| Date and Time | Speaker and Title | Abstract | Reference |
|---|---|---|---|
| 11th of November 14:15 (Helsinki time) |
Przemek Uznański: Cardinality estimation using Gumbel distribution |
Cardinality estimation is the task of approximating the number of distinct elements in a large dataset with possibly repeating elements. LogLog and HyperLogLog (c.f. Durand and Flajolet [ESA 2003], Flajolet et al. [Discrete Math Theor. 2007]) are small space sketching schemes for cardinality estimation, which have both strong theoretical guarantees of performance and are highly effective in practice. This makes them a highly popular solution with many implementations in big-data systems (e.g. Algebird, Apache DataSketches, BigQuery, Presto and Redis). However, despite having simple and elegant formulation, both the analysis of LogLog and HyperLogLog are extremely involved -- spanning over tens of pages of analytic combinatorics and complex function analysis. We propose a modification to both LogLog and HyperLogLog that replaces discrete geometric distribution with a continuous Gumbel distribution. This leads to a very short, simple and elementary analysis of estimation guarantees, and smoother behavior of the estimator. |
ArXiv |
| 28th of October 2020 14:15 (Helsinki time) |
Philipp Schneider: Shortest Paths in the HYBRID Network Model |
The HYBRID model, introduced in [Augustine et al., SODA '20], provides a theoretical foundation for networks that allow multiple communication modes. The model follows the principles of synchronous message passing, where nodes are allowed to use two fundamentally different communication modes. First, a local mode where nodes may exchange arbitrary information per round over edges of a local communication graph G (akin to the LOCAL model). Second, a global mode where every node may exchange log n messages of size O(log n) bits per round with arbitrary nodes in the network. The HYBRID model intends to reflect the conditions of many real hybrid networks, where high-bandwidth but inherently local communication is combined with highly flexible global communication with restricted bandwidth. We explore the power and limitations of the HYBRID model by investigating the complexity of computing shortest paths of the local communication graph G. The aim of the talk is to give an overview of the known techniques for information dissemination in the HYBRID model. Subsequently, we will look into how these techniques can be used to obtain algorithmic upper bounds for various shortest paths problems and how these compare to the known lower bounds. As a sideffect we will also learn how the HYBRID model is related to other models of computation. |
ArXiv |
| 21th of October 2020 14:15 (Helsinki time) |
Darya Melnyk: TBA |
TBA | |
| 7th of October 2020 14:15 (Helsinki time) |
Aleksander Łukasiewicz: All-Pairs LCA in DAGs: Breaking through the O(n^2.5) barrier |
Let G=(V,E) be an n-vertex directed acyclic graph (DAG). A lowest common ancestor (LCA) of two vertices u and v is a common ancestor w of u and v such that no descendant of w has the same property. In this paper, we consider the problem of computing an LCA, if any, for all pairs of vertices in a DAG. The fastest known algorithms for this problem exploit fast matrix multiplication subroutines and have running times ranging from O(n^2.687) [Bender et al. SODA'01] down to O(n^2.615) [Kowaluk and Lingas~ICALP'05] and O(n^2.569) [Czumaj et al. TCS'07]. Somewhat surprisingly, all those bounds would still be Ω(n^2.5) even if matrix multiplication could be solved optimally (i.e., ω=2). This appears to be an inherent barrier for all the currently known approaches, which raises the natural question on whether one could break through the O(n^2.5) barrier for this problem. In this paper, we answer this question affirmatively: in particular, we present an O(n^2.447) (O(n^7/3) for ω=2) algorithm for finding an LCA for all pairs of vertices in a DAG, which represents the first improvement on the running times for this problem in the last 13 years. A key tool in our approach is a fast algorithm to partition the vertex set of the transitive closure of G into a collection of O(ℓ) chains and O(n/ℓ) antichains, for a given parameter ℓ. As usual, a chain is a path while an antichain is an independent set. We then find, for all pairs of vertices, a candidate} LCA among the chain and antichain vertices, separately. The first set is obtained via a reduction to min-max matrix multiplication. The computation of the second set can be reduced to Boolean matrix multiplication similarly to previous results on this problem. We finally combine the two solutions together in a careful (non-obvious) manner. To appear in SODA 2021. |
ArXiv |
| 30th September 2020 14:15 (Helsinki time) |
Julian Portmann: Tight Bounds for Deterministic High-Dimensional Grid Exploration |
