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1. A framework for quadratic form maximization over convex sets through nonconvex relaxations

   https://doi.org/10.1145/3406325.3451128  

   Bhattiprolu, Vijay ; Lee, Euiwoong ; Naor, Assaf ( January 2021 , STOC'21)

   We investigate the approximability of the following optimization problem. The input is an n× n matrix A=(Aij) with real entries and an origin-symmetric convex body K⊂ ℝn that is given by a membership oracle. The task is to compute (or approximate) the maximum of the quadratic form ∑i=1n∑j=1n Aij xixj=⟨ x,Ax⟩ as x ranges over K. This is a rich and expressive family of optimization problems; for different choices of matrices A and convex bodies K it includes a diverse range of optimization problems like max-cut, Grothendieck/non-commutative Grothendieck inequalities, small set expansion and more. While the literature studied these special cases using case-specific reasoning, here we develop a general methodology for treatment of the approximability and inapproximability aspects of these questions. The underlying geometry of K plays a critical role; we show under commonly used complexity assumptions that polytime constant-approximability necessitates that K has type-2 constant that grows slowly with n. However, we show that even when the type-2 constant is bounded, this problem sometimes exhibits strong hardness of approximation. Thus, even within the realm of type-2 bodies, the approximability landscape is nuanced and subtle. However, the link that we establish between optimization and geometry of Banach spaces allows usmore »to devise a generic algorithmic approach to the above problem. We associate to each convex body a new (higher dimensional) auxiliary set that is not convex, but is approximately convex when K has a bounded type-2 constant. If our auxiliary set has an approximate separation oracle, then we design an approximation algorithm for the original quadratic optimization problem, using an approximate version of the ellipsoid method. Even though our hardness result implies that such an oracle does not exist in general, this new question can be solved in specific cases of interest by implementing a range of classical tools from functional analysis, most notably the deep factorization theory of linear operators. Beyond encompassing the scenarios in the literature for which constant-factor approximation algorithms were found, our generic framework implies that that for convex sets with bounded type-2 constant, constant factor approximability is preserved under the following basic operations: (a) Subspaces, (b) Quotients, (c) Minkowski Sums, (d) Complex Interpolation. This yields a rich family of new examples where constant factor approximations are possible, which were beyond the reach of previous methods. We also show (under commonly used complexity assumptions) that for symmetric norms and unitarily invariant matrix norms the type-2 constant nearly characterizes the approximability of quadratic maximization.« less
2. The Communication Complexity of Optimization

   https://doi.org/10.1137/1.9781611975994.106  

   Vempala, Santosh S. ; Wang, Ruosong ; Woodruff, David P. ( January 2020 , ACM-SIAM Symposium on Discrete Algorithms)

   We consider the communication complexity of a number of distributed optimization problems. We start with the problem of solving a linear system. Suppose there is a coordinator together with s servers P1, …, Ps, the i-th of which holds a subset A(i) x = b(i) of ni constraints of a linear system in d variables, and the coordinator would like to output an x ϵ ℝd for which A(i) x = b(i) for i = 1, …, s. We assume each coefficient of each constraint is specified using L bits. We first resolve the randomized and deterministic communication complexity in the point-to-point model of communication, showing it is (d2 L + sd) and (sd2L), respectively. We obtain similar results for the blackboard communication model. As a result of independent interest, we show the probability a random matrix with integer entries in {–2L, …, 2L} is invertible is 1–2−Θ(dL), whereas previously only 1 – 2−Θ(d) was known. When there is no solution to the linear system, a natural alternative is to find the solution minimizing the ℓp loss, which is the ℓp regression problem. While this problem has been studied, we give improved upper or lower bounds for every value ofmore »p ≥ 1. One takeaway message is that sampling and sketching techniques, which are commonly used in earlier work on distributed optimization, are neither optimal in the dependence on d nor on the dependence on the approximation ε, thus motivating new techniques from optimization to solve these problems. Towards this end, we consider the communication complexity of optimization tasks which generalize linear systems, such as linear, semi-definite, and convex programming. For linear programming, we first resolve the communication complexity when d is constant, showing it is (sL) in the point-to-point model. For general d and in the point-to-point model, we show an Õ(sd3L) upper bound and an (d2 L + sd) lower bound. In fact, we show if one perturbs the coefficients randomly by numbers as small as 2−Θ(L), then the upper bound is Õ(sd2L) + poly(dL), and so this bound holds for almost all linear programs. Our study motivates understanding the bit complexity of linear programming, which is related to the running time in the unit cost RAM model with words of O(log(nd)) bits, and we give the fastest known algorithms for linear programming in this model. Read More: https://epubs.siam.org/doi/10.1137/1.9781611975994.106« less
3. A Complete Linear Programming Hierarchy for Linear Codes

   https://doi.org/10.4230/LIPIcs.ITCS.2022.51  

   Coregliano, Leonardo Nagami ( January 2022 , Leibniz international proceedings in informatics)

   Braverman, Mark (Ed.)

   A longstanding open problem in coding theory is to determine the best (asymptotic) rate R₂(δ) of binary codes with minimum constant (relative) distance δ. An existential lower bound was given by Gilbert and Varshamov in the 1950s. On the impossibility side, in the 1970s McEliece, Rodemich, Rumsey and Welch (MRRW) proved an upper bound by analyzing Delsarte’s linear programs. To date these results remain the best known lower and upper bounds on R₂(δ) with no improvement even for the important class of linear codes. Asymptotically, these bounds differ by an exponential factor in the blocklength. In this work, we introduce a new hierarchy of linear programs (LPs) that converges to the true size A^{Lin}₂(n,d) of an optimum linear binary code (in fact, over any finite field) of a given blocklength n and distance d. This hierarchy has several notable features: 1) It is a natural generalization of the Delsarte LPs used in the first MRRW bound. 2) It is a hierarchy of linear programs rather than semi-definite programs potentially making it more amenable to theoretical analysis. 3) It is complete in the sense that the optimum code size can be retrieved from level O(n²). 4) It provides an answer inmore »the form of a hierarchy (in larger dimensional spaces) to the question of how to cut Delsarte’s LP polytopes to approximate the true size of linear codes. We obtain our hierarchy by generalizing the Krawtchouk polynomials and MacWilliams inequalities to a suitable "higher-order" version taking into account interactions of 𝓁 words. Our method also generalizes to translation schemes under mild assumptions.« less
4. 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)

   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 themore »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 .« less
5. 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 hardnessmore »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.« less