Sharp Bounds and Precise Values for the Ni-Chromatic Number of Graphs

We consider the problem of designing deterministic graph algorithms for the model of Massively Parallel Computation (MPC) that improve with the sparsity of the input graph, as measured by the notion of arboricity. For the problems of maximal independent set (MIS), maximal matching (MM), and vertex coloring, we improve the state of the art as follows. Let $\lambda$ denote the arboricity of the $n$-node input graph with maximum degree $\Delta$. MIS and MM: We develop a deterministic low-space MPC algorithm that reduces the maximum degree to $poly(\lambda)$ in $O(\log \log n)$ rounds, improving and simplifying the randomized $O(\log \log n)$-round $poly(\max(\lambda, \log n))$-degree reduction of Ghaffari, Grunau, Jin [DISC'20]. Our approach when combined with the state-of-the-art $O(\log \Delta + \log \log n)$-round algorithm by Czumaj, Davies, Parter [SPAA'20, TALG'21] leads to an improved deterministic round complexity of $O(\log \lambda + \log \log n)$ for MIS and MM in low-space MPC. We also extend above MIS and MM algorithms to work with linear global memory. Specifically, we show that both problems can be solved in deterministic time $O(\min(\log n, \log \lambda \cdot \log \log n))$, and even in $O(\log \log n)$ time for graphs with arboricity at most $\log^{O(1)} \log n$. In this setting, only a $O(\log^2 \log n)$-running time bound for trees was known due to Latypov and Uitto [ArXiv'21]. Vertex Coloring: We present a $O(1)$-round deterministic algorithm for the problem of $O(\lambda)$-coloring in linear-memory MPC with relaxed global memory of $n \cdot poly(\lambda)$ that solves the problem after just one single graph partitioning step. This matches the state-of-the-art randomized round complexity by Ghaffari and Sayyadi [ICALP'19] and improves upon the deterministic $O(\lambda^{\epsilon})$-round algorithm by Barenboim and Khazanov [CSR'18].

In a proper vertex coloring of a graph, a Grundy vertex is a vertex colored with color c and adjacent to any color 1 to c-1. A partial Grundy coloring of a graph is a proper coloring of its vertices where any color admits at least one Grundy vertex. The partial Grundy number of a graph is the maximum number of colors used in a partial Grundy coloring of this graph. In this article we consider this parameter for some classes of graphs, in particular $K_{1,s}$-free graphs, regular graphs, Cartesian and direct products of graphs.

Over the past 30 years numerous algorithms have been designed for symmetry breaking problems in the LOCAL model, such as maximal matching, MIS, vertex coloring, and edge-coloring. For most problems the best randomized algorithm is at least exponentially faster than the best deterministic algorithm. In this paper we prove that these exponential gaps are necessary and establish numerous connections between the deterministic and randomized complexities in the LOCAL model. Each of our results has a very compelling take-away message: Fast Δ-coloring of trees requires random bits. Building on the recent randomized lower bounds of Brandt et al. [6], we prove that the randomized complexity of Δ-coloring a tree with maximum degree Δ is θ(logΔ log n), for any Δ ≥ 55, whereas its deterministic complexity is θ(logΔn) for any Δ ≥ 3. This also establishes a large separation between the deterministic complexity of Δ-coloring and (Δ+1)-coloring trees.Randomized lower bounds imply deterministic lower bounds. We prove that any deterministic algorithm for a natural class of problems that runs in O(1) + o(logΔn) rounds can be transformed to run in O(log* n -- log*Δ + 1) rounds. If the transformed algorithm violates a lower bound (even allowing randomization), then one can conclude that the problem requires Ω(logΔn) time deterministically. (This gives an alternate proof that deterministically Δ-coloring a tree with small Δ takes Ω(logΔn) rounds.)Deterministic lower bounds imply randomized lower bounds. We prove that the randomized complexity of any natural problem on instances of size n is at least its deterministic complexity on instances of size √log n}. This shows that a deterministic Ω(logΔn) lower bound for any problem (Δ-coloring a tree, for example) implies a randomized Ω(logΔ log n) lower bound. It also illustrates that the graph shattering technique employed in recent randomized symmetry breaking algorithms is absolutely essential to the LOCAL model. For example, it is provably impossible to improve the 2O(√log log n) terms in the complexities of the best MIS and (Δ+1)-coloring algorithms without also improving the 2O(√log n})-round Panconesi-Srinivasan algorithms.

