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1. Fast Approximate Counting of Cycles

   https://doi.org/10.4230/LIPIcs.ICALP.2024.37  

   Censor-Hillel, Keren; Even, Tomer; Vassilevska_Williams, Virginia (January 2024, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Bringmann, Karl; Grohe, Martin; Puppis, Gabriele; Svensson, Ola (Ed.)

   We consider the problem of approximate counting of triangles and longer fixed length cycles in directed graphs. For triangles, Tětek [ICALP'22] gave an algorithm that returns a (1±ε)-approximation in Õ(n^ω/t^{ω-2}) time, where t is the unknown number of triangles in the given n node graph and ω < 2.372 is the matrix multiplication exponent. We obtain an improved algorithm whose running time is, within polylogarithmic factors the same as that for multiplying an n× n/t matrix by an n/t × n matrix. We then extend our framework to obtain the first nontrivial (1± ε)-approximation algorithms for the number of h-cycles in a graph, for any constant h ≥ 3. Our running time is Õ(MM(n,n/t^{1/(h-2)},n)), the time to multiply n × n/(t^{1/(h-2)}) by n/(t^{1/(h-2)) × n matrices. Finally, we show that under popular fine-grained hypotheses, this running time is optimal. 

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2. Space Complexity of Minimum Cut Problems in Single-Pass Streams

   https://doi.org/10.4230/LIPIcs.ITCS.2025.43  

   Ding, Matthew; Garces, Alexandro; Li, Jason; Lin, Honghao; Nelson, Jelani; Shah, Vihan; Woodruff, David P (January 2025, Schloss Dagstuhl – Leibniz-Zentrum für Informatik)

   Meka, Raghu (Ed.)

   We consider the problem of finding a minimum cut of a weighted graph presented as a single-pass stream. While graph sparsification in streams has been intensively studied, the specific application of finding minimum cuts in streams is less well-studied. To this end, we show upper and lower bounds on minimum cut problems in insertion-only streams for a variety of settings, including for both randomized and deterministic algorithms, for both arbitrary and random order streams, and for both approximate and exact algorithms. One of our main results is an Õ(n/ε) space algorithm with fast update time for approximating a spectral cut query with high probability on a stream given in an arbitrary order. Our result breaks the Ω(n/ε²) space lower bound required of a sparsifier that approximates all cuts simultaneously. Using this result, we provide streaming algorithms with near optimal space of Õ(n/ε) for minimum cut and approximate all-pairs effective resistances, with matching space lower-bounds. The amortized update time of our algorithms is Õ(1), provided that the number of edges in the input graph is at least (n/ε²)^{1+o(1)}. We also give a generic way of incorporating sketching into a recursive contraction algorithm to improve the post-processing time of our algorithms. In addition to these results, we give a random-order streaming algorithm that computes the exact minimum cut on a simple, unweighted graph using Õ(n) space. Finally, we give an Ω(n/ε²) space lower bound for deterministic minimum cut algorithms which matches the best-known upper bound up to polylogarithmic factors. 

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3. Adversarially Robust Coloring for Graph Streams

   Chakrabarti, Amit; Ghosh, Prantar; Stoeckl, Manuel (January 2022, Leibniz international proceedings in informatics)

   A streaming algorithm is considered to be adversarially robust if it provides correct outputs with high probability even when the stream updates are chosen by an adversary who may observe and react to the past outputs of the algorithm. We grow the burgeoning body of work on such algorithms in a new direction by studying robust algorithms for the problem of maintaining a valid vertex coloring of an n-vertex graph given as a stream of edges. Following standard practice, we focus on graphs with maximum degree at most Δ and aim for colorings using a small number f(Δ) of colors. A recent breakthrough (Assadi, Chen, and Khanna; SODA 2019) shows that in the standard, non-robust, streaming setting, (Δ+1)-colorings can be obtained while using only Õ(n) space. Here, we prove that an adversarially robust algorithm running under a similar space bound must spend almost Ω(Δ²) colors and that robust O(Δ)-coloring requires a linear amount of space, namely Ω(nΔ). We in fact obtain a more general lower bound, trading off the space usage against the number of colors used. From a complexity-theoretic standpoint, these lower bounds provide (i) the first significant separation between adversarially robust algorithms and ordinary randomized algorithms for a natural problem on insertion-only streams and (ii) the first significant separation between randomized and deterministic coloring algorithms for graph streams, since deterministic streaming algorithms are automatically robust. We complement our lower bounds with a suite of positive results, giving adversarially robust coloring algorithms using sublinear space. In particular, we can maintain an O(Δ²)-coloring using Õ(n √Δ) space and an O(Δ³)-coloring using Õ(n) space. 

