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1. Improving the Smoothed Complexity of FLIP for Max Cut Problems

   https://doi.org/10.1145/3454125  

   Bibak, Ali ; Carlson, Charles ; Chandrasekaran, Karthekeyan ( August 2021 , ACM Transactions on Algorithms)

   null (Ed.)

   Finding locally optimal solutions for MAX-CUT and MAX- k -CUT are well-known PLS-complete problems. An instinctive approach to finding such a locally optimum solution is the FLIP method. Even though FLIP requires exponential time in worst-case instances, it tends to terminate quickly in practical instances. To explain this discrepancy, the run-time of FLIP has been studied in the smoothed complexity framework. Etscheid and Röglin (ACM Transactions on Algorithms, 2017) showed that the smoothed complexity of FLIP for max-cut in arbitrary graphs is quasi-polynomial. Angel, Bubeck, Peres, and Wei (STOC, 2017) showed that the smoothed complexity of FLIP for max-cut in complete graphs is ( O Φ 5 n 15.1 ), where Φ is an upper bound on the random edge-weight density and Φ is the number of vertices in the input graph. While Angel, Bubeck, Peres, and Wei’s result showed the first polynomial smoothed complexity, they also conjectured that their run-time bound is far from optimal. In this work, we make substantial progress toward improving the run-time bound. We prove that the smoothed complexity of FLIP for max-cut in complete graphs is O (Φ n 7.83 ). Our results are based on a carefully chosen matrix whose rank captures the run-time of the method along with improved rank bounds for this matrix and an improved union bound based on this matrix. In addition, our techniques provide a general framework for analyzing FLIP in the smoothed framework. We illustrate this general framework by showing that the smoothed complexity of FLIP for MAX-3-CUT in complete graphs is polynomial and for MAX - k - CUT in arbitrary graphs is quasi-polynomial. We believe that our techniques should also be of interest toward showing smoothed polynomial complexity of FLIP for MAX - k - CUT in complete graphs for larger constants k . 

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2. Smoothed analysis for low-rank solutions to semidefinite programs in quadratic penalty form

   Bhojanapalli, Srinadh ; Boumal, Nicolas ; Jain, Prateek ; Netrapalli, Praneeth ( January 2018 , Proceedings of the 31st Conference On Learning Theory, PMLR)

   Semidefinite programs (SDP) are important in learning and combinatorial optimization with numerous applications. In pursuit of low-rank solutions and low complexity algorithms, we consider the Burer–Monteiro factorization approach for solving SDPs. For a large class of SDPs, upon random perturbation of the cost matrix, with high probability, we show that all approximate second-order stationary points are approximate global optima for the penalty formulation of appropriately rank-constrained SDPs, as long as the number of constraints scales sub-quadratically with the desired rank. Our result is based on a simple penalty function formulation of the rank-constrained SDP along with a smoothed analysis to avoid worst-case cost matrices. We particularize our results to two applications, namely, Max-Cut and matrix completion. 

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3. Local algorithms for maximum cut and minimum bisection on locally treelike regular graphs of large degree

   https://doi.org/10.1002/rsa.21149  

   El Alaoui, Ahmed ; Montanari, Andrea ; Sellke, Mark ( May 2023 , Random Structures & Algorithms)

   Given a graph of degree over vertices, we consider the problem of computing a near maximum cut or a near minimum bisection in polynomial time. For graphs of girth , we develop a local message passing algorithm whose complexity is , and that achieves near optimal cut values among all ‐local algorithms. Focusing on max‐cut, the algorithm constructs a cut of value , where is the value of the Parisi formula from spin glass theory, and (subscripts indicate the asymptotic variables). Our result generalizes to locally treelike graphs, that is, graphs whose girth becomes after removing a small fraction of vertices. Earlier work established that, for random ‐regular graphs, the typical max‐cut value is . Therefore our algorithm is nearly optimal on such graphs. An immediate corollary of this result is that random regular graphs have nearly minimum max‐cut, and nearly maximum min‐bisection among all regular locally treelike graphs. This can be viewed as a combinatorial version of the near‐Ramanujan property of random regular graphs.

    

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4. Empirical study on sufficient numbers of minimum cuts in strongly connected directed random graphs

   https://doi.org/10.1002/net.21943  

   Chang, Eric ; Cheng, C. K. ; Gupta, Anushka ; Hsu, Po‐Ya ; Moffitt, Amanda ; Ren, Alissa ; Tsaur, Irene ; Wang, Samuel ( May 2020 , Networks)

   We focus on the all‐pairs minimum cut (APMC) problem, a graph partitioning problem whose solution requires finding the minimum cut for every pair of nodes in a given graph. While it is solved for undirected graphs, a solution for APMC in directed graphs still requires anO(n²)brute force approach. We show that the empirical number of distinct minimum cuts in randomly generated strongly connected directed graphs is proportional tonrather than the theoretical value ofn², suggesting the possibility of an algorithm which finds all minimum cuts in less thanO(n²)time. We also provide an example of the strict upper bound on the number of cuts in graphs with three nodes. We model the distributions with the Generalized extreme value (GEV) distribution and enable the possibility of using a GEV distribution to predict the probability of achieving a certain number of minimum cuts, given the number of nodes and edges. Finally, we contribute to the notion of symmetric cuts by showing that there can beO(n²)symmetric cuts in graphs when node replication is allowed.

    

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5. A unified view of graph regularity via matrix decompositions

   https://doi.org/10.1002/rsa.21053  

   Bodwin, Greg ; Vempala, Santosh ( October 2021 , Random Structures & Algorithms)

   We give a unified proof of algorithmic weak and Szemerédi regularity lemmas for several well‐studied classes of sparse graphs, for which only weak regularity lemmas were previously known. These include core‐dense graphs, low threshold rank graphs, and (a version of)upper regular graphs. More precisely, we definecut pseudorandom graphs, we prove our regularity lemmas for these graphs, and then we show that cut pseudorandomness captures all of the above graph classes as special cases. The core of our approach is an abstracted matrix decomposition, which can be computed by a simple algorithm by Charikar. Using work of Oveis Gharan and Trevisan, it also implies new PTASes for MAX‐CUT, MAX‐BISECTION, MIN‐BISECTION for a significantly expanded class of input graphs. (It is NP Hard to get PTASes for these graphs in general.)

    

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