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1. Bipartite Matching in Nearly-linear Time on Moderately Dense Graphs

   https://doi.org/10.1109/FOCS46700.2020.00090  

   van den Brand, Jan; Lee, Yin-Tat; Nanongkai, Danupon; Peng, Richard; Saranurak, Thatchaphol; Sidford, Aaron; Song, Zhao; Wang, Di (November 2020, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   null (Ed.)

   We present an $$\tilde O(m+n^{1.5})$$-time randomized algorithm for maximum cardinality bipartite matching and related problems (e.g. transshipment, negative-weight shortest paths, and optimal transport) on $$m$$-edge, $$n$$-node graphs. For maximum cardinality bipartite matching on moderately dense graphs, i.e. $$m = \Omega(n^{1.5})$$, our algorithm runs in time nearly linear in the input size and constitutes the first improvement over the classic $$O(m\sqrt{n})$$-time [Dinic 1970; Hopcroft-Karp 1971; Karzanov 1973] and $$\tilde O(n^\omega)$$-time algorithms [Ibarra-Moran 1981] (where currently $$\omega\approx 2.373$$). On sparser graphs, i.e. when $$m = n^{9/8 + \delta}$$ for any constant $$\delta>0$$, our result improves upon the recent advances of [Madry 2013] and [Liu-Sidford 2020b, 2020a] which achieve an $$\tilde O(m^{4/3+o(1)})$$ runtime. We obtain these results by combining and advancing recent lines of research in interior point methods (IPMs) and dynamic graph algorithms. First, we simplify and improve the IPM of [v.d.Brand-Lee-Sidford-Song 2020], providing a general primal-dual IPM framework and new sampling-based techniques for handling infeasibility induced by approximate linear system solvers. Second, we provide a simple sublinear-time algorithm for detecting and sampling high-energy edges in electric flows on expanders and show that when combined with recent advances in dynamic expander decompositions, this yields efficient data structures for maintaining the iterates of both [v.d.Brand~et~al.] and our new IPMs. Combining this general machinery yields a simpler $$\tilde O(n \sqrt{m})$$ time algorithm for matching based on the logarithmic barrier function, and our state-of-the-art $$\tilde O(m+n^{1.5})$$ time algorithm for matching based on the [Lee-Sidford 2014] barrier (as regularized in [v.d.Brand~et~al.]). 

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2. Fast Dynamic Cuts, Distances and Effective Resistances via Vertex Sparsifiers

   https://doi.org/10.1109/FOCS46700.2020.00109  

   Chen, Li; Goranci, Gramoz; Henzinger, Monika; Peng, Richard; Saranurak, Thatchaphol (November 2020, 61st IEEE Annual Symposium on Foundations of Computer Science, FOCS 2020, Durham, NC, USA, November 16-19, 2020)

   null (Ed.)

   We present a general framework of designing efficient dynamic approximate algorithms for optimization on undirected graphs. In particular, we develop a technique that, given any problem that admits a certain notion of vertex sparsifiers, gives data structures that maintain approximate solutions in sub-linear update and query time. We illustrate the applicability of our paradigm to the following problems. (1) A fully-dynamic algorithm that approximates all-pair maximum-flows/minimum-cuts up to a nearly logarithmic factor in $$\tilde{O}(n^{2/3})$$ amortized time against an oblivious adversary, and $$\tilde{O}(m^{3/4})$$ time against an adaptive adversary. (2) An incremental data structure that maintains $O(1)$-approximate shortest path in $$n^{o(1)}$$ time per operation, as well as fully dynamic approximate all-pair shortest path and transshipment in $$\tilde{O}(n^{2/3+o(1)})$$ amortized time per operation. (3) A fully-dynamic algorithm that approximates all-pair effective resistance up to an $$(1+\eps)$$ factor in $$\tilde{O}(n^{2/3+o(1)} \epsilon^{-O(1)})$$ amortized update time per operation. The key tool behind result (1) is the dynamic maintenance of an algorithmic construction due to Madry [FOCS' 10], which partitions a graph into a collection of simpler graph structures (known as j-trees) and approximately captures the cut-flow and metric structure of the graph. The $O(1)$-approximation guarantee of (2) is by adapting the distance oracles by [Thorup-Zwick JACM `05]. Result (3) is obtained by invoking the random-walk based spectral vertex sparsifier by [Durfee et al. STOC `19] in a hierarchical manner, while carefully keeping track of the recourse among levels in the hierarchy. 

