Distributed Coloring of Hypergraphs

Abstract

For any integer $r \ge 2$, a linear r-uniform hypergraph is a generalization of ordinary graphs, where edges contain r vertices and two edges intersect in at most one node. We consider the problem of coloring such hypergraphs in several constrained models of computing, i.e., computing a partition such that no edge is fully contained in the same class. In particular, we give a $\textrm{poly}(\log \log n)$-round randomized Local algorithm that computes a $O(\varDelta ^{1/(r-1)})$-coloring w.h.p. This is tight up to polynomial factors of the time complexity as $\varOmega (\log _\varDelta \log n)$ distributed rounds are necessary for even obtaining a $\varDelta$-coloring, where $\varDelta$ is the maximum degree, and tight up to logarithmic factors of the number of colors, as $\varTheta ((\varDelta /\log \varDelta )^{1/(r-1)})$ colors are necessary for existence. We also give simple algorithms that run in O(1)-rounds of the Congested Clique model and in a single-pass of the semi-streaming model.

Appendices

A Missing Proofs of Main Algorithm

1.1 A.1 Proof of Lemma 8 (Degrees Decrease in GeometricTrials)

Lemma 8. Let $C \ge 2^{1/(r-2)}$ be a constant, and $G=(V,E)$ be a triangle-free graph of rank r and maximum degree $\varDelta$. Let each node v try a random color in $[K]=[C\cdot \varDelta ^{1/(r-1)}]$ and uncolor itself if it is part of a monochromatic edge. Let $s\in [2 \varDelta ^{1/(r-1)}/C^{r-2}, K]$ and $t\ge 2(s/C)^{r-1}$. Then w.p. at least $1-2\exp (-t/6)-(r-1)\varDelta \exp (-s/6)$, v’s degree (number of fully uncolored incident edges) becomes at most 2t.

Proof

As explained in the main text, an edge e incident on a node v has two ways of surviving this process: by being monochromatic itself, or by having each of its nodes be part of a monochromatic edge distinct from e.

The number of monochromatic edges incident on v corresponds to its defect in the tentative coloring, which we previously analyzed. As in the proof of Lemma 5, at most t edges survive that way, w.p. $1-\exp (-t/6)$.

We turn to the second type of surviving edges. Let us say an edge e incident on v is forbidding to v if the nodes in e other than v all have the same color. We say that a color c is forbidden to a node v if v has an incident forbidding edge whose nodes other than v are all colored c. Let F(v) be the set of edges forbidding to v. The second type of surviving edge occurs when each of its nodes selects a forbidden colors. We show that $| F(v) |$ is concentrated.

Claim

For a node v and integers $x > 0$, $t \ge 2 d_v/x^{r-2}$, $\Pr [| F(v) | \ge t] \le \exp (-t/6)$.

Proof

(Proof of Appendix A.1). An edge e is forbidding to $v\in e$ with probability $1/x^{r-2}$. Therefore, $\mathop {\mathrm {\mathbb {E}}}\limits [| F(v) |] = d_v / x^{r-2}$, and $t \ge 2\mathop {\mathrm {\mathbb {E}}}\limits [| F(v) |]$. Because v’s neighborhood is triangle-free, edges incident on v share no other vertex. Therefore, whether each edge is forbidding to v is independent of whether other edges are, and by Lemma 4 (Chernoff bound), the probability ensues.

We now bound the probability that many edges survive due to all of its nodes picking a forbidden color.

Recall $s \ge 2 \varDelta ^{1/(r-1)}/C^{r-2}$. By Appendix A.1, a node u has less than s forbidden colors w.p. $1-\exp (-s/6)$. Therefore, all the neighbors of v have less than s forbidden colors w.p. $1-(r-1)\varDelta \exp (-s/6)$. In the rest of the argument, we condition on the event that nodes at distance 2 from v forbade at most s colors to each direct neighbor of v, and fix the random choices of nodes at distance 2 from v to a specific assignment satisfying this conditioning.

Consider an edge e incident on v with vertices $v,u_1,\ldots ,u_{r-1}$. The probability that $u_i$ picks a color forbidden by the distance 2 neighbors of v it is adjacent to is at most s/K. The probability that the $r-1$ $u_i$’s do so is at most $(s/K)^{r-1}$ by the independence of their choices. Therefore, the expected number of edges that remain uncolored due to this second argument is at most $\varDelta (s/K)^{r-1}$.

Finally, for each e incident on v let $X_e$ be the indicator random variable of the event that all its nodes other than v picked a color forbidden by the nodes at distance 2 from v. Let X be the sum $\sum _{e\ni v} X_e$. The $X_e$’s are all independent once the random choices of nodes at distance 2 are fixed. Therefore, by Lemma 4 (Chernoff bound), for $t \ge 2\varDelta (s/K)^{r-1} = 2 (s/C)^{r-1}$, $\Pr [X \ge t] \le 1 - \exp (-t/6)$.