We study the problem of exploring an oriented grid with autonomous agents governed by finite automata. In the case of a 2-dimensional grid, the question how many agents are required to explore the grid, or equivalently, find a hidden treasure in the grid, is fully understood in both the synchronous and the semi-synchronous setting. For higher dimensions, Dobrev, Narayanan, Opatrny, and Pankratov [ICALP'19] showed very recently that, surprisingly, a (small) constant number of agents suffices to find the treasure, independent of the number of dimensions, thereby disproving a conjecture by Cohen, Emek, Louidor, and Uitto [SODA'17]. Dobrev et al. left as an open question whether their bounds on the number of agents can be improved. We answer this question in the affirmative for deterministic finite automata: we show that 3 synchronous and 4 semi-synchronous agents suffice to explore an n-dimensional grid for any constant n. The bounds are optimal and notably, the matching lower bounds already hold in the 2-dimensional case. Our techniques can also be used to make progress on other open questions asked by Dobrev et al.: we prove that 4 synchronous and 5 semi-synchronous agents suffice for polynomial-time exploration, and we show that, under a natural assumption, 3 synchronous and 4 semi-synchronous agents suffice to explore unoriented grids of arbitrary dimension (which, again, is tight). |
ArXiv |
| Date and Time | Speaker and Title | Abstract | Reference |
|---|---|---|---|
| 16th September 2020 14:15 (Helsinki time) |
Janne Korhonen: Input-dynamic distributed graph algorithms for congested networks |
Consider a distributed system, where the topology of the communication network remains fixed, but local inputs given to nodes may change over time. In this work, we explore the following question: if some of the local inputs change, can an existing solution be updated efficiently, in a dynamic and distributed manner?
To address this question, we define the batch dynamic CONGEST model, where the communication network G=(V,E) remains fixed and a dynamic edge labelling defines the problem input. The task is to maintain a solution to a graph problem on the labeled graph under batch changes. We investigate, when a batch of α
edge label changes arrive, — how much time as a function of α we need to update an existing solution, and — how much information the nodes have to keep in local memory between batches in order to update the solution quickly. We give a general picture of the complexity landscape in this model, including a general framework for lower bounds. In particular, we prove non-trivial upper bounds for two selected, contrasting problems: maintaining a minimum spanning tree and detecting cliques. |
ArXiv |
| 9nd September 2020 14:15 (Helsinki time) |
Yannic Maus: Distributed Graph Coloring: Linial for Lists |
Linial's famous color reduction algorithm reduces a given m-coloring of a graph with maximum degree Δ to a O(Δ² log m)-coloring, in a single round in the LOCAL model. We show a similar result when nodes are restricted to choose their color from a list of allowed colors: given an m-coloring in a directed graph of maximum outdegree β, if every node has a list of size Ω(β² (log β + log log m + log log |C|)) from a color space C then they can select a color in two rounds in the LOCAL model. Moreover, the communication of a node essentially consists of sending its list to the neighbors. This is obtained as part of a framework that also contains Linial's color reduction (with an alternative proof) as a special case. Our result also leads to a defective list coloring algorithm. As a corollary, we improve the state-of-the-art truly local (deg+1)-list coloring algorithm from Barenboim et al. [PODC'18] by slightly reducing the runtime to O(sqrt(Δ log Δ) + log* n) and significantly reducing the message size (from huge to roughly Δ). Our techniques are inspired by the local conflict coloring framework of Fraigniaud et al. [FOCS'16]. | ArXiv |
| 26th August 2020 14:15 (Helsinki time) |
Ebrahim Ghorbani: Spectral gap of regular graphs and a conjecture by Aldous-Fill on the relaxation time of the random walk |
Aldous and Fill conjectured that the maximum relaxation time for the random walk on a connected regular graph with n vertices is (1+o(1)) \frac{3n^2}{2\pi^2}. This conjecture can be rephrased in terms of the spectral gap as follows: the spectral gap (algebraic connectivity) of a connected k-regular graph on n vertices is at least (1+o(1))\frac{2k\pi^2}{3n^2}, and the bound is attained for at least one value of k. Brand, Guiduli, and Imrich determined the structure of connected cubic graphs on n vertices with minimum spectral gap. We investigate the structure of quartic (i.e.~4-regular) graphs with the minimum spectral gap among all connected quartic graphs. We show that they must have a path-like structure built from specific blocks. Based on these results, we prove the Aldous--Fill conjecture follows for k=3 and 4. This talk is based on joint works with M. Abdi and W. Imrich. | |
| 19th August 2020 14:15 (Helsinki time) |
Klaus-Tycho Förster: On the Feasibility of Perfect Resilience with Local Fast Failover |