The Maximal Independent Set (MIS) formulation tackles the graph coloring problem (GCP) as the partitioning of vertices of a graph into a minimum number of maximal independent sets as each MIS can be assigned a unique color. Mehrotra and Trick [5] solved the MIS formulation with an exact IP approach, but they were restricted to solving smaller or easier instances. For harder instances, it might be impossible to get the optimal solution within a reasonable computation time. We develop a heuristic algorithm, hoping that we can solve these problems in more reasonable time. However, though heuristics can find a near-optimal solution extremely fast compared to the exact approaches, there is still significant variations in performance that can only be explained by the fact that certain structures or properties in graphs may be better suited to some heuristics more than others. Selecting the best algorithm on average across all instances does not help us pick the best one for a particular instance. The need to understand how the best heuristic for a particular class of instance depends on these graph properties is an important issue. In this research, we use data mining to select the best solution strategies for classes of graph coloring instances.

Maximum Face-Constrained Coloring of Plane Graphs

The rapid development of network technology and the advent of various smart devices can cause the rapid growth of network data traffic in the hot areas. However, the existing cellular network capacity cannot meet the needs of network users. Therefore, the small cellular base stations (SCBSs) are proposed and widely used. Nevertheless, the physical cell identifications (PCI) are extremely limited, thus it is difficult to reasonably allocate the limited PCIs to the massive SCBSs, which greatly affect the quality of service (QoS) for users in hot areas. In this paper, a PCI allocation and reuse scheme based on the maximum degree first coloring (MDFC) algorithm is proposed to allocate the limited PCIs for SCBSs in hot areas. Moreover, the K-means clustering algorithm is adopted to divide and identify different hot areas with the mobility data of massive users, which also contributes to the allocation of PCIs. The proposed scheme reduces the rate of PCI conflict and confusion, and improves the users’ QoS. Simulation results show that the MDFC algorithm has better PCI allocation efficiency, and shorter PCI allocation time. Thus, it is more suitable for the deployment of cellular networks in hot areas.

Oriented cliques and colorings of graphs with low maximum degree

We present improved results for approximating maximum-weight independent set (MaxIS) in the CONGEST and LOCAL models of distributed computing. Given an input graph, let n and Δ be the number of nodes and maximum degree, respectively, and let MIS(n,Δ) be the running time of finding a maximal independent set (MIS) in the CONGEST model. Bar-Yehuda et al. [PODC 2017] showed that there is an algorithm in the CONGEST model that finds a Δ-approximation for MaxIS in O(MIS(n,Δ)log W) rounds, where W is the maximum weight of a node in the graph, which can be as large as poly (n). Whether their algorithm is deterministic or randomized that succeeds with high probability depends on the MIS algorithm that is used as a black-box. Our results: 1) A deterministic O(MIS(n,Δ)/e)-round algorithm that finds a (1+e)Δ-approximation for MaxIS in the CONGEST model. 2) A randomized (poly(log log n)/e)-round algorithm that finds, with high probability, a (1+e)Δ-approximation for MaxIS in the CONGEST model. That is, by sacrificing only a tiny fraction of the approximation guarantee, we achieve an exponential speed-up in the running time over the previous best known result. 3) A randomized O(log n⋅ poly(log log n)/e)-round algorithm that finds, with high probability, a 8(1+e)α-approximation for MaxIS in the CONGEST model, where α is the arboricity of the graph. For graphs of arboricity α < Δ/(8(1+e)), this result improves upon the previous best known result in both the approximation factor and the running time. One may wonder whether it is possible to approximate MaxIS with high probability in fewer than poly(log log n) rounds. Interestingly, a folklore randomized ranking algorithm by Boppana implies a single round algorithm that gives an expected Δ-approximation in the CONGEST model. However, it is unclear how to convert this algorithm to one that succeeds with high probability without sacrificing a large number of rounds. For unweighted graphs of maximum degree Δ ≤ n/log n, we show a new analysis of the randomized ranking algorithm, which we combine with the local-ratio technique, to provide a O(1/e)-round algorithm in the CONGEST model that, with high probability, finds an independent set of size at least n/((1+e)(Δ+1)). This result cannot be extended to very high degree graphs, as we show a lower bound of Ω(log^*n) rounds for any randomized algorithm that with probability at least 1-1/log n finds an independent set of size Ω(n/Δ). This lower bound holds even for the LOCAL model. The hard instances that we use to prove our lower bound are graphs of maximum degree Δ = Ω(n/log^*n).

