Testing the diameter of graphs

On the core of the multicommodity flow game

General Partitioning on Random Graphs

Robust sunflowers are a generalization of combinatorial sunflowers that have applications in monotone circuit complexity Rossman (SIAM J. Comput. 43:256–279, 2014), DNF sparsification Gopalan et al. (Comput. Complex. 22:275–310 2013), randomness extractors Li et al. (In: APPROX-RANDOM, LIPIcs 116:51:1–13, 2018), and recent advances on the Erdős-Rado sunflower conjecture Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) Lovett et al. (From dnf compression to sunflower theorems via regularity, 2019) Rao (Discrete Anal. 8,2020). The recent breakthrough of Alweiss, Lovett, Wu and Zhang Alweiss et al. (In: Proceedings of the 52nd Annual ACM SIGACT Symposium on Theory of Computing, STOC. Association for Computing Machinery, New York, NY, USA, 2020) gives an improved bound on the maximum size of a w-set system that excludes a robust sunflower. In this paper, we use this result to obtain an exp (n^{1/2-o(1)}) lower bound on the monotone circuit size of an explicit n-variate monotone function, improving the previous best known exp (n^{1/3-o(1)}) due to Andreev (Algebra and Logic, 26:1–18, 1987) and Harnik and Raz (In: Proceedings of the Thirty-Second Annual ACM Symposium on Theory of Computing, ACM, New York, 2000). We also show an exp (varOmega (n)) lower bound on the monotone arithmetic circuit size of a related polynomial via a very simple proof. Finally, we introduce a notion of robust clique-sunflowers and use this to prove an n^{varOmega (k)} lower bound on the monotone circuit size of the CLIQUE function for all k leqslant n^{1/3-o(1)}, strengthening the bound of Alon and Boppana (Combinatorica, 7:1–22, 1987).

Testing Eulerianity and connectivity in directed sparse graphs

Consider property testing on bounded degree graphs and let $\\varepsilon>0$\ndenote the proximity parameter. A remarkable theorem of Newman-Sohler (SICOMP\n2013) asserts that all properties of planar graphs (more generally hyperfinite)\nare testable with query complexity only depending on $\\varepsilon$. Recent\nadvances in testing minor-freeness have proven that all additive and monotone\nproperties of planar graphs can be tested in $poly(\\varepsilon^{-1})$ queries.\nSome properties falling outside this class, such as Hamiltonicity, also have a\nsimilar complexity for planar graphs. Motivated by these results, we ask: can\nall properties of planar graphs can be tested in $poly(\\varepsilon^{-1})$\nqueries? Is there a uniform query complexity upper bound for all planar\nproperties, and what is the "hardest" such property to test? We discover a\nsurprisingly clean and optimal answer. Any property of bounded degree planar\ngraphs can be tested in $\\exp(O(\\varepsilon^{-2}))$ queries. Moreover, there is\na matching lower bound, up to constant factors in the exponent. The natural\nproperty of testing isomorphism to a fixed graph needs\n$\\exp(\\Omega(\\varepsilon^{-2}))$ queries, thereby showing that (up to\npolynomial dependencies) isomorphism to an explicit fixed graph is the hardest\nproperty of planar graphs. The upper bound is a straightforward adapation of\nthe Newman-Sohler analysis that tracks dependencies on $\\varepsilon$ carefully.\nThe main technical contribution is the lower bound construction, which is\nachieved by a special family of planar graphs that are all mutually far from\neach other. We can also apply our techniques to get analogous results for\nbounded treewidth graphs. We prove that all properties of bounded treewidth\ngraphs can be tested in $\\exp(O(\\varepsilon^{-1}\\log \\varepsilon^{-1}))$\nqueries. Moreover, testing isomorphism to a fixed forest requires\n$\\exp(\\Omega(\\varepsilon^{-1}))$ queries.\n

We present a deterministic $O(\log n \log \log n)$ space algorithm for undirected st-connectivity. It is based on a space-efficient simulation of the deterministic EREW algorithm of Chong and Lam [J. Algorithms, 18 (1995), pp. 378-402], an approach suggested by Prof. Vijaya Ramachandran, and uses the universal exploration sequences for trees constructed by Koucký in [Proceedings of the 16th Annual IEEE Conference on Computational Complexity, 2001, pp. 21-27]. Our result improves the $O(\log^{4/3} n)$ bound of Armoni et al. in [Proceedings of the 20th Annual ACM Symposium on Theory of Computing, 1997, pp. 230-239] and is a big step towards the optimal $O(\log n)$. Independently of our result and using a different set of techniques, the optimal bound was achieved by Reingold in [Proceedings of the 37th Annual ACM Symposium on Theory of Computing, 2005, pp. 376-385].

