Deterministic Distributed Edge-Coloring via Hypergraph Maximal Matching

We present a deterministic distributed algorithm that computes a $(2\Delta-1)$-edge-coloring, or even list-edge-coloring, in any $n$-node graph with maximum degree $\Delta$, in $O(\log^7 \Delta \log n)$ rounds. This answers one of the long-standing open questions of \emph{distributed graph algorithms} from the late 1980s, which asked for a polylogarithmic-time algorithm. See, e.g., Open Problem 4 in the Distributed Graph Coloring book of Barenboim and Elkin. The previous best round complexities were $2^{O(\sqrt{\log n})}$ by Panconesi and Srinivasan [STOC'92] and $\tilde{O}(\sqrt{\Delta}) + O(\log^* n)$ by Fraigniaud, Heinrich, and Kosowski [FOCS'16]. A corollary of our deterministic list-edge-coloring also improves the randomized complexity of $(2\Delta-1)$-edge-coloring to poly$(\log\log n)$ rounds. The key technical ingredient is a deterministic distributed algorithm for \emph{hypergraph maximal matching}, which we believe will be of interest beyond this result. In any hypergraph of rank $r$ --- where each hyperedge has at most $r$ vertices --- with $n$ nodes and maximum degree $\Delta$, this algorithm computes a maximal matching in $O(r^5 \log^{6+\log r } \Delta \log n)$ rounds. This hypergraph matching algorithm and its extensions lead to a number of other results. In particular, a polylogarithmic-time deterministic distributed maximal independent set algorithm for graphs with bounded neighborhood independence, hence answering Open Problem 5 of Barenboim and Elkin's book, a $((\log \Delta/\varepsilon)^{O(\log (1/\varepsilon))})$-round deterministic algorithm for $(1+\varepsilon)$-approximation of maximum matching, and a quasi-polylogarithmic-time deterministic distributed algorithm for orienting $\lambda$-arboricity graphs with out-degree at most $(1+\varepsilon)\lambda$, for any constant $\varepsilon>0$, hence partially answering Open Problem 10 of Barenboim and Elkin's book

Rainbow Coloring Hardness via Low Sensitivity Polymorphisms

A k-uniform hypergraph is said to be r-rainbow colorable if there is an r-coloring of its vertices such that every hyperedge intersects all r color classes. Given as input such a hypergraph, finding a r-rainbow coloring of it is NP-hard for all k >= 3 and r >= 2. Therefore, one settles for finding a rainbow coloring with fewer colors (which is an easier task). When r=k (the maximum possible value), i.e., the hypergraph is k-partite, one can efficiently 2-rainbow color the hypergraph, i.e., 2-color its vertices so that there are no monochromatic edges. In this work we consider the next smaller value of r=k-1, and prove that in this case it is NP-hard to rainbow color the hypergraph with q := ceil[(k-2)/2] colors. In particular, for k <=6, it is NP-hard to 2-color (k-1)-rainbow colorable k-uniform hypergraphs. Our proof follows the algebraic approach to promise constraint satisfaction problems. It proceeds by characterizing the polymorphisms associated with the approximate rainbow coloring problem, which are rainbow colorings of some product hypergraphs on vertex set [r]^n. We prove that any such polymorphism f: [r]^n -> [q] must be C-fixing, i.e., there is a small subset S of C coordinates and a setting a in [q]^S such that fixing x_{|S} = a determines the value of f(x). The key step in our proof is bounding the sensitivity of certain rainbow colorings, thereby arguing that they must be juntas. Armed with the C-fixing characterization, our NP-hardness is obtained via a reduction from smooth Label Cover

