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1. Optimal Bounds for the k -cut Problem

   https://doi.org/10.1145/3478018  

   Gupta, Anupam; Harris, David G.; Lee, Euiwoong; Li, Jason (February 2022, Journal of the ACM)

   In the k -cut problem, we want to find the lowest-weight set of edges whose deletion breaks a given (multi)graph into k connected components. Algorithms of Karger and Stein can solve this in roughly O ( n 2k ) time. However, lower bounds from conjectures about the k -clique problem imply that Ω ( n (1- o (1)) k ) time is likely needed. Recent results of Gupta, Lee, and Li have given new algorithms for general k -cut in n 1.98k + O(1) time, as well as specialized algorithms with better performance for certain classes of graphs (e.g., for small integer edge weights). In this work, we resolve the problem for general graphs. We show that the Contraction Algorithm of Karger outputs any fixed k -cut of weight α λ k with probability Ω k ( n - α k ), where λ k denotes the minimum k -cut weight. This also gives an extremal bound of O k ( n k ) on the number of minimum k -cuts and an algorithm to compute λ k with roughly n k polylog( n ) runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight k -clique. The first main ingredient in our result is an extremal bound on the number of cuts of weight less than 2 λ k / k , using the Sunflower lemma. The second ingredient is a fine-grained analysis of how the graph shrinks—and how the average degree evolves—in the Karger process. 

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2. Bilu-Linial Stability, Certified Algorithms and the Independent Set Problem

   Angelidakis, H; Awasthi, P.; Blum, A.; Chatziafratis, V.; Dan, C. (September 2019, 27th Annual European Symposium on Algorithms (ESA 2019))

   We study the classic Maximum Independent Set problem under the notion of stability introduced by Bilu and Linial (2010): a weighted instance of Independent Set is γ-stable if it has a unique optimal solution that remains the unique optimal solution under multiplicative perturbations of the weights by a factor of at most γ ≥ 1. The goal then is to efficiently recover this “pronounced” optimal solution exactly. In this work, we solve stable instances of Independent Set on several classes of graphs: we improve upon previous results by solving \tilde{O}(∆/sqrt(log ∆))-stable instances on graphs of maximum degree ∆, (k − 1)-stable instances on k-colorable graphs and (1 + ε)-stable instances on planar graphs (for any fixed ε > 0), using both combinatorial techniques as well as LPs and the Sherali-Adams hierarchy. For general graphs, we give an algorithm for (εn)-stable instances, for any fixed ε > 0, and lower bounds based on the planted clique conjecture. As a by-product of our techniques, we give algorithms as well as lower bounds for stable instances of Node Multiway Cut (a generalization of Edge Multiway Cut), by exploiting its connections to Vertex Cover. Furthermore, we prove a general structural result showing that the integrality gap of convex relaxations of several maximization problems reduces dramatically on stable instances. Moreover, we initiate the study of certified algorithms for Independent Set. The notion of a γ-certified algorithm was introduced very recently by Makarychev and Makarychev (2018) and it is a class of γ-approximation algorithms that satisfy one crucial property: the solution returned is optimal for a perturbation of the original instance, where perturbations are again multiplicative up to a factor of γ ≥ 1 (hence, such algorithms not only solve γ-stable instances optimally, but also have guarantees even on unstable instances). Here, we obtain ∆-certified algorithms for Independent Set on graphs of maximum degree ∆, and (1 + ε)-certified algorithms on planar graphs. Finally, we analyze the algorithm of Berman and Fürer (1994) and prove that it is a ((∆+1)/3 + ε)-certified algorithm for Independent Set on graphs of maximum degree ∆ where all weights are equal to 1. 