In order to provide a high resilience and to react quickly to link failures, modern computer networks support fully decentralized flow rerouting, also known as local fast failover. In a nutshell, the task of a local fast failover algorithm is to pre-define fast failover rules for each node using locally available information only. These rules determine for each incoming link from which a packet may arrive and the set of local link failures (i.e., the failed links incident to a node), on which outgoing link a packet should be forwarded. Ideally, such a local fast failover algorithm provides a perfect resilience deterministically: a packet emitted from any source can reach any target, as long as the underlying network remains connected. Feigenbaum et al. showed that it is not always possible to provide perfect resilience and showed how to tolerate a single failure in any network. Interestingly, not much more is known currently about the feasibility of perfect resilience. This paper revisits perfect resilience with local fast failover, both in a model where the source can and cannot be used for forwarding decisions. We first derive several fairly general impossibility results: By establishing a connection between graph minors and resilience, we prove that it is impossible to achieve perfect resilience on any non-planar graph; furthermore, while planarity is necessary, it is also not sufficient for perfect resilience. On the positive side, we show that graph families which are closed under the subdivision of links, can allow for simple and efficient failover algorithms which simply skip failed links. We demonstrate this technique by deriving perfect resilience for outerplanar graphs and related scenarios, as well as for scenarios where the source and target are topologically close after failures. |
Paper |
| 12th August 2020 14:15 (Helsinki time) |
Chris Brzuska: On Building Fine-Grained One-Way Functions from Strong Average-Case Hardness |
Constructing one-way function from average-case hardness is a long-standing open problem
A positive result would exclude Pessiland (Impagliazzo '95) and establish a highly desirable win-win situation: Either (symmetric) cryptography exists unconditionally, enabling many of the important primitives which are used to secure our communications, or all NP problems can be solved efficiently on average, which would be a revolution in algorithms. Motivated by the interest of establishing such a win-win result and the lack of progress on this seemingly very hard question, we initiate the investigation of weaker yet meaningful candidate win-win results. Specifically, we study the following type of win-win results: either there are fine-grained one-way functions (FGOWF), which relax the standard notion of a one-way function by requiring only a fixed polynomial gap (as opposed to superpolynomial) between the running time of the function and the running time of an inverter, or nontrivial speedups can be obtained for all NP problems on average. We obtain three main results: - We introduce the random language model (RLM), which captures idealized average-case hard languages, analogous to how the random oracle model captures idealized one-way functions. We provide a construction of a FGOWF (with quadratic hardness gap) and prove its security in the RLM. This rules out an idealized version of Pessiland, where ideally hard languages would exist yet even weak forms of cryptography would fail. - On the negative side, we prove a strong oracle separation: we show that there is no black-box proof that either FGOWF exist, or non-trivial speedup can be obtained for all NP languages on average (i.e., there is no exponentially average-case hard NP languages). - We provide a second strong negative result for an even weaker candidate win-win result: there is no black-box proof that either FGOWF exist, or non-trivial speedups can be obtained for all NP languages on average when amortizing over many instances (i.e., there is no exponentially average-case hard NP languages whose hardness amplifies optimally through parallel repetitions). This separation forms the core technical contribution of our work. Our results lay the foundations for a program towards building fine-grained one-way functions from strong forms of average-case hardness, following the template of constructions in the random language model. We provide a preliminary investigation of this program, showing black-box barriers toward instantiating our idealized constructions from natural hardness properties. Joint work with Geoffroy Couteau. |
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| 29th July 2020 14:15 (Helsinki time) |
Davin Choo: k-means++: few more steps yield constant approximation |
The k-means++ algorithm of Arthur and Vassilvitskii (SODA 2007) is a state-of-the-art algorithm for solving the k-means clustering problem and is known to give an O(log k)-approximation in expectation. Recently, Lattanzi and Sohler (ICML 2019) proposed augmenting k-means++ with O(k log log k) local search steps to yield a constant approximation (in expectation) to the k-means clustering problem. In this paper, we improve their analysis to show that, for any arbitrarily small constant $\eps > 0$, with only $\eps k$ additional local search steps, one can achieve a constant approximation guarantee (with high probability in k), resolving an open problem in their paper. | Paper |
| 22nd July 2020 14:15 (Helsinki time) |
Yuval Efron: Beyond Alice and Bob: Improved Inapproximability for Maximum Independent Set in CONGEST |
By far the most fruitful technique for showing lower bounds for the CONGEST model is reductions to two-party communication complexity. This technique has yielded nearly tight results for various fundamental problems such as distance computations, minimum spanning tree, minimum vertex cover, and more.