Approximate Hypergraph Coloring under Low-discrepancy and Related Promises

A hypergraph is said to be $\chi$-colorable if its vertices can be colored with $\chi$ colors so that no hyperedge is monochromatic. $2$-colorability is a fundamental property (called Property B) of hypergraphs and is extensively studied in combinatorics. Algorithmically, however, given a $2$-colorable $k$-uniform hypergraph, it is NP-hard to find a $2$-coloring miscoloring fewer than a fraction $2^{-k+1}$ of hyperedges (which is achieved by a random $2$-coloring), and the best algorithms to color the hypergraph properly require $\approx n^{1-1/k}$ colors, approaching the trivial bound of $n$ as $k$ increases. In this work, we study the complexity of approximate hypergraph coloring, for both the maximization (finding a $2$-coloring with fewest miscolored edges) and minimization (finding a proper coloring using fewest number of colors) versions, when the input hypergraph is promised to have the following stronger properties than $2$-colorability: (A) Low-discrepancy: If the hypergraph has discrepancy $\ell \ll \sqrt{k}$, we give an algorithm to color the it with $\approx n^{O(\ell^2/k)}$ colors. However, for the maximization version, we prove NP-hardness of finding a $2$-coloring miscoloring a smaller than $2^{-O(k)}$ (resp. $k^{-O(k)}$) fraction of the hyperedges when $\ell = O(\log k)$ (resp. $\ell=2$). Assuming the UGC, we improve the latter hardness factor to $2^{-O(k)}$ for almost discrepancy-$1$ hypergraphs. (B) Rainbow colorability: If the hypergraph has a $(k-\ell)$-coloring such that each hyperedge is polychromatic with all these colors, we give a $2$-coloring algorithm that miscolors at most $k^{-\Omega(k)}$ of the hyperedges when $\ell \ll \sqrt{k}$, and complement this with a matching UG hardness result showing that when $\ell =\sqrt{k}$, it is hard to even beat the $2^{-k+1}$ bound achieved by a random coloring.Comment: Approx 201

Topological lower bounds for the chromatic number: A hierarchy

This paper is a study of ``topological'' lower bounds for the chromatic number of a graph. Such a lower bound was first introduced by Lov\'asz in 1978, in his famous proof of the \emph{Kneser conjecture} via Algebraic Topology. This conjecture stated that the \emph{Kneser graph} \KG_{m,n}, the graph with all $k$-element subsets of $\{1,2,...,n\}$ as vertices and all pairs of disjoint sets as edges, has chromatic number $n-2k+2$. Several other proofs have since been published (by B\'ar\'any, Schrijver, Dolnikov, Sarkaria, Kriz, Greene, and others), all of them based on some version of the Borsuk--Ulam theorem, but otherwise quite different. Each can be extended to yield some lower bound on the chromatic number of an arbitrary graph. (Indeed, we observe that \emph{every} finite graph may be represented as a generalized Kneser graph, to which the above bounds apply.) We show that these bounds are almost linearly ordered by strength, the strongest one being essentially Lov\'asz' original bound in terms of a neighborhood complex. We also present and compare various definitions of a \emph{box complex} of a graph (developing ideas of Alon, Frankl, and Lov\'asz and of \kriz). A suitable box complex is equivalent to Lov\'asz' complex, but the construction is simpler and functorial, mapping graphs with homomorphisms to $\Z_2$-spaces with $\Z_2$-maps.Comment: 16 pages, 1 figure. Jahresbericht der DMV, to appea

Online and quasi-online colorings of wedges and intervals

We consider proper online colorings of hypergraphs defined by geometric regions. We prove that there is an online coloring algorithm that colors $N$ intervals of the real line using $\Theta(\log N/k)$ colors such that for every point $p$, contained in at least $k$ intervals, not all the intervals containing $p$ have the same color. We also prove the corresponding result about online coloring a family of wedges (quadrants) in the plane that are the translates of a given fixed wedge. These results contrast the results of the first and third author showing that in the quasi-online setting 12 colors are enough to color wedges (independent of $N$ and $k$). We also consider quasi-online coloring of intervals. In all cases we present efficient coloring algorithms