Putting everything together, w.p. at least $1 - 2\exp (-t/6) - (r-1)\varDelta \exp (-s/6)$, each of the two sources of surviving edges contributes at most t edges, for a total of at most 2t.

1.2 A.2 Proof of Lemma 10

Lemma 10. For each node v, the probability that v is in $V_{\textsf{Quit}}$ is at most

$$\begin{aligned} 3(r-1)\varDelta (\log \log \varDelta ) \cdot \exp (-(\varDelta _{\textsf{goal}}/4)^{1/(r-1)}/6). \end{aligned}$$

Proof

Recall the values of variables which dictate how nodes behave in GeometricTrials,

• $C=4, \alpha =1/2, \varDelta _{\textsf{goal}}= \varDelta ^{1/(r-1)}$,

• $C_i = C^{2^i} = 4^{2^i}, K_i = \alpha ^i \varDelta ^{1/(r-1)}$,

• $\varDelta _i = \max \lbrace (K_i/C_i)^{r-1}, \varDelta _{\textsf{goal}} \rbrace , i_{\textsf{last}}= \max \lbrace i \mid \varDelta _i > \varDelta _{\textsf{goal}} \rbrace$.

Let us analyze the probability that a live node v in the i-th iteration of GeometricTrials decreases its degree to less than $\varDelta _{i+1}$ (or gets colored). By Lemma 9, if $\varDelta _{i+1} \ge 4 \varDelta _i/C_i^{(r-1)^2}$, then v’s degree decreases to less than $\varDelta _{i+1}$ with probability

$$\begin{aligned} 1-3(r-1)\varDelta _i\exp (-(\varDelta _{i+1}/4)^{1/(r-1)}C_i/6). \end{aligned}$$

We verify that $\varDelta _{i+1} \ge 4 \varDelta _i/C_i^{(r-1)^2}$ indeed holds. Note that, by definition,

$$\begin{aligned}\varDelta _{i+1}&\ge (K_{i+1}/C_{i+1})^{r-1} = \alpha ^{(i+1)(r-1)} K C^{-(r-1)2^{i+1}} = \varDelta _i \cdot (\alpha ^{r-1} C^{-(r-1)2^{i}})\\&= \varDelta _i \cdot 4^{-(r-1)/2 -(r-1)2^i} \ge 4\varDelta _i /C_i^{(r-1)^2}. \end{aligned}$$

For each node active in iteration $i\le i_{\textsf{last}}$ (which has therefore degree at most $\varDelta _i$), by Lemma 9, the probability that its degree fails to decrease to $\varDelta _{i+1}$ or less after each live node tries a color is at most

$$\begin{aligned} 3(r-1)\varDelta _i e^{-(\varDelta _{i+1}/4)^{1/(r-1)}/6}. \end{aligned}$$

Summing over all the rounds, the probability that a node joins $V_\textsf{Quit}$ during the $i_{\textsf{last}}+1 \le \log \log \varDelta$ loop iterations of GeometricTrials is at most

$$3(r-1)\sum _{i=0}^{i_{\textsf{last}}} \varDelta _i e^{-(\varDelta _{i+1}/4)^{1/(r-1)}/6} \quad \le \quad 3(r-1) \varDelta (\log \log \varDelta ) e^{-(\varDelta _{\textsf{goal}}/4)^{1/(r-1)}/6}. $$

B Missing Proofs for the $\varTheta (\log ^* n)$ Algorithm and Lower Bound

We give two results on deterministic algorithms. Firstly, we give an $O\left( \log ^* n \right)$ rounds algorithm for $\tilde{O}\left( \varDelta ^{r/(r - 1)} \right)$-coloring any r-uniform hypergraph. Secondly, we complement this algorithm with a lower bound of $\varOmega \left( \frac{\log ^* n}{r} \right)$ on finding such a coloring.

1.1 B.1 Finding a $\tilde{O}\left( \varDelta ^{r/(r - 1)} \right)$-Coloring

In this section, we give a one round algorithm for transforming a strong $\tilde{O}(r^2 \varDelta ^2)$-coloring into a weak $\tilde{O}\left( r \cdot \varDelta ^{r/(r - 1)} \right)$-coloring for an r-regular hypergraph. This result can be viewed as an addendum to Linial’s algorithm for finding a $O(\varDelta ^2)$-coloring in $O(\log ^* n)$ rounds. This reduction is performed via a combinatorial argument extending the notion of a $\varDelta$-cover free family to an r-weak $\varDelta$-cover free family. Note that a $\tilde{O}(r^2\varDelta ^2)$ strong coloring can be found in $O(\log ^* n)$ rounds by using Linial’s algorithm on the underlying graph. In order to obtain such a coloring, it is useful to introduce r-weak $\varDelta$-cover free families of sets. This generalization of $\varDelta$-cover free families serves to relax coloring constraint to the problem of finding a weak coloring.