List star edge coloring of sparse graphs

This issue of SICOMP contains six specially selected papers from STOC 2020, the Fifty-second Annual ACM Symposium on the Theory of Computing, which was held June 22--26, 2020, initially planned at Chicago, Illinois, but due to COVID-19 was an online conference in the end. The papers here were chosen to represent the range and quality of the STOC program. These papers have been revised and extended by their authors and subjected to the standard thorough reviewing process of SICOMP. The program committee for STOC 2020 consisted of an executive committee made up of Nima Anari, Boaz Barak, Sébastien Bubeck, Mark Bun, Arkadev Chattopadhyay, Chandra Chekuri, Julia Chuzhoy, Marek Cygan, Ilias Diakonikolas, Yevgeniy Dodis, Sebastian Forster, Ankit Garg, Nika Haghtalab, Prahladh Harsha, Justin Holmgren, Piotr Indyk, Rahul Jain, Sanjeev Khanna, Dakshita Khurana, Pravesh Kothari, Robert Krauthgamer, Marvin Künnemann, Tengyu Ma, Rafael Oliveira, Merav Parter, Sofya Raskhodnikova, Robert Robere, Dana Ron, Noga Ron-Zewi, Thatchaphol Saranurak, Balasubramanian Sivan, Christian Sohler, Madhur Tulsiani, Omri Weinstein, Christian Wulff-Nilsen, and Henry Yuen. The program chair was Julia Chuzhoy. Included in this issue are the following papers: ``Explicit Near-Ramanujan Graphs of Every Degree" by Sidhanth Mohanty, Ryan O'Donnell, and Pedro Paredes shows a deterministic poly$(n)$-time algorithm that outputs a $d$-regular graph on $\Theta(n)$ vertices that is $\epsilon$-near-Ramanujan. ``Reducing Path TSP to TSP" by Vera Traub, Jens Vygen, and Rico Zenklusen presents a black-box reduction from the path version of the traveling salesman problem (Path TSP) to the classical tour version (TSP). ``Nearly Optimal Static Las Vegas Succinct Dictionary" by Huacheng Yu obtains a randomized dictionary data structure using ${OPT}+{poly}\lg n+O(\lg^{(\ell)} U)$ bits of space with expected constant query time for the static dictionary problem. ``Separating the Communication Complexity of Truthful and Nontruthful Algorithms for Combinatorial Auctions" by Sepehr Assadi, Hrishikesh Khandeparkar, Raghuvansh Raj Saxena, and S. Matthew Weinberg provides the first separation in the approximation guarantee achievable by truthful and nontruthful combinatorial auctions with polynomial communication. ``Strong Average-Case Circuit Lower Bounds from Nontrivial Derandomization" by Lijie Chen and Hanlin Ren establish a connection between nondeterministic algorithms estimating the acceptance probability of a given circuit and average-case lower bounds for nondeterministic time classes. ``Improved Bounds for Perfect Sampling of $k$-Colorings in Graphs" by Siddharth Bhandari and Sayantan Chakraborty presents a randomized algorithm that takes as input an undirected $n$-vertex graph $G$ with maximum degree $\Delta$ and an integer $k>3\Delta$ and returns a random proper $k$-coloring of $G$. We thank the authors, the STOC 2020 program committee, the STOC 2020 external reviewers, and the SICOMP referees for all of their hard work. Arkadev Chattopadhyay, Marek Cygan, Noga Ron-Zewi, Christian Wulff-Nilsen - Guest editors

