@inproceedings{1175, abstract = {We study space complexity and time-space trade-offs with a focus not on peak memory usage but on overall memory consumption throughout the computation. Such a cumulative space measure was introduced for the computational model of parallel black pebbling by [Alwen and Serbinenko ’15] as a tool for obtaining results in cryptography. We consider instead the non- deterministic black-white pebble game and prove optimal cumulative space lower bounds and trade-offs, where in order to minimize pebbling time the space has to remain large during a significant fraction of the pebbling. We also initiate the study of cumulative space in proof complexity, an area where other space complexity measures have been extensively studied during the last 10–15 years. Using and extending the connection between proof complexity and pebble games in [Ben-Sasson and Nordström ’08, ’11] we obtain several strong cumulative space results for (even parallel versions of) the resolution proof system, and outline some possible future directions of study of this, in our opinion, natural and interesting space measure.}, author = {Alwen, Joel F and De Rezende, Susanna and Nordstrom, Jakob and Vinyals, Marc}, editor = {Papadimitriou, Christos}, issn = {18688969}, location = {Berkeley, CA, United States}, pages = {38:1--38--21}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Cumulative space in black-white pebbling and resolution}}, doi = {10.4230/LIPIcs.ITCS.2017.38}, volume = {67}, year = {2017}, } @inproceedings{11772, abstract = {A dynamic graph algorithm is a data structure that supports operations on dynamically changing graphs.}, author = {Henzinger, Monika H}, booktitle = {44th International Conference on Current Trends in Theory and Practice of Computer Science}, isbn = {9783319731162}, issn = {0302-9743}, location = {Krems, Austria}, pages = {40–44}, publisher = {Springer Nature}, title = {{The state of the art in dynamic graph algorithms}}, doi = {10.1007/978-3-319-73117-9_3}, volume = {10706}, year = {2017}, } @inproceedings{11829, abstract = {In recent years it has become popular to study dynamic problems in a sensitivity setting: Instead of allowing for an arbitrary sequence of updates, the sensitivity model only allows to apply batch updates of small size to the original input data. The sensitivity model is particularly appealing since recent strong conditional lower bounds ruled out fast algorithms for many dynamic problems, such as shortest paths, reachability, or subgraph connectivity. In this paper we prove conditional lower bounds for these and additional problems in a sensitivity setting. For example, we show that under the Boolean Matrix Multiplication (BMM) conjecture combinatorial algorithms cannot compute the (4/3-\varepsilon)-approximate diameter of an undirected unweighted dense graph with truly subcubic preprocessing time and truly subquadratic update/query time. This result is surprising since in the static setting it is not clear whether a reduction from BMM to diameter is possible. We further show under the BMM conjecture that many problems, such as reachability or approximate shortest paths, cannot be solved faster than by recomputation from scratch even after only one or two edge insertions. We extend our reduction from BMM to Diameter to give a reduction from All Pairs Shortest Paths to Diameter under one deletion in weighted graphs. This is intriguing, as in the static setting it is a big open problem whether Diameter is as hard as APSP. We further get a nearly tight lower bound for shortest paths after two edge deletions based on the APSP conjecture. We give more lower bounds under the Strong Exponential Time Hypothesis. Many of our lower bounds also hold for static oracle data structures where no sensitivity is required. Finally, we give the first algorithm for the (1+\varepsilon)-approximate radius, diameter, and eccentricity problems in directed or undirected unweighted graphs in case of single edges failures. The algorithm has a truly subcubic running time for graphs with a truly subquadratic number of edges; it is tight w.r.t. the conditional lower bounds we obtain.}, author = {Henzinger, Monika H and Lincoln, Andrea and Neumann, Stefan and Vassilevska Williams, Virginia}, booktitle = {8th Innovations in Theoretical Computer Science Conference}, isbn = {9783959770293}, issn = {1868-8969}, location = {Berkley, CA, United States}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Conditional hardness for sensitivity problems}}, doi = {10.4230/LIPICS.ITCS.2017.26}, volume = {67}, year = {2017}, } @inproceedings{11833, abstract = {We introduce a new algorithmic framework for designing dynamic graph algorithms in minor-free graphs, by exploiting the structure of such graphs and a tool called vertex sparsification, which is a way to compress large graphs into small ones that well preserve relevant properties among a subset of vertices and has previously mainly been used in the design of approximation algorithms. Using this framework, we obtain a Monte Carlo randomized fully dynamic algorithm for (1 + epsilon)-approximating the energy of electrical flows in n-vertex planar graphs with tilde{O}(r epsilon^{-2}) worst-case update time and tilde{O}((r + n / sqrt{r}) epsilon^{-2}) worst-case query time, for any r larger than some constant. For r=n^{2/3}, this gives tilde{O}(n^{2/3} epsilon^{-2}) update time and tilde{O}(n^{2/3} epsilon^{-2}) query time. We also extend this algorithm to work for minor-free graphs with similar approximation and running time guarantees. Furthermore, we illustrate our framework on the all-pairs max flow and shortest path problems by giving corresponding dynamic algorithms in minor-free graphs with both sublinear update and query times. To the best of our knowledge, our results are the first to systematically establish such a connection between dynamic graph algorithms and vertex sparsification. We also present both upper bound and lower bound for maintaining the energy of electrical flows in the incremental subgraph model, where updates consist of only vertex activations, which might be of independent interest.