Generating all maximal induced subgraphs for hereditary and connected-hereditary graph properties

The distributed $(\Delta + 1)$-coloring problem is one of the most fundamental and well-studied problems in distributed algorithms. Starting with the work of Cole and Vishkin in 1986, a long line of gradually improving algorithms has been published. The state-of-the-art running time, prior to our work, is $O(\Delta \log \Delta + \log^* n)$, due to Kuhn and Wattenhofer [Proceedings of the $25$th Annual ACM Symposium on Principles of Distributed Computing, Denver, CO, 2006, pp. 7--15]. Linial [Proceedings of the $28$th Annual IEEE Symposium on Foundation of Computer Science, Los Angeles, CA, 1987, pp. 331--335] proved a lower bound of $\frac{1}{2} \log^* n$ for the problem, and Szegedy and Vishwanathan [Proceedings of the 25th Annual ACM Symposium on Theory of Computing, San Diego, CA, 1993, pp. 201--207] provided a heuristic argument that shows that algorithms from a wide family of locally iterative algorithms are unlikely to achieve a running time smaller than $\Theta(\Delta \log \Delta)$. We present a deterministic $(\Delta + 1)$-coloring distributed algorithm with running time $O(\Delta) + \frac{1}{2} \log^* n$. We also present a trade-off between the running time and the number of colors, and devise an $O(\lambda\cdot\Delta)$-coloring algorithm, with running time $O(\Delta / \lambda + \log^* n)$, for any parameter $\lambda > 1$. Our algorithm breaks the heuristic barrier of Szegedy and Vishwanathan and achieves running time which is linear in the maximum degree $\Delta$. On the other hand, the conjecture of Szegedy and Vishwanathan may still be true, as our algorithm does not belong to the family of locally iterative algorithms. On the way to this result we study a generalization of the notion of graph coloring, which is called defective coloring [L. Cowen, R. Cowen, and D. Woodall, J. Graph Theory, 10 (1986), pp. 187--195]. In an $m$-defective $p$-coloring the vertices are colored with $p$ colors so that each vertex has up to $m$ neighbors with the same color. We show that an $m$-defective $p$-coloring with reasonably small $m$ and $p$ can be computed very efficiently in the distributed setting. We also develop a technique to employ multiple defective colorings of various subgraphs of the original graph $G$ for computing a $(\Delta+1)$-coloring of $G$. We believe that these techniques are of independent interest.

Locally Checkable Labeling (LCL) problems include essentially all the classic problems of LOCAL distributed algorithms. In a recent enlightening revelation, Chang and Pettie [FOCS'17] showed that any LCL (on bounded degree graphs) that has an o(log n)-round randomized algorithm can be solved in T_(LLL)(n) rounds, which is the randomized complexity of solving (a relaxed variant of) the Lovasz Local Lemma (LLL) on bounded degree n-node graphs. Currently, the best known upper bound on T_(LLL)(n) is O(log n), by Chung, Pettie, and Su [PODC'14], while the best known lower bound is Omega(log log n), by Brandt et al. [STOC'16]. Chang and Pettie conjectured that there should be an O(log log n)-round algorithm (on bounded degree graphs). Making the first step of progress towards this conjecture, and providing a significant improvement on the algorithm of Chung et al. [PODC'14], we prove that T_(LLL)(n)= 2^O(sqrt(log log n)). Thus, any o(log n)-round randomized distributed algorithm for any LCL problem on bounded degree graphs can be automatically sped up to run in 2^O(sqrt(log log n)) rounds. Using this improvement and a number of other ideas, we also improve the complexity of a number of graph coloring problems (in arbitrary degree graphs) from the O(log n)-round results of Chung, Pettie and Su [PODC'14] to 2^O(sqrt(log log n)). These problems include defective coloring, frugal coloring, and list vertex-coloring.