Definition 4

(r -weak $\varDelta$ -cover free families). Let $\mathcal {F}$ be a family of sets. The family $\mathcal {F}$ is a r-weak $\varDelta$-cover free family if, for every set $S_0 \in \mathcal {F}$, and $\varDelta$ subfamilies $\mathcal {S}_j = \lbrace S_{j,1}, \dots , S_{j,r-1} \rbrace \subseteq \mathcal {F}\setminus \lbrace S_0 \rbrace$, each of size $r-1$, the following holds:

$$ S_0 \not \subseteq \bigcup \limits _{j = 1}^{\varDelta } \bigcap \limits _{k = 1}^{r - 1} S_{j,k} $$

Note that a 2-weak $\varDelta$-cover free family is equivalent to the classical definition of an $\varDelta$-cover free family.

Lemma 13

(Lower bound on the size of r -weak $\varDelta$ -cover free families). For three integers $n, \varDelta , r \in \mathbb {N}$ such that $n \ge r \ge 2$ and $\varDelta \ge 1$, there exists an r-weak $\varDelta$-cover free family $\mathcal {F}$ of size n, where each $S \in \mathcal {F}$ is a subset of a ground set [m], $m = 5\lceil r \varDelta ^{r/(r - 1)} \ln (n) \rceil$.

Proof

In the proof that follows, we use that $e^{-s} \ge \left( 1 - \frac{s}{r} \right) ^r$ for all $1 \le s \le r$. For some m, consider a random collection $\mathcal {F} = \lbrace S_1,\ldots ,S_n \rbrace$ of subsets of [m] constructed the following way: for every element $x \in [m]$ and index $i \in [n]$, x belongs to $S_i$ with some fixed probability p, independently of every other pair $(x',i') \ne (x,i)$. For any given element x, index $i_0 \in [n]$, and $\varDelta$ sets of $r-1$ indices $\lbrace i_{j,1},\dots ,i_{j,r-1} \rbrace \subseteq [n]\setminus \lbrace i_0 \rbrace$, the probability of x being in the set $S_{i_0}$ but out of $\bigcup _{j = 1}^{\varDelta } \left( \bigcap _{k = 1}^{r - 1} S_{i_{j,k}} \right)$ is:

$$\begin{aligned} \Pr \left[ x \in S_{i_0} \setminus \bigcup _{j = 1}^{\varDelta } \left( \bigcap _{k = 1}^{r - 1} S_{i_{j,k}}\right) \right]&\ge \Pr \left[ x \in S_{i_0}\right] \left( 1 - \sum _{j = 1}^{\varDelta } \Pr \left[ x \in \bigcap _{k = 1}^{r - 1} S_{i_{j,k}}\right] \right) \\&= p(1-\varDelta p^{r-1}) \end{aligned}$$

Setting $p = (2\varDelta )^{-1/(r-1)}$, this probability is at least $\frac{1}{4\varDelta ^{1/(r-1)}}$. Therefore the probability that for every $x \in [m]$, $x \notin S_{i_0} \setminus \bigcup _{j = 1}^{\varDelta } \left( \bigcap _{k = 1}^{r - 1} S_{i_{j,k}} \right)$ is no more than $\left( 1 - \frac{1}{4\varDelta ^{1/(r - 1)}} \right) ^{m} \le e^{-m / (4\varDelta ^{1/(r - 1)})} \le n^{-5r\varDelta /4}$. The probability that valid multiset of indices $i_0, i_{1,1}, \dots , i_{\varDelta ,(r-1)}$ exists such that $S_{i_0} \subseteq \bigcup _{j = 1}^{\varDelta } \left( \bigcap _{k = 1}^{r - 1} S_{i_{j,k}} \right)$ is no more than $n^{(r - 1)\varDelta + 1} n^{-5 r \varDelta /4} < 1$. Therefore, an r-weak $\varDelta$-cover free family of n sets with no such indices exists.

Theorem 6. There is a deterministic $O(\log ^* n)$-round Congest algorithm to $O ( r \cdot \varDelta ^{r/(r - 1)} \log (r\varDelta ))$-color hypergraphs of maximum degree $\varDelta$ and rank r.