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4. On the fine-grained complexity of small-size geometric set cover and discrete k-center for small k

   https://doi.org/10.4230/LIPIcs.ICALP.2023.34  

   Chan, Timothy M.; He, Qizheng; Yu, Yuancheng (July 2023, Proc. 50th International Colloquium on Automata, Languages, and Programming (ICALP 2023))

   Etessami, Kousha; Feige, Uriel; Puppis, Gabriele (Ed.)

   We study the time complexity of the discrete k-center problem and related (exact) geometric set cover problems when k or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for rectilinear discrete 3-center in 2D, running in Õ(n^{3/2}) time. - We prove a lower bound of Ω(n^{4/3-δ}) for rectilinear discrete 3-center in 4D, for any constant δ > 0, under a standard hypothesis about triangle detection in sparse graphs. - Given n points and n weighted axis-aligned unit squares in 2D, we give the first subquadratic algorithm for finding a minimum-weight cover of the points by 3 unit squares, running in Õ(n^{8/5}) time. We also prove a lower bound of Ω(n^{3/2-δ}) for the same problem in 2D, under the well-known APSP Hypothesis. For arbitrary axis-aligned rectangles in 2D, our upper bound is Õ(n^{7/4}). - We prove a lower bound of Ω(n^{2-δ}) for Euclidean discrete 2-center in 13D, under the Hyperclique Hypothesis. This lower bound nearly matches the straightforward upper bound of Õ(n^ω), if the matrix multiplication exponent ω is equal to 2. - We similarly prove an Ω(n^{k-δ}) lower bound for Euclidean discrete k-center in O(k) dimensions for any constant k ≥ 3, under the Hyperclique Hypothesis. This lower bound again nearly matches known upper bounds if ω = 2. - We also prove an Ω(n^{2-δ}) lower bound for the problem of finding 2 boxes to cover the largest number of points, given n points and n boxes in 12D . This matches the straightforward near-quadratic upper bound. 

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5. Streaming algorithms for the missing item finding problem

   Stoeckl, Manuel (January 2023, SODA 2023)

   Many problems on data streams have been studied at two extremes of difficulty: either allowing randomized algorithms, in the static setting (where they should err with bounded probability on the worst case stream); or when only deterministic and infallible algorithms are required. Some recent works have considered the adversarial setting, in which a randomized streaming algorithm must succeed even on data streams provided by an adaptive adversary that can see the intermediate outputs of the algorithm. In order to better understand the differences between these models, we study a streaming task called “Missing Item Finding”. In this problem, for r < n, one is given a data stream a1 , . . . , ar of elements in [n], (possibly with repetitions), and must output some x ∈ [n] which does not equal any of the ai. We prove that, for r = nΘ(1) and δ = 1/poly(n), the space required for randomized algorithms that solve this problem in the static setting with error δ is Θ(polylog(n)); for algorithms in the adversarial setting with error δ, Θ((1 + r2/n)polylog(n)); and for deterministic algorithms, Θ(r/polylog(n)). Because our adversarially robust algorithm relies on free access to a string of O(r log n) random bits, we investigate a “random start” model of streaming algorithms where all random bits used are included in the space cost. Here we find a conditional lower bound on the space usage, which depends on the space that would be needed for a pseudo-deterministic algorithm to solve the problem. We also prove an Ω(r/polylog(n)) lower bound for the space needed by a streaming algorithm with < 1/2polylog(n) error against “white-box” adversaries that can see the internal state of the algorithm, but not predict its future random decisions. 

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