In this work, we take this technique a step further, and we introduce a framework of reductions to t-party communication complexity, for every t≥2. Our framework enables us to show improved hardness results for maximum independent set. Recently, Bachrach et al.[PODC 2019] used the two-party framework to show hardness of approximation for maximum independent set. They show that finding a (5/6+ϵ)-approximation requires Ω(n/log6n) rounds, and finding a (7/8+ϵ)-approximation requires Ω(n2/log7n) rounds, in the CONGEST model where n in the number of nodes in the network.
We improve the results of Bachrach et al. by using reductions to multi-party communication complexity. Our results: (1) Any algorithm that finds a (1/2+ϵ)-approximation for maximum independent set in the CONGEST model requires Ω(n/log3n) rounds. (2) Any algorithm that finds a (3/4+ϵ)-approximation for maximum independent set in the CONGEST model requires Ω(n2/log3n) rounds. |
Paper |
| 15th July 2020 14:15 (Helsinki time) |
Yi-Jun Chang: Simple Contention Resolution via Multiplicative Weight Updates |
We consider the classic contention resolution problem, in which devices conspire to share some common resource, for which they each need temporary and exclusive access. To ground the discussion, suppose (identical) devices wake up at various times, and must send a single packet over a shared multiple-access channel. In each time step they may attempt to send their packet; they receive ternary feedback {0,1,2^+} from the channel, 0 indicating silence (no one attempted transmission), 1 indicating success (one device successfully transmitted), and 2^+ indicating noise. We prove that a simple strategy suffices to achieve a channel utilization rate of 1/e-O(epsilon), for any epsilon > 0. In each step, device i attempts to send its packet with probability p_i, then applies a rudimentary multiplicative weight-type update to p_i. p_i ← { p_i * e^{epsilon} upon hearing silence (0), p_i upon hearing success (1), p_i * e^{-epsilon/(e-2)} upon hearing noise (2^+) }. This scheme works well even if the introduction of devices/packets is adversarial, and even if the adversary can jam time slots (make noise) at will. We prove that if the adversary jams J time slots, then this scheme will achieve channel utilization 1/e-epsilon, excluding O(J) wasted slots. Results similar to these (Bender, Fineman, Gilbert, Young, SODA 2016) were already achieved, but with a lower constant efficiency (less than 0.05) and a more complex algorithm. | Paper |
| 8th July 2020 14:15 (Helsinki time) |
Diksha Gupta: Sybil Defense in the Presence of Churn Using Resource Burning |
In this talk, I will begin by discussing a recent result for defending against Sybil attacks in dynamic permissionless systems, in the presence of a computationally bounded adversary - TOtal Good COMputation (ToGCom). This technique guarantees a majority of honest identities (IDs) at all times, at a computational cost to the protocol that is comparable to that of the attacker. Next, I will talk about the concept of resource burning - the verifiable consumption of resources. This resource does not necessarily need to be computational power, but can also be communication capacity, computer memory, and human effort, subject to some constraints. Using this insight, I will conclude with a generalizing of our Sybil defense techniques to a variety of systems with churn, in the presence of a resource-bounded adversary. | |
| 1st July 2020 14:15 (Helsinki time) |
Sebastian Brandt: Lower Bounds for Ruling Sets in the LOCAL Model |
Given a graph G = (V,E), an (a,b)-ruling set is a subset S of V such
that the distance between any two vertices in S is at least a, and the
distance between any vertex in V and the closest vertex in S is at most
b. Ruling sets are a generalization of maximal independent sets (which
are (2,1)-ruling sets) and constitute an important building block of
many distributed algorithms. The recent breakthrough on network
decompositions by Rozhon and Ghaffari [STOC'20] implies that, in the
distributed LOCAL model, ruling sets can be computed deterministically
in polylogarithmic time, for a wide range of parameters a, b. In this talk, we present polylogarithmic lower bounds for the deterministic computation of ruling sets, and Omega(poly(log log n)) lower bounds for the randomized computation of ruling sets, both in the LOCAL model, improving on the previously best known lower bounds of Omega(log*n)) by Linial [FOCS'87] and Naor [J.Disc.Math.'91]. In the special case of maximal independent sets, such lower bounds were already known; however, our lower bounds are the first (beyond Omega(log*n)) that are also applicable on trees. |
Paper |
| 17th June 2020 14:15 (Helsinki time) |
Frederik Mallman-Trenn: Finding the best papers with noisy reviews |
Say you are tasked to find the best 150 papers among more than 550 papers.