Proof

Let $\phi$ be a strong $O(r^2 \varDelta ^2)$-coloring of the graph, computed using Linial’s algorithm on the underlying graph. We show that $\phi$ can be converted in to a weak $\tilde{O}\left( r \cdot \varDelta ^{r/(r - 1)} \right)$-coloring in a single round. From Lemma 13, there must be an r-weak $\varDelta$-cover free family $\mathcal {F}$ of $O(r^2 \varDelta ^2)$ sets from a universe of $O(r \cdot \varDelta ^{r/(r - 1)} \log (r \varDelta ))$ elements. By indexing these sets in some universal order, each vertex v can choose the set $S_v$ at index $\phi (v)$. As the set is a member of $\mathcal {F}$, following Definition 4 there must exist at least one element $c_v \in S_v$, such that for every edge e incident to v, $c_v \notin \bigcap \limits _{u \in e \setminus \{v\}} S_u$. Therefore, coloring v with $c_v$, v can not be incident to any monochromatic edge. As computing the value of $c_v$ only requires the color of each neighbor in the $O(r^2 \varDelta ^2)$-coloring, this can be done in a single round from the $O(r^2 \varDelta ^2)$-strong coloring. Hence the total round complexity of this process is $O(\log ^* n)$, dominated by the process of finding the initial strong coloring. Further, as $c_v$ is selected from a universe of size $O(r \cdot \varDelta ^{r/(r - 1)} \log (r \varDelta ))$, the coloring of G from this process corresponds to a weak $O(r \cdot \varDelta ^{r/(r - 1)} \log (r \varDelta )) = \tilde{O}(r \cdot \varDelta ^{r/(r - 1)})$-coloring of G.

1.2 B.2 Lower Bounds on Finding Polynomial Colorings

We show that there exists a lower bound of $\varOmega \left( \frac{\log ^*n}{r} \right)$ for finding on finding a $\textrm{poly}(\varDelta )$-coloring, by generalizing the classic lower bound due to Linial [33]. Rather than using a simple n-cycle, we construct a strongly connected n-hyper-cycle. A strongly connected n-hyper-cycle with minimum rank r can be derived from an n-cycle C by constructing an edge for each connected component of size r in C. Note that the degree of a vertex in such a graph is $2 (r - 1)$. We provide a lower bound using this construction be reduction from the problem of $O(\varDelta ^c)$-coloring an n-cycle.

Theorem 7. For any pair of constants r and c, no Local algorithm can find a $O(\varDelta ^c / r)$ coloring of a rank r hypergraph in fewer than $\varOmega \left( (\log ^* n) / r\right)$ rounds.

Proof

For the sake of contradiction, let $\mathcal {A}$ be an algorithm that can weakly color a strongly connected n-hyper-cycle with $O(\varDelta ^c)$ colors for some pair of constants r and c. Let $T_\mathcal {A}$ be its complexity. Let $G = (V,E)$ be a cycle graph with n vertices. It is known that no algorithm can find a $O(\varDelta ^2)$-coloring on G in fewer than $\varOmega (\log ^* n)$ rounds. We show that an algorithm for coloring G in $O(T_\mathcal {A})$ rounds exists, implying the lower bound on $T_\mathcal {A}$.

Let $G' = (V, H)$ be an r-uniform hypergraph constructed from G, with edge set $H = \{(v_1, \ldots , v_r) \in V^r \mid (v_i,v_{i + 1}) \in E, \forall i \in 1,2,\ldots ,r - 1 \}$. Note that $G'$ corresponds to a strongly connected n-cycle. Observe that any algorithm on $G'$ can be simulated on G in at most a factor of r additional rounds. Let $\phi$ be the coloring on V after running $\mathcal {A}$. Given any vertex $v \in V$, let $h_1,h_2 \in H$ be the pair of hyperedges incident to v such that $N(v) = h_1 \cup h_2$. In other words, $h_1$ and $h_2$ are the hyperedges that include v and the two vertices at a distance of $r - 1$ from v in G. As $\phi$ is a weak coloring of $G'$, there must exists two vertices $u_1,u_2 \in h_1 \times h_2$ such that $\phi (v) \not \in \lbrace \phi (u_1),\phi (u_2) \rbrace$.

Let $C = (V', E')$ be a connected component in G such that $\forall v,u \in V', \phi (v) = \phi (u)$. Following the above observation, the maximum length of such a component is $2r - 3$. As the number of colors assigned by $\phi$ is at most $O(r^c)$ for some pair of constants r and c, $\phi$ can be turned into a proper coloring of G by going through each color class and coloring the nodes in each component in order of decreasing ID. Therefore $\phi$ can be transformed into a proper coloring of G in at most $O(r^{c+1})$ rounds, and hence $r\cdot (T_\mathcal {A}+ O(r^{c+1})) \in \varOmega (\log ^*n)$, i.e. $\mathcal {A}$ must take at least $\varOmega \left( \log ^* n \right)$ rounds, since r and c are constants.

Note that the hypergraph used for the lower bound is not linear, i.e., the theorem does not rule out the existence of an $o(\log ^* n)$ round algorithm whose scope is limited to linear hypergraphs.