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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)

   Abstract 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. Minimum Cuts in Directed Graphs via Partial Sparsification

   https://doi.org/10.1109/FOCS52979.2021.00113  

   Cen, Ruoxu; Li, Jason; Nanongkai, Danupon; Panigrahi, Debmalya; Saranurak, Thatchaphol; Quanrud, Kent (February 2022, 62nd IEEE Annual Symposium on Foundations of Computer Science, FOCS 2021, Denver, CO, USA, February 7-10, 2022)

   We give an algorithm to find a minimum cut in an edge-weighted directed graph with n vertices and m edges in O ̃(n · max{m^{2/3}, n}) time. This improves on the 30 year old bound of O ̃(nm) obtained by Hao and Orlin for this problem. Using similar techniques, we also obtain O ̃ (n^2 /ε^2 )-time (1+ε)-approximation algorithms for both the minimum edge and minimum vertex cuts in directed graphs, for any fixed ε. Before our work, no (1+ε)-approximation algorithm better than the exact runtime of O ̃(nm) is known for either problem. Our algorithms follow a two-step template. In the first step, we employ a partial sparsification of the input graph to preserve a critical subset of cut values approximately. In the second step, we design algorithms to find the (edge/vertex) mincut among the preserved cuts from the first step. For edge mincut, we give a new reduction to O ̃ (min{n/m^{1/3} , √n}) calls of any maxflow subroutine, via packing arborescences in the sparsifier. For vertex mincut, we develop new local flow algorithms to identify small unbalanced cuts in the sparsified graph. 

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5. An Experimental Evaluation of Semidefinite Programming and Spectral Algorithms for Max Cut

   https://doi.org/10.4230/LIPIcs.SEA.2022.19  

   Mirka, Renee; Williamson, David P. (July 2022, Leibniz international proceedings in informatics)

   Schulz, Christian; Ucar, Bora (Ed.)

   We experimentally evaluate the performance of several Max Cut approximation algorithms. In particular, we compare the results of the Goemans and Williamson algorithm using semidefinite programming with Trevisan’s algorithm using spectral partitioning. The former algorithm has a known .878 approximation guarantee whereas the latter has a .614 approximation guarantee. We investigate whether this gap in approximation guarantees is evident in practice or whether the spectral algorithm performs as well as the SDP. We also compare the performances to the standard greedy Max Cut algorithm which has a .5 approximation guarantee and two additional spectral algorithms. The algorithms are tested on Erdős-Renyi random graphs, complete graphs from TSPLIB, and real-world graphs from the Network Repository. We find, unsurprisingly, that the spectral algorithms provide a significant speed advantage over the SDP. In our experiments, the spectral algorithms return cuts with values which are competitive with those of the SDP. 

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