You can ask people to review a given paper by either asking 1) Is paper A better than paper B or 2) What’s the score of paper A? The problem is that each review returns an incorrect response with a small probability, say 2/3. How can you assign reviews so that you the total number of queries is small and the number of review rounds is small? |
Paper |
| 10th June 2020 14:15 (Helsinki time) |
Joachim Spoerhase: Approximation Algorithms for some Submodular and Graph-based Optimization Problems |
In this talk, I will give an overview over three recent results on approximation algorithms for discrete optimization problems.
The first result concerns optimizing a monotone submodular function with respect to an arbitrary (constant) number of packing and covering constraints. We give a randomized polynomial-time approximation algorithm that is essentially tight with respect to approximation ratio, running time, and constraint violation. The algorithm is based on rounding a multi-linear relaxation of the problem. We also discuss special cases where we can give *deterministic* algorithms based on a novel combinatorial approach.
The second set of results concerns optimization problems for graphs. We will briefly look at a polynomial-time approximation scheme (PTAS) for Steiner tree on map graphs, which generalizes a known PTAS planar (edge-weighted) Steiner tree, and which is motivated by the study of planar node-weighted Steiner trees. We will also touch upon a result for the sparsest cut problem in tree-width bounded graphs. REMARK: This talk is given as a memorial for Sumedha Uniyal who was a postdoc at Aalto for 3 years until she passed away on February 19, 2020. I will give an overview over our three joint results. Additional contributors include Jaroslaw Byrka (Wroclaw), Parinya Chalermsook (Aalto), Mateusz Lewandowski (Wroclaw), Syed Mohammad Meesum (Wroclaw), Eyal Mizrachi (Technion), Matthias Mnich (TU Hamburg) Roy Schwartz (Technion), Daniel Vaz (TU Munich) |
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| 3rd June 2020 14:15 (Helsinki time) |
Parinya Chalermsook: Some New Algorithmic Min-Max Relations via Local Search |
The study of min-max relations has been a cornerstone in combinatorial optimization. Classic examples include, for instance, LP duality relations, Tutte-Berge formula, and the notion of perfect graphs. These relations have yielded many ground-breaking algorithmic results. Algorithmic uses of these bounds often involve two steps: (1) Prove a non-constructive bound and (2) Use another technique (e.g. convex programs) to turn the first step algorithmic. In this talk, I will report our recent use of local search arguments to derive two new algorithmic min-max relations in one go. In these results, a bound is proved by showing that a locally optimal solution must satisfy such bound, therefore yielding immediately an efficient algorithm.