}, author = {Goranci, Gramoz and Henzinger, Monika H and Peng, Pan}, booktitle = {25th Annual European Symposium on Algorithms}, isbn = {978-3-95977-049-1}, issn = {1868-8969}, location = {Vienna, Austria}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{The power of vertex sparsifiers in dynamic graph algorithms}}, doi = {10.4230/LIPICS.ESA.2017.45}, volume = {87}, year = {2017}, } @inproceedings{11832, abstract = {In this paper, we study the problem of opening centers to cluster a set of clients in a metric space so as to minimize the sum of the costs of the centers and of the cluster radii, in a dynamic environment where clients arrive and depart, and the solution must be updated efficiently while remaining competitive with respect to the current optimal solution. We call this dynamic sum-of-radii clustering problem. We present a data structure that maintains a solution whose cost is within a constant factor of the cost of an optimal solution in metric spaces with bounded doubling dimension and whose worst-case update time is logarithmic in the parameters of the problem.}, author = {Henzinger, Monika H and Leniowski, Dariusz and Mathieu, Claire}, booktitle = {25th Annual European Symposium on Algorithms}, isbn = {978-3-95977-049-1}, issn = {1868-8969}, location = {Vienna, Austria}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Dynamic clustering to minimize the sum of radii}}, doi = {10.4230/LIPICS.ESA.2017.48}, volume = {87}, year = {2017}, } @inproceedings{11874, abstract = {We consider the problem of maintaining an approximately maximum (fractional) matching and an approximately minimum vertex cover in a dynamic graph. Starting with the seminal paper by Onak and Rubinfeld [STOC 2010], this problem has received significant attention in recent years. There remains, however, a polynomial gap between the best known worst case update time and the best known amortised update time for this problem, even after allowing for randomisation. Specifically, Bernstein and Stein [ICALP 2015, SODA 2016] have the best known worst case update time. They present a deterministic data structure with approximation ratio (3/2 + ∊) and worst case update time O(m1/4/ ∊2), where m is the number of edges in the graph. In recent past, Gupta and Peng [FOCS 2013] gave a deterministic data structure with approximation ratio (1+ ∊) and worst case update time No known randomised data structure beats the worst case update times of these two results. In contrast, the paper by Onak and Rubinfeld [STOC 2010] gave a randomised data structure with approximation ratio O(1) and amortised update time O(log2 n), where n is the number of nodes in the graph. This was later improved by Baswana, Gupta and Sen [FOCS 2011] and Solomon [FOCS 2016], leading to a randomised date structure with approximation ratio 2 and amortised update time O(1). We bridge the polynomial gap between the worst case and amortised update times for this problem, without using any randomisation. We present a deterministic data structure with approximation ratio (2 + ∊) and worst case update time O(log3 n), for all sufficiently small constants ∊.}, author = {Bhattacharya, Sayan and Henzinger, Monika H and Nanongkai, Danupon}, booktitle = {28th Annual ACM-SIAM Symposium on Discrete Algorithms}, location = {Barcelona, Spain}, pages = {470 -- 489}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Fully dynamic approximate maximum matching and minimum vertex cover in o(log3 n) worst case update time}}, doi = {10.1137/1.9781611974782.30}, year = {2017}, } @inproceedings{11873, abstract = {We study the problem of computing a minimum cut in a simple, undirected graph and give a deterministic O(m log2 n log log2 n) time algorithm. This improves both on the best previously known deterministic running time of O(m log12 n) (Kawarabayashi and Thorup [12]) and the best previously known randomized running time of O(mlog3n) (Karger [11]) for this problem, though Karger's algorithm can be further applied to weighted graphs. Our approach is using the Kawarabayashi and Tho- rup graph compression technique, which repeatedly finds low-conductance cuts. To find these cuts they use a diffusion-based local algorithm. We use instead a flow- based local algorithm and suitably adjust their framework to work with our flow-based subroutine. Both flow and diffusion based methods have a long history of being applied to finding low conductance cuts. Diffusion algorithms have several variants that are naturally local while it is more complicated to make flow methods local. Some prior work has proven nice properties for local flow based algorithms with respect to improving or cleaning up low conductance cuts. Our flow subroutine, however, is the first that is both local and produces low conductance cuts. Thus, it may be of independent interest.