More Like this

1. Computing Nash Equilibria in Multidimensional Congestion Games (Extended Abstract)

   Irfan, Mohammad T; Chan, Hau; Soundy, Jared (May 2024, Proceedings of the 23rd International Conference on Autonomous Agents and Multiagent Systems (AAMAS 2024))

   We study pure-strategy Nash equilibrium (PSNE) computation in 𝑘-dimensional congestion games (𝑘-DCGs) where the weights or demands of the players are 𝑘-dimensional vectors. We first show that deciding the existence of a PSNE in a 𝑘-DCG is NP-complete even for games when players have binary and unit demand vectors. We then focus on computing PSNE for 𝑘-DCGs and their variants with general, linear, and exponential cost functions. For general cost functions (potentially non-monotonic), we provide the first configuration-space framework to find a PSNE if one exists. For linear and exponential cost functions, we provide potential function-based algorithms to find a PSNE. These algorithms run in polynomial time under certain assumptions. We also study structured demands and cost functions, giving polynomial-time algorithms to compute PSNE for several cases. For general cost functions, we give a constructive proof of existence for an (𝛼, 𝛽)-PSNE (for certain 𝛼 and 𝛽), where 𝛼 and 𝛽 are multiplicative and additive approximation factors, respectively. 

   more » « less
2. 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 . 

   more » « less
3. On Sketching Approximations for Symmetric Boolean CSPs

   Boyland, Joanna; Hwang, Michael; Prasad, Tarun; Singer, Noah; Velusamy, Santhoshini (September 2022, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022))

   Chakrabarti, Amit; Swamy, Chaitanya (Ed.)

   A Boolean maximum constraint satisfaction problem, Max-CSP(f), is specified by a predicate f:{-1,1}^k → {0,1}. An n-variable instance of Max-CSP(f) consists of a list of constraints, each of which applies f to k distinct literals drawn from the n variables. For k = 2, Chou, Golovnev, and Velusamy [Chou et al., 2020] obtained explicit ratios characterizing the √ n-space streaming approximability of every predicate. For k ≥ 3, Chou, Golovnev, Sudan, and Velusamy [Chou et al., 2022] proved a general dichotomy theorem for √ n-space sketching algorithms: For every f, there exists α(f) ∈ (0,1] such that for every ε > 0, Max-CSP(f) is (α(f)-ε)-approximable by an O(log n)-space linear sketching algorithm, but (α(f)+ε)-approximation sketching algorithms require Ω(√n) space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting α'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}, we show that for odd k ≥ 3, α(kAND) = α'_k, and for even k ≥ 2, α(kAND) = 2α'_{k+1}. Thus, for every k, kAND can be (2-o(1))2^{-k}-approximated by O(log n)-space sketching algorithms; we contrast this with a lower bound of Chou, Golovnev, Sudan, Velingker, and Velusamy [Chou et al., 2022] implying that streaming (2+ε)2^{-k}-approximations require Ω(n) space! We also resolve the ratio for the "at-least-(k-1)-1’s" function for all even k; the "exactly-(k+1)/2-1’s" function for odd k ∈ {3,…,51}; and fifteen other functions. We stress here that for general f, the dichotomy theorem in [Chou et al., 2022] only implies that α(f) can be computed to arbitrary precision in PSPACE, and thus closed-form expressions need not have existed a priori. Our analyses involve identifying and exploiting structural "saddle-point" properties of this dichotomy. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [Chou et al., 2020] while simplifying [Chou et al., 2022]. Finally, we investigate the √ n-space streaming lower bounds in [Chou et al., 2022], and show that they are incomplete for 3AND, i.e., they fail to rule out (α(3AND})-ε)-approximations in o(√ n) space. 

   more » « less
4. Reconstruction algorithms for low-rank tensors and depth-3 multilinear circuits

   https://doi.org/10.1145/3406325.3451096  

   Bhargava, Vishwas; Saraf, Shubhangi; Volkovich, Ilya (June 2021, STOC 2021: Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing)

   null (Ed.)

   We give new and efficient black-box reconstruction algorithms for some classes of depth-3 arithmetic circuits. As a consequence, we obtain the first efficient algorithm for computing the tensor rank and for finding the optimal tensor decomposition as a sum of rank-one tensors when then input is a constant-rank tensor. More specifically, we provide efficient learning algorithms that run in randomized polynomial time over general fields and in deterministic polynomial time over and for the following classes: 1) Set-multilinear depth-3 circuits of constant top fan-in ((k) circuits). As a consequence of our algorithm, we obtain the first polynomial time algorithm for tensor rank computation and optimal tensor decomposition of constant-rank tensors. This result holds for d dimensional tensors for any d, but is interesting even for d=3. 2) Sums of powers of constantly many linear forms ((k) circuits). As a consequence we obtain the first polynomial-time algorithm for tensor rank computation and optimal tensor decomposition of constant-rank symmetric tensors. 3) Multilinear depth-3 circuits of constant top fan-in (multilinear (k) circuits). Our algorithm works over all fields of characteristic 0 or large enough characteristic. Prior to our work the only efficient algorithms known were over polynomially-sized finite fields (see. Karnin-Shpilka 09’). Prior to our work, the only polynomial-time or even subexponential-time algorithms known (deterministic or randomized) for subclasses of (k) circuits that also work over large/infinite fields were for the setting when the top fan-in k is at most 2 (see Sinha 16’ and Sinha 20’). 

   more » « less
5. Universal Algorithms for Clustering

   Bruce Maggs, Arun Ganesh (April 2021, Leibniz international proceedings in informatics)

   Bansal, Nikhil and (Ed.)

   his paper presents universal algorithms for clustering problems, including the widely studied k-median, k-means, and k-center objectives. The input is a metric space containing all potential client locations. The algorithm must select k cluster centers such that they are a good solution for any subset of clients that actually realize. Specifically, we aim for low regret, defined as the maximum over all subsets of the difference between the cost of the algorithm’s solution and that of an optimal solution. A universal algorithm’s solution sol for a clustering problem is said to be an (α, β)-approximation if for all subsets of clients C', it satisfies sol(C') ≤ α ⋅ opt(C') + β ⋅ mr, where opt(C') is the cost of the optimal solution for clients C' and mr is the minimum regret achievable by any solution. Our main results are universal algorithms for the standard clustering objectives of k-median, k-means, and k-center that achieve (O(1), O(1))-approximations. These results are obtained via a novel framework for universal algorithms using linear programming (LP) relaxations. These results generalize to other 𝓁_p-objectives and the setting where some subset of the clients are fixed. We also give hardness results showing that (α, β)-approximation is NP-hard if α or β is at most a certain constant, even for the widely studied special case of Euclidean metric spaces. This shows that in some sense, (O(1), O(1))-approximation is the strongest type of guarantee obtainable for universal clustering. 

   more » « less