REMARK: This talk is given as a memorial for Sumedha Uniyal who was a postdoc at Aalto for 3 years until she passed away on February 19, 2020. She played a central role in this project. Additional contributors include Samir Khuller (Northwestern), Andreas Schmid (Aalto), and Pattara Sukprasert (Northwestern). |
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| 27th May 2020 14:15 (Helsinki time) |
Krzysztof Nowicki: Faster Algorithms for Edge Connectivity via Random 2-Out Contractions |
We provide a simple new randomized contraction approach to the global minimum cut problem for simple undirected graphs. The contractions exploit 2-out edge sampling from each vertex rather than the standard uniform edge sampling. Our new approach yields better algorithms for sequential, distributed, and parallel models of computation. | Paper |
| 20th May 2020 15:15 (Helsinki time, one hour later than usually!) |
Jan Studený: On the Hardness of Classifying Distributed Complexity of Graph Problems on Paths, Cycles, and Trees |
Given the description of a locally checkable graph problem Π for paths or cycles or trees, find out how hard is to determine what the distributed complexity of solving Π in the usual LOCAL model of distributed computing. In this talk I will present our results on answering this question for paths and cycles and discuss the case of trees. | Paper |
| 13th May 2020 14:15 (Helsinki time) |
Manuela Fischer: Local Glauber Dynamics |
Sampling constitutes an important tool in a variety of areas: from machine learning and combinatorial optimization to computational physics and biology. A central class of sampling algorithms is the Markov Chain Monte Carlo method, based on the construction of a Markov chain with the desired sampling distribution as its stationary distribution. Many of the traditional Markov chains, such as the Glauber dynamics, do not scale well with increasing dimension. To address this shortcoming, we propose a simple local update rule based on the Glauber dynamics that leads to efficient parallel and distributed algorithms for sampling from Gibbs distributions. Concretely, we present a Markov chain that mixes in O(logn) rounds when Dobrushin's condition for the Gibbs distribution is satisfied. This improves over the LubyGlauber algorithm by Feng, Sun, and Yin [PODC'17], which needs O(Δ log n) rounds, and their LocalMetropolis algorithm, which converges in O(log n) rounds but requires a considerably stronger mixing condition. Here, n denotes the number of nodes in the graphical model inducing the Gibbs distribution, and Δ its maximum degree. In particular, our method can sample a uniform proper coloring with α Δ colors in O(log n) rounds for any α > 2, which almost matches the threshold of the sequential Glauber dynamics and improves on the α > 2+sqrt(2)- threshold of Feng et al. | Paper |
| 6th May 2020 14:15 (Helsinki time) |
Jesper Nederlof: Bipartite TSP in O(1.9999^n) Time, Assuming Quadratic Time Matrix Multiplication |
The symmetric traveling salesman problem (TSP) is the problem of
finding the shortest Hamiltonian cycle in an edge-weighted undirected
graph. In 1962 Bellman, and independently Held and Karp, showed that
TSP instances with n cities can be solved in O(n^2*2^n) time. Since
then it has been a notorious problem to improve the runtime to
O((2-eps)^n) for some constant eps >0. In this work we establish the
following progress: If (s x s)-matrices can be multiplied in
s^{2+o(1)} time, than all instances of TSP in bipartite graphs can be
solved in O(1.9999^n) time by a randomized algorithm with constant
error probability. We also indicate how our methods may be useful to
solve TSP in non-bipartite graphs. On a high level, our approach is via a new problem called the MINHAMPAIR problem: Given two families of weighted perfect matchings, find a combination of minimum weight that forms a Hamiltonian cycle. As our main technical contribution, we give a fast algorithm for MINHAMPAIR based on a new sparse cut-based factorization of the `matchings connectivity matrix', introduced by Cygan et al. [JACM'18]. |
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| 29th April 2020 14:15 (Helsinki time) |
Sorrachai Yingchareonthawornchai: Computing and Testing Small Connectivity in Near-Linear Time and Queries via Fast Local Cut Algorithms |
Consider the following "local" cut-detection problem in a directed graph: We are given a seed vertex x and need to remove at most k edges so that at most ν edges can be reached from x (a "local" cut) or output ⊥ to indicate that no such cut exists. If we are given query access to the input graph, then this problem can in principle be solved without reading the whole graph and with query complexity depending on k and ν. In this talk, I will present a simple randomized algorithm spending O(νk^2) time and O(kν) queries for the slight variant of the above problem, improving in particular a previous time bound of O(k^O(k) ν) by Chechik et al. [SODA '17]. I will then present two key applications of the local cut algorithm. The first application is a fast randomized algorithm for the classic k-vertex connectivity problem that takes near-linear time when k = O(polylog(n)). Second is property testing algorithms for k-edge and -vertex connectivity with query complexities that are near-linear in k, exponentially improving the state-of-the-art. This resolves two open problems, one by Goldreich and Ron [STOC '97] and one by Orenstein and Ron [Theor. Comput Sci. '11]. | Paper |
