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1. Minimum L_∞ Hausdorff distance of point sets under translation: Generalizing Klee's measure problem

   https://doi.org/10.4230/LIPIcs.SoCG.2023.24  

   Chan, Timothy M. ( June 2023 , Proc. 39th Sympos. Computational Geometry (SoCG))

   Chambers, Erin W. ; Gudmundsson, Joachim (Ed.)

   We present a (combinatorial) algorithm with running time close to O(n^d) for computing the minimum directed L_∞ Hausdorff distance between two sets of n points under translations in any constant dimension d. This substantially improves the best previous time bound near O(n^{5d/4}) by Chew, Dor, Efrat, and Kedem from more than twenty years ago. Our solution is obtained by a new generalization of Chan’s algorithm [FOCS'13] for Klee’s measure problem. To complement this algorithmic result, we also prove a nearly matching conditional lower bound close to Ω(n^d) for combinatorial algorithms, under the Combinatorial k-Clique Hypothesis. 

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2. Towards a Decomposition-Optimal Algorithm for Counting and Sampling Arbitrary Motifs in Sublinear Time

   Biswas, A. S. ; Eden, T. ; Rubinfeld, R. ( January 2021 , Random)

   Counting and uniformly sampling motifs in a graph are fundamental algorithmic tasks with numerous applications across multiple fields. Since these problems are computationally expensive, recent efforts have focused on devising sublinear-time algorithms for these problems. We consider the model where the algorithm gets a constant size motif H and query access to a graph G, where the allowed queries are degree, neighbor, and pair queries, as well as uniform edge sample queries. In the sampling task, the algorithm is required to output a uniformly distributed copy of H in G (if one exists), and in the counting task it is required to output a good estimate to the number of copies of H in G. Previous algorithms for the uniform sampling task were based on a decomposition of H into a collection of odd cycles and stars, denoted D∗(H) = {Ok1 , ...,Okq , Sp1 , ..., Spℓ19 }. These algorithms were shown to be optimal for the case where H is a clique or an odd-length cycle, but no other lower bounds were known. We present a new algorithm for sampling arbitrary motifs which, up to poly(log n) factors, for any motif H whose decomposition contains at least two components or at least one star, is always preferable. The main ingredient leading to this improvement is an improved uniform algorithm for sampling stars, which might be of independent interest, as it allows to sample vertices according to the p-th moment of the degree distribution. We further show how to use our sampling algorithm to get an approximate counting algorithm, with essentially the same complexity. Finally, we prove that this algorithm is decomposition-optimal for decompositions that contain at least one odd cycle. That is, we prove that for any decomposition D that contains at least one odd cycle, there exists a motif HD 30 with decomposition D, and a family of graphs G, so that in order to output a uniform copy of H in a uniformly chosen graph in G, the number of required queries matches our upper bound. These are the first lower bounds for motifs H with a nontrivial decomposition, i.e., motifs that have more than a single component in their decomposition. 

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3. Why Is Maximum Clique Often Easy in Practice?

   https://doi.org/10.1287/opre.2019.1970  

   Walteros, Jose L. ; Buchanan, Austin ( November 2020 , Operations Research)

   null (Ed.)

   To this day, the maximum clique problem remains a computationally challenging problem. Indeed, despite researchers’ best efforts, there exist unsolved benchmark instances with 1,000 vertices. However, relatively simple algorithms solve real-life instances with millions of vertices in a few seconds. Why is this the case? Why is the problem apparently so easy in many naturally occurring networks? In this paper, we provide an explanation. First, we observe that the graph’s clique number ω is very near to the graph’s degeneracy d in most real-life instances. This observation motivates a main contribution of this paper, which is an algorithm for the maximum clique problem that runs in time polynomial in the size of the graph, but exponential in the gap [Formula: see text] between the clique number ω and its degeneracy-based upper bound d+1. When this gap [Formula: see text] can be treated as a constant, as is often the case for real-life graphs, the proposed algorithm runs in time [Formula: see text]. This provides a rigorous explanation for the apparent easiness of these instances despite the intractability of the problem in the worst case. Further, our implementation of the proposed algorithm is actually practical—competitive with the best approaches from the literature. 