Let $$k\ge 4$$ be an integer, and let G be a graph with maximum degree $$\Delta $$ . In 1991, Alon, McDiarmid and Reed proved that G has a proper coloring using $$O(\Delta ^{(k-1)/(k-2)})$$ colors such that G does not have bichromatic paths with k vertices. In this paper, we improve this result by showing G has a proper coloring using $$(1+\lceil k/2\rceil ^{1/(k-3)})\Delta ^{(k-1)/(k-2)}+\Delta +1$$ colors such that G does not have bichromatic paths with k vertices. We remark that there exists a graph G with maximum degree $$\Delta $$ such that for any proper coloring of G using $$\Omega (\frac{\Delta ^{(k-1)/(k-2)}}{(\log \Delta )^{1/(k-2)}})$$ colors, there is always a bichromatic path with k vertices. Using the similar method, we also show that G has a proper coloring using $$O(\Delta ^{4/3})$$ colors such that G contains neither bichromatic cycles nor bichromatic paths with order 5.

Let G be a graph and let its maximum degree and maximum average degree be denoted by Δ(G) and mad(G), respectively. A neighbor sum distinguishing k-edge colorings of graph G is a proper k-edge coloring of graph G such that, for any edge uv ∈ E(G), the sum of colors assigned on incident edges of u is different from the sum of colors assigned on incident edges of v. The smallest value of k in such a coloring of G is denoted by χ′∑(G). Flandrin et al. proposed the following conjecture that χ′∑ (G) ≤ Δ(G) + 2 for any connected graph with at least 3 vertices and G ≠ C5. In this paper, we prove that the conjecture holds for a normal graph with mad(G) < $$\tfrac{{37}} {{12}}$$ and Δ(G) ≥ 7.

Lokshtanov, Marx, and Saurabh SODA 2011 proved that there is no (k-epsilon)^pw(G)poly(n) time algorithm for deciding if an n-vertex graph G with pathwidth pw admits a proper vertex coloring with k colors unless the Strong Exponential Time Hypothesis (SETH) is false, for any constant epsilon>0. We show here that nevertheless, when k>lfloor Delta/2 rfloor + 1, where Delta is the maximum degree in the graph G, there is a better algorithm, at least when there are few colorings. We present a Monte Carlo algorithm that given a graph G along with a path decomposition of G with pathwidth pw(G) runs in (lfloor Delta/2 rfloor + 1)^pw(G)poly(n)s time, that distinguishes between k-colorable graphs having at most s proper k-colorings and non-k-colorable graphs. We also show how to obtain a k-coloring in the same asymptotic running time. Our algorithm avoids violating SETH for one since high degree vertices still cost too much and the mentioned hardness construction uses a lot of them. We exploit a new variation of the famous Alon--Tarsi theorem that has an algorithmic advantage over the original form. The original theorem shows a graph has an orientation with outdegree less than k at every vertex, with a different number of odd and even Eulerian subgraphs only if the graph is k-colorable, but there is no known way of efficiently finding such an orientation. Our new form shows that if we instead count another difference of even and odd subgraphs meeting modular degree constraints at every vertex picked uniformly at random, we have a fair chance of getting a non-zero value if the graph has few k-colorings. Yet every non-k-colorable graph gives a zero difference, so a random set of constraints stands a good chance of being useful for separating the two cases.

Equitable colorings of planar graphs with maximum degree at least nine

In the vertex cover problem the input is a graph G and an integer k, and the goal is to decide whether there is a set of vertices S of size at most k such that every edge of G is incident on at least one vertex in S. We study the vertex cover problem on graphs with maximum degree 4 and minimum degree at least 2, parameterized by $$r = k-n/3$$ . We give an algorithm for this problem whose running time is $$O^*(1.6253^r)$$ . As a corollary, we obtain an $$O^*(1.2403^k)$$ -time algorithm for vertex cover on graphs with maximum degree 4.