On the Complexity of Digraph Colourings and Vertex Arboricity

It has been shown by Bokal et al. that deciding 2-colourability of digraphs is an NP-complete problem. This result was later on extended by Feder et al. to prove that deciding whether a digraph has a circular $p$-colouring is NP-complete for all rational $p>1$. In this paper, we consider the complexity of corresponding decision problems for related notions of fractional colourings for digraphs and graphs, including the star dichromatic number, the fractional dichromatic number and the circular vertex arboricity. We prove the following results: Deciding if the star dichromatic number of a digraph is at most $p$ is NP-complete for every rational $p>1$. Deciding if the fractional dichromatic number of a digraph is at most $p$ is NP-complete for every $p>1, p \neq 2$. Deciding if the circular vertex arboricity of a graph is at most $p$ is NP-complete for every rational $p>1$. To show these results, different techniques are required in each case. In order to prove the first result, we relate the star dichromatic number to a new notion of homomorphisms between digraphs, called circular homomorphisms, which might be of independent interest. We provide a classification of the computational complexities of the corresponding homomorphism colouring problems similar to the one derived by Feder et al. for acyclic homomorphisms.Comment: 21 pages, 1 figur

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

Fully Dynamic Matching in Bipartite Graphs

Maximum cardinality matching in bipartite graphs is an important and well-studied problem. The fully dynamic version, in which edges are inserted and deleted over time has also been the subject of much attention. Existing algorithms for dynamic matching (in general graphs) seem to fall into two groups: there are fast (mostly randomized) algorithms that do not achieve a better than 2-approximation, and there slow algorithms with \O(\sqrt{m}) update time that achieve a better-than-2 approximation. Thus the obvious question is whether we can design an algorithm -- deterministic or randomized -- that achieves a tradeoff between these two: a $o(\sqrt{m})$ approximation and a better-than-2 approximation simultaneously. We answer this question in the affirmative for bipartite graphs. Our main result is a fully dynamic algorithm that maintains a 3/2 + \eps approximation in worst-case update time O(m^{1/4}\eps^{/2.5}). We also give stronger results for graphs whose arboricity is at most \al, achieving a (1+ \eps) approximation in worst-case time O(\al (\al + \log n)) for constant \eps. When the arboricity is constant, this bound is $O(\log n)$ and when the arboricity is polylogarithmic the update time is also polylogarithmic. The most important technical developement is the use of an intermediate graph we call an edge degree constrained subgraph (EDCS). This graph places constraints on the sum of the degrees of the endpoints of each edge: upper bounds for matched edges and lower bounds for unmatched edges. The main technical content of our paper involves showing both how to maintain an EDCS dynamically and that and EDCS always contains a sufficiently large matching. We also make use of graph orientations to help bound the amount of work done during each update.Comment: Longer version of paper that appears in ICALP 201

Algorithms to Approximate Column-Sparse Packing Problems

Column-sparse packing problems arise in several contexts in both deterministic and stochastic discrete optimization. We present two unifying ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain improved approximation algorithms for some well-known families of such problems. As three main examples, we attain the integrality gap, up to lower-order terms, for known LP relaxations for k-column sparse packing integer programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set packing (Bansal et al., Algorithmica, 2012), and go "half the remaining distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM Transactions of Algorithm

Adaptive Out-Orientations with Applications

We give simple algorithms for maintaining edge-orientations of a fully-dynamic graph, such that the out-degree of each vertex is bounded. On one hand, we show how to orient the edges such that the out-degree of each vertex is proportional to the arboricity $\alpha$ of the graph, in a worst-case update time of $O(\log^2 n \log \alpha)$. On the other hand, motivated by applications in dynamic maximal matching, we obtain a different trade-off, namely the improved worst case update time of $O(\log n \log \alpha)$ for the problem of maintaining an edge-orientation with at most $O(\alpha + \log n)$ out-edges per vertex. Since our algorithms have update times with worst-case guarantees, the number of changes to the solution (i.e. the recourse) is naturally limited. Our algorithms make choices based entirely on local information, which makes them automatically adaptive to the current arboricity of the graph. In other words, they are arboricity-oblivious, while they are arboricity-sensitive. This both simplifies and improves upon previous work, by having fewer assumptions or better asymptotic guarantees. As a consequence, one obtains an algorithm with improved efficiency for maintaining a $(1+\varepsilon)$ approximation of the maximum subgraph density, and an algorithm for dynamic maximal matching whose worst-case update time is guaranteed to be upper bounded by $O(\alpha + \log n\log \alpha)$, where $\alpha$ is the arboricity at the time of the update

The Arboricity Captures the Complexity of Sampling Edges

In this paper, we revisit the problem of sampling edges in an unknown graph G = (V, E) from a distribution that is (pointwise) almost uniform over E. We consider the case where there is some a priori upper bound on the arboriciy of G. Given query access to a graph G over n vertices and of average degree {d} and arboricity at most alpha, we design an algorithm that performs O(alpha/d * {log^3 n}/epsilon) queries in expectation and returns an edge in the graph such that every edge e in E is sampled with probability (1 +/- epsilon)/m. The algorithm performs two types of queries: degree queries and neighbor queries. We show that the upper bound is tight (up to poly-logarithmic factors and the dependence in epsilon), as Omega(alpha/d) queries are necessary for the easier task of sampling edges from any distribution over E that is close to uniform in total variational distance. We also prove that even if G is a tree (i.e., alpha = 1 so that alpha/d = Theta(1)), Omega({log n}/{loglog n}) queries are necessary to sample an edge from any distribution that is pointwise close to uniform, thus establishing that a poly(log n) factor is necessary for constant alpha. Finally we show how our algorithm can be applied to obtain a new result on approximately counting subgraphs, based on the recent work of Assadi, Kapralov, and Khanna (ITCS, 2019)

Distributed local approximation algorithms for maximum matching in graphs and hypergraphs

We describe approximation algorithms in Linial's classic LOCAL model of distributed computing to find maximum-weight matchings in a hypergraph of rank $r$. Our main result is a deterministic algorithm to generate a matching which is an $O(r)$-approximation to the maximum weight matching, running in $\tilde O(r \log \Delta + \log^2 \Delta + \log^* n)$ rounds. (Here, the $\tilde O()$ notations hides $\text{polyloglog } \Delta$ and $\text{polylog } r$ factors). This is based on a number of new derandomization techniques extending methods of Ghaffari, Harris & Kuhn (2017). As a main application, we obtain nearly-optimal algorithms for the long-studied problem of maximum-weight graph matching. Specifically, we get a $(1+\epsilon)$ approximation algorithm using $\tilde O(\log \Delta / \epsilon^3 + \text{polylog}(1/\epsilon, \log \log n))$ randomized time and $\tilde O(\log^2 \Delta / \epsilon^4 + \log^*n / \epsilon)$ deterministic time. The second application is a faster algorithm for hypergraph maximal matching, a versatile subroutine introduced in Ghaffari et al. (2017) for a variety of local graph algorithms. This gives an algorithm for $(2 \Delta - 1)$-edge-list coloring in $\tilde O(\log^2 \Delta \log n)$ rounds deterministically or $\tilde O( (\log \log n)^3 )$ rounds randomly. Another consequence (with additional optimizations) is an algorithm which generates an edge-orientation with out-degree at most $\lceil (1+\epsilon) \lambda \rceil$ for a graph of arboricity $\lambda$; for fixed $\epsilon$ this runs in $\tilde O(\log^6 n)$ rounds deterministically or $\tilde O(\log^3 n )$ rounds randomly