}, author = {Henzinger, Monika H and Rao, Satish and Wang, Di}, booktitle = {28th Annual ACM-SIAM Symposium on Discrete Algorithms}, location = {Barcelona, Spain}, pages = {1919--1938}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Local flow partitioning for faster edge connectivity}}, doi = {10.1137/1.9781611974782.125}, year = {2017}, } @inproceedings{11831, abstract = {Graph Sparsification aims at compressing large graphs into smaller ones while (approximately) preserving important characteristics of the input graph. In this work we study Vertex Sparsifiers, i.e., sparsifiers whose goal is to reduce the number of vertices. Given a weighted graph G=(V,E), and a terminal set K with |K|=k, a quality-q vertex cut sparsifier of G is a graph H with K contained in V_H that preserves the value of minimum cuts separating any bipartition of K, up to a factor of q. We show that planar graphs with all the k terminals lying on the same face admit quality-1 vertex cut sparsifier of size O(k^2) that are also planar. Our result extends to vertex flow and distance sparsifiers. It improves the previous best known bound of O(k^2 2^(2k)) for cut and flow sparsifiers by an exponential factor, and matches an Omega(k^2) lower-bound for this class of graphs. We also study vertex reachability sparsifiers for directed graphs. Given a digraph G=(V,E) and a terminal set K, a vertex reachability sparsifier of G is a digraph H=(V_H,E_H), K contained in V_H that preserves all reachability information among terminal pairs. We introduce the notion of reachability-preserving minors, i.e., we require H to be a minor of G. Among others, for general planar digraphs, we construct reachability-preserving minors of size O(k^2 log^2 k). We complement our upper-bound by showing that there exists an infinite family of acyclic planar digraphs such that any reachability-preserving minor must have Omega(k^2) vertices.}, author = {Goranci, Gramoz and Henzinger, Monika H and Peng, Pan}, booktitle = {25th Annual European Symposium on Algorithms}, isbn = {978-3-95977-049-1}, issn = {1868-8969}, location = {Vienna, Austria}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Improved guarantees for vertex sparsification in planar graphs}}, doi = {10.4230/LIPICS.ESA.2017.44}, volume = {87}, year = {2017}, } @article{11903, abstract = {Online social networks allow the collection of large amounts of data about the influence between users connected by a friendship-like relationship. When distributing items among agents forming a social network, this information allows us to exploit network externalities that each agent receives from his neighbors that get the same item. In this paper we consider Friends-of-Friends (2-hop) network externalities, i.e., externalities that not only depend on the neighbors that get the same item but also on neighbors of neighbors. For these externalities we study a setting where multiple different items are assigned to unit-demand agents. Specifically, we study the problem of welfare maximization under different types of externality functions. Let n be the number of agents and m be the number of items. Our contributions are the following: (1) We show that welfare maximization is APX-hard; we show that even for step functions with 2-hop (and also with 1-hop) externalities it is NP-hard to approximate social welfare better than (1−1/e). (2) On the positive side we present (i) an 𝑂(𝑛√)-approximation algorithm for general concave externality functions, (ii) an O(log m)-approximation algorithm for linear externality functions, and (iii) a 518(1−1/𝑒)-approximation algorithm for 2-hop step function externalities. We also improve the result from [7] for 1-hop step function externalities by giving a 12(1−1/𝑒)-approximation algorithm.}, author = {Bhattacharya, Sayan and Dvořák, Wolfgang and Henzinger, Monika H and Starnberger, Martin}, issn = {1433-0490}, journal = {Theory of Computing Systems}, number = {4}, pages = {948--986}, publisher = {Springer Nature}, title = {{Welfare maximization with friends-of-friends network externalities}}, doi = {10.1007/s00224-017-9759-8}, volume = {61}, year = {2017}, } @article{1191, abstract = {Variation in genotypes may be responsible for differences in dispersal rates, directional biases, and growth rates of individuals. These traits may favor certain genotypes and enhance their spatiotemporal spreading into areas occupied by the less advantageous genotypes. We study how these factors influence the speed of spreading in the case of two competing genotypes under the assumption that spatial variation of the total population is small compared to the spatial variation of the frequencies of the genotypes in the population. In that case, the dynamics of the frequency of one of the genotypes is approximately described by a generalized Fisher–Kolmogorov–Petrovskii–Piskunov (F–KPP) equation. This generalized F–KPP equation with (nonlinear) frequency-dependent diffusion and advection terms admits traveling wave solutions that characterize the invasion of the dominant genotype. Our existence results generalize the classical theory for traveling waves for the F–KPP with constant coefficients. Moreover, in the particular case of the quadratic (monostable) nonlinear growth–decay rate in the generalized F–KPP we study in detail the influence of the variance in diffusion and mean displacement rates of the two genotypes on the minimal wave propagation speed.}, author = {Kollár, Richard and Novak, Sebastian}, journal = {Bulletin of Mathematical Biology}, number = {3}, pages = {525--559}, publisher = {Springer}, title = {{Existence of traveling waves for the generalized F–KPP equation}}, doi = {10.1007/s11538-016-0244-3}, volume = {79}, year = {2017}, }