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3. Mining Largest Maximal Quasi-Cliques

   https://doi.org/10.1145/3446637  

   Sanei-Mehri, Seyed-Vahid; Das, Apurba; Hashemi, Hooman; Tirthapura, Srikanta (June 2021, ACM Transactions on Knowledge Discovery from Data)

   Quasi-cliques are dense incomplete subgraphs of a graph that generalize the notion of cliques. Enumerating quasi-cliques from a graph is a robust way to detect densely connected structures with applications in bioinformatics and social network analysis. However, enumerating quasi-cliques in a graph is a challenging problem, even harder than the problem of enumerating cliques. We consider the enumeration of top- k degree-based quasi-cliques and make the following contributions: (1) we show that even the problem of detecting whether a given quasi-clique is maximal (i.e., not contained within another quasi-clique) is NP-hard. (2) We present a novel heuristic algorithm K ernel QC to enumerate the k largest quasi-cliques in a graph. Our method is based on identifying kernels of extremely dense subgraphs within a graph, followed by growing subgraphs around these kernels, to arrive at quasi-cliques with the required densities. (3) Experimental results show that our algorithm accurately enumerates quasi-cliques from a graph, is much faster than current state-of-the-art methods for quasi-clique enumeration (often more than three orders of magnitude faster), and can scale to larger graphs than current methods. 

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4. Pairwise Consistent Measurement Set Maximization for Robust Multi-Robot Map Merging

   https://doi.org/10.1109/ICRA.2018.8460217  

   Mangelson, Joshua G.; Dominic, Derrick; Eustice, Ryan M.; Vasudevan, Ram (May 2018, 2018 IEEE International Conference on Robotics and Automation (ICRA))

   This paper reports on a method for robust selection of inter-map loop closures in multi-robot simultaneous localization and mapping (SLAM). Existing robust SLAM methods assume a good initialization or an “odometry backbone” to classify inlier and outlier loop closures. In the multi-robot case, these assumptions do not always hold. This paper presents an algorithm called Pairwise Consistency Maximization (PCM) that estimates the largest pairwise internally consistent set of measurements. Finding the largest pairwise internally consistent set can be transformed into an instance of the maximum clique problem from graph theory, and by leveraging the associated literature it can be solved in real time. This paper evaluates how well PCM approximates the combinatorial gold standard using simulated data. It also evaluates the performance of PCM on synthetic and real-world data sets in comparison with DCS, SCGP, and RANSAC, and shows that PCM significantly outperforms these methods. 

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5. YACC: A Framework Generalizing TuránShadow for Counting Large Cliques

   Shweta Jain, Hanghang Tong (April 2022, SDM2022)

   Clique-counting is a fundamental problem that has application in many areas eg. dense subgraph discovery, community detection, spam detection, etc. The problem of k-clique-counting is difficult because as k increases, the number of k-cliques goes up exponentially. Enumeration algorithms (even parallel ones) fail to count k-cliques beyond a small k. Approximation algorithms, like TuránShadow have been shown to perform well upto k = 10, but are inefficient for larger cliques. The recently proposed Pivoter algorithm significantly improved the state-of-the-art and was able to give exact counts of all k-cliques in a large number of graphs. However, the clique counts of some graphs (for example, com-lj) are still out of reach of these algorithms. We revisit the TuránShadow algorithm and propose a generalized framework called YACC that leverages several insights about real-world graphs to achieve faster clique-counting. The bottleneck in TuránShadow is a recursive subroutine whose stopping condition is based on a classic result from extremal combinatorics called Turán's theorem. This theorem gives a lower bound for the k-clique density in a subgraph in terms of its edge density. However, this stopping condition is based on a worst-case graph that does not reflect the nature of real-world graphs. Using techniques for quickly discovering dense subgraphs, we relax the stopping condition in a systematic way such that we get a smaller recursion tree while still maintaining the guarantees provided by TuránShadow. We deploy our algorithm on several real-world data sets and show that YACC reduces the size of the recursion tree and the running time by over an order of magnitude. Using YACC, we are able to obtain clique counts for several graphs for which clique-counting was infeasible before, including com-lj. 

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