequentially in any order. When a node v is processed, it can access its t-neighborhood and eventually additional information stored by vertices in this neighborhood that have been processed before v. In this model, symmetry breaking becomes free (the given order already breaks the symmetry) and only locality remains challenging.

Although finding an approximate maximum cut is not locally checkable, one can make an interesting parallel in this case. A very simple deterministic approximation algorithm for the maximum cut can be run in the SLOCAL model: simply process vertices in any order and when a vertex is processed put it on the side maximizing its local cut according to already processed neighbors. Note that each vertex loses at most half of its edges at each step, and each edge is counted once, so this is indeed a \(\frac{1}{2}\)-approximation with locality 1 and although we are studying a problem that is not locally checkable, this suggests that symmetry breaking (and not locality) is the bottleneck in our case.