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4. Finding a planted clique by adaptive probing

   https://doi.org/10.30757/ALEA.v17-30  

   Racz, Miklos Z ; Schiffer, Benjamin ( August 2020 , Alea)

   null (Ed.)

   We consider a variant of the planted clique problem where we are allowed unbounded computational time but can only investigate a small part of the graph by adaptive edge queries. We determine (up to logarithmic factors) the number of queries necessary both for detecting the presence of a planted clique and for finding the planted clique. Specifically, let $G \sim G(n,1/2,k)$ be a random graph on $n$ vertices with a planted clique of size $k$. We show that no algorithm that makes at most $q = o(n^2 / k^2 + n)$ adaptive queries to the adjacency matrix of $G$ is likely to find the planted clique. On the other hand, when $k \geq (2+\epsilon) \log_2 n$ there exists a simple algorithm (with unbounded computational power) that finds the planted clique with high probability by making $q = O( (n^2 / k^2) \log^2 n + n \log n)$ adaptive queries. For detection, the additive $n$ term is not necessary: the number of queries needed to detect the presence of a planted clique is $n^2 / k^2$ (up to logarithmic factors). 

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5. The Parallel Reversible Pebbling Game: Analyzing the Post-quantum Security of iMHFs

   Blocki, Jeremiah ; Holman, Blake ; Lee, Seunghoon ( December 2022 , Theory of Cryptography Conference (TCC 2022))

   Kiltz, E. (Ed.)

   The classical (parallel) black pebbling game is a useful abstraction which allows us to analyze the resources (space, space-time, cumulative space) necessary to evaluate a function f with a static data-dependency graph G. Of particular interest in the field of cryptography are data-independent memory-hard functions fG,H which are defined by a directed acyclic graph (DAG) G and a cryptographic hash function H. The pebbling complexity of the graph G characterizes the amortized cost of evaluating fG,H multiple times as well as the total cost to run a brute-force preimage attack over a fixed domain X, i.e., given y∈{0,1}∗ find x∈X such that fG,H(x)=y. While a classical attacker will need to evaluate the function fG,H at least m=|X| times a quantum attacker running Grover’s algorithm only requires O(m−−√) blackbox calls to a quantum circuit CG,H evaluating the function fG,H. Thus, to analyze the cost of a quantum attack it is crucial to understand the space-time cost (equivalently width times depth) of the quantum circuit CG,H. We first observe that a legal black pebbling strategy for the graph G does not necessarily imply the existence of a quantum circuit with comparable complexity—in contrast to the classical setting where any efficient pebbling strategy for G corresponds to an algorithm with comparable complexity for evaluating fG,H. Motivated by this observation we introduce a new parallel reversible pebbling game which captures additional restrictions imposed by the No-Deletion Theorem in Quantum Computing. We apply our new reversible pebbling game to analyze the reversible space-time complexity of several important graphs: Line Graphs, Argon2i-A, Argon2i-B, and DRSample. Specifically, (1) we show that a line graph of size N has reversible space-time complexity at most O(N^{1+2/√logN}). (2) We show that any (e, d)-reducible DAG has reversible space-time complexity at most O(Ne+dN2^d). In particular, this implies that the reversible space-time complexity of Argon2i-A and Argon2i-B are at most O(N^2 loglogN/√logN) and O(N^2/(log N)^{1/3}), respectively. (3) We show that the reversible space-time complexity of DRSample is at most O((N^2loglog N)/log N). We also study the cumulative pebbling cost of reversible pebblings extending a (non-reversible) pebbling attack of Alwen and Blocki on depth-reducible graphs. 

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