For general antiferromagnetic 2-spin systems, including the hardcore model on weighted independent sets and the antiferromagnetic Ising model, there is an for the partition function on graphs of maximum degree when the infinite regular tree lies in the uniqueness region by Li, Lu, and Yin [Correlation Decay up to Uniqueness in Spin Systems, preprint, https://arxiv.org/abs/1111.7064, 2021]. Moreover, in the tree nonuniqueness region, Sly in [Computational transition at the uniqueness threshold, in Proceedings of the 51st Annual IEEE Symposium on Foundations of Computer Science, 2010, pp. 287–296] showed that there is no to estimate the partition function unless . The algorithmic results follow from the correlation decay approach due to Weitz [Counting independent sets up to the tree threshold, in Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 140–149] or the polynomial interpolation approach developed by Barvinok [Combinatorics and Complexity of Partition Functions, Springer, 2016]. However, the running time is only polynomial for constant . For the hardcore model, recent work of Anari, Liu, and Oveis Gharan [Spectral independence in high-dimensional expanders and applications to the hardcore model, in Proceedings of the 61st Annual IEEE Symposium on Foundations of Computer Science, 2020, pp. 1319–1330] establishes rapid mixing of the simple single-site Markov chain, known as the Glauber dynamics, in the tree uniqueness region. Our work simplifies their analysis of the Glauber dynamics by considering the total pairwise influence of a fixed vertex on other vertices, as opposed to the total influence of other vertices on , thereby extending their work to all 2-spin models and improving the mixing time. More important, our proof ties together the three disparate algorithmic approaches: we show that contraction of the so-called tree recursions with a suitable potential function, which is the primary technique for establishing efficiency of Weitz’s correlation decay approach and Barvinok’s polynomial interpolation approach, also establishes rapid mixing of the Glauber dynamics. We emphasize that this connection holds for all 2-spin models (both antiferromagnetic and ferromagnetic), and existing proofs for the correlation decay and polynomial interpolation approaches immediately imply rapid mixing of the Glauber dynamics. Our proof utilizes the fact that the graph partition function is a divisor of the partition function for Weitz’s self-avoiding walk tree. This fact leads to new tools for the analysis of the influence of vertices and may be of independent interest for the study of complex zeros.

Scheduling multicasts on unit-capacity trees and meshes

The papers in this volume were presented at the 43rd ACM Symposium on Theory of Computing (STOC), held June 6-8, 2011 in San Jose, California, as part of the fifth Federated Computing Research Conference (FCRC). STOC is sponsored by the ACM Special Interest Group on Algorithms and Computation Theory (SIGACT). The papers at STOC were selected by the program committee from among 304 submissions received by November 4, 2010 (5 of which were withdrawn by authors during the review process). The committee did its work by a combination of extensive electronic discussion and a physical meeting of the entire committee in San Francisco on January 26-27, 2011. These proceedings contain the 84 papers that were accepted to STOC, one of which is a merge of two related submissions. Three pairs of papers having similar results shared presentation slots at the conference, for a total of 81 talks. All submissions received careful consideration, but they were not refereed in a formal sense. Authors are encouraged to submit full versions of their papers to journals for a complete review and archival publication. The committee selected two papers for a Best Paper Award: "Electrical Flows, Laplacian Systems, and Faster Approximation of Maximum Flow in Undirected Graphs," by Paul Christiano, Jonathan A. Kelner, Aleksander Madry, Daniel A. Spielman, and Shang-Hua Teng, and "Subexponential Lower Bounds for Randomized Pivoting Rules for the Simplex Algorithm," by Oliver Friedmann, Thomas Dueholm Hansen, and Uri Zwick. The committee selected "Analyzing Network Coding Gossip Made Easy" by Bernhard Haeupler for the Danny Lewin Best Student Paper Award. STOC was preceded by an FCRC plenary talk on June 5 given by Leslie Valiant as recipient of the 2011 ACM Turing Award for his transformative contributions to the theory of computation, including the theory of probably approximately correct (PAC) learning, the complexity of enumeration and of algebraic computation, and the theory of parallel and distributed computing. At the conference, the 11th Knuth Prize was awarded to Ravi Kannan for his work on algorithmic techniques that have fundamentally contributed to computational complexity, discrete mathematics, geometry, and operations research. He presented the Knuth Prize Lecture as an FCRC plenary talk on the morning of June 7. The program committee also decided to add a Poster Session, held on the evening of June 6, to increase the opportunities for disseminating and discussing research at the conference. Posters were not refereed; any registered FCRC attendee could present a poster on research in any aspect of theoretical computer science.

Nonlinear codes from points of bounded height

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

We study the asymptotic growth of the diameter of a graph obtained by adding sparse “long” edges to a square box in \documentclass{article} \usepackage{amsmath,amsfonts}\pagestyle{empty}\begin{document} ${\mathbb Z}^d$ \end{document}. We focus on the cases when an edge between x and y is added with probability decaying with the Euclidean distance as |x − y|−s+o(1) when |x − y| → ∞. For s ∈ (d, 2d) we show that the graph diameter for the graph reduced to a box of side L scales like (log L)Δ+o(1) where Δ−1 := log2(2d/s). In particular, the diameter grows about as fast as the typical graph distance between two vertices at distance L. We also show that a ball of radius r in the intrinsic metric on the (infinite) graph will roughly coincide with a ball of radius exp{r1/Δ+o(1)} in the Euclidean metric. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 39, 210‐227, 2011