| 22th April 2020 15:15 (Helsinki time |
Václav Rozhoň: Polylogarithmic-Time Deterministic Network Decomposition and Distributed Derandomization |
We present a simple polylogarithmic-time deterministic distributed algorithm for network decomposition. This improves on a celebrated 2^O(sqrt(log n))-time algorithm of Panconesi and Srinivasan [STOC'92] and settles a long-standing question in distributed graph algorithms. It also leads to the first polylogarithmic-time deterministic distributed algorithms for numerous other problems, hence resolving several well-known open problems, including Linial's question about the deterministic complexity of maximal independent set [FOCS'87; SICOMP'92]. By known connections, our result leads also to substantially faster randomized distributed algorithms for a number of well-studied problems including (Delta+1)-coloring, maximal independent set, and Lovász Local Lemma, as well as massively parallel algorithms for (Delta+1)-coloring. | Paper |
| 15th April 2020 14:15 (Helsinki time |
Przemek Uznański: Hardness of Exact Distance Queries in Sparse Graphs Through Hub Labeling |
A distance labeling scheme is an assignment of bit-labels to the vertices of an undirected, unweighted graph such that the distance between any pair of vertices can be decoded solely from their labels. An important class of distance labeling schemes is that of hub labelings, where a node v∈G stores its distance to the so-called hubs S(v)⊆V, chosen so that for any u,v∈V there is w∈S(u)∩S(v) belonging to some shortest uv path. Notice that for most existing graph classes, the best distance labelling constructions existing use at some point a hub labeling scheme at least as a key building block. Our interest lies in hub labelings of sparse graphs, i.e., those with |E(G)|=O(n), for which we show a lowerbound of n/2^O(√logn) for the average size of the hubsets. Additionally, we show a hub-labeling construction for sparse graphs of average size O(n/RS(n)^c) for some 0 < c < 1, where RS(n) is the so-called Ruzsa-Szemerédi function, linked to structure of induced matchings in dense graphs. This implies that further improving the lower bound on hub labeling size to n/2^(logn)^o(1) would require a breakthrough in the study of lower bounds on RS(n), which have resisted substantial improvement in the last 70 years. For general distance labeling of sparse graphs, we show a lowerbound of 1/2O(√logn) * SumIndex(n), where SumIndex(n) is the communication complexity of the Sum-Index problem over Z_n. Our results suggest that the best achievable hub-label size and distance-label size in sparse graphs may be Θ(n/2^(log n)^c) for some 0 < c < 1. | Paper |
| 8th April 2020 14:15 (Helsinki time |
Alex Jung: On the Duality between Network Flows and Network Lasso |
Many applications generate data with an intrinsic network structure such as time series data, image data or social network data. The network Lasso (nLasso) has been proposed recently as a method for joint clustering and optimization of machine learning models for networked data. The nLasso extends the Lasso from sparse linear models to clustered graph signals. This paper explores the duality of nLasso and network flow optimization. We show that, in a very precise sense, nLasso is equivalent to a minimum-cost flow problem on the data network structure. Our main technical result is a concise characterization of nLasso solutions via existence of certain network flows. The main conceptual result is a useful link between nLasso methods and basic graph algorithms such as clustering or maximum flow. | Paper |
| 1st April 2020 14:15 (Helsinki time) |
Petteri Kaski: Probabilistic Tensors and Opportunistic Boolean Matrix Multiplication |
We introduce probabilistic extensions of classical deterministic measures of algebraic complexity of a tensor, such as the rank and the border rank. We show that these probabilistic extensions satisfy various natural and algorithmically serendipitous properties, such as submultiplicativity under taking of Kronecker products. Furthermore, the probabilistic extensions enable strictly lower rank over their deterministic counterparts for specific tensors of interest, starting from the tensor <2,2,2> that represents 2-by-2 matrix multiplication. By submultiplicativity, this leads immediately to novel randomized algorithm designs, such as algorithms for Boolean matrix multiplication as well as detecting and estimating the number of triangles and other subgraphs in graphs. | Paper |
| 25th March 2020 14:15 (Helsinki time) |
Jukka Suomela: Foundations of Distributed Computing in the 2020s |
In this talk I will review some major advances in the theory of distributed computing in the past decade and discuss key research challenges for the 2020s, with the main focus on distributed graph algorithms. I will present promising new approaches for doing research in the field, and I will also highlight examples of seemingly elementary questions that are still beyond the reach of our current techniques. | Slides |