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3. Quantum SDP Solvers: Large Speed-Ups, Optimality, and Applications to Quantum Learning

   Brand; Kalev, Amir; Li, Tongyang; Lin, Cedric Yen-Yu; Svore, Krysta M.; Wu, Xiaodi (July 2019, Leibniz international proceedings in informatics)

   We give two new quantum algorithms for solving semidefinite programs (SDPs) providing quantum speed-ups. We consider SDP instances with m constraint matrices, each of dimension n, rank at most r, and sparsity s. The first algorithm assumes an input model where one is given access to an oracle to the entries of the matrices at unit cost. We show that it has run time O~(s^2 (sqrt{m} epsilon^{-10} + sqrt{n} epsilon^{-12})), with epsilon the error of the solution. This gives an optimal dependence in terms of m, n and quadratic improvement over previous quantum algorithms (when m ~~ n). The second algorithm assumes a fully quantum input model in which the input matrices are given as quantum states. We show that its run time is O~(sqrt{m}+poly(r))*poly(log m,log n,B,epsilon^{-1}), with B an upper bound on the trace-norm of all input matrices. In particular the complexity depends only polylogarithmically in n and polynomially in r. We apply the second SDP solver to learn a good description of a quantum state with respect to a set of measurements: Given m measurements and a supply of copies of an unknown state rho with rank at most r, we show we can find in time sqrt{m}*poly(log m,log n,r,epsilon^{-1}) a description of the state as a quantum circuit preparing a density matrix which has the same expectation values as rho on the m measurements, up to error epsilon. The density matrix obtained is an approximation to the maximum entropy state consistent with the measurement data considered in Jaynes' principle from statistical mechanics. As in previous work, we obtain our algorithm by "quantizing" classical SDP solvers based on the matrix multiplicative weight update method. One of our main technical contributions is a quantum Gibbs state sampler for low-rank Hamiltonians, given quantum states encoding these Hamiltonians, with a poly-logarithmic dependence on its dimension, which is based on ideas developed in quantum principal component analysis. We also develop a "fast" quantum OR lemma with a quadratic improvement in gate complexity over the construction of Harrow et al. [Harrow et al., 2017]. We believe both techniques might be of independent interest. 

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4. Massively Parallel Algorithms for Small Subgraph Counting

   Biswas, Amartya Shankha; Eden, Talya; Liu, Quanquan C.; Rubinfeld, Ronitt Slobodan (January 2022, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022))

   Over the last two decades, frameworks for distributed-memory parallel computation, such as MapReduce, Hadoop, Spark and Dryad, have gained significant popularity with the growing prevalence of large network datasets. The Massively Parallel Computation (MPC) model is the de-facto standard for studying graph algorithms in these frameworks theoretically. Subgraph counting is one such fundamental problem in analyzing massive graphs, with the main algorithmic challenges centering on designing methods which are both scalable and accurate. Given a graph G = (V, E) with n vertices, m edges and T triangles, our first result is an algorithm that outputs a (1+ε)-approximation to T, with asymptotically optimal round and total space complexity provided any S ≥ max{(√ m, n²/m)} space per machine and assuming T = Ω(√{m/n}). Our result gives a quadratic improvement on the bound on T over previous works. We also provide a simple extension of our result to counting any subgraph of k size for constant k ≥ 1. Our second result is an O_δ(log log n)-round algorithm for exactly counting the number of triangles, whose total space usage is parametrized by the arboricity α of the input graph. We extend this result to exactly counting k-cliques for any constant k. Finally, we prove that a recent result of Bera, Pashanasangi and Seshadhri (ITCS 2020) for exactly counting all subgraphs of size at most 5 can be implemented in the MPC model in Õ_δ(√{log n}) rounds, O(n^δ) space per machine and O(mα³) total space. In addition to our theoretical results, we simulate our triangle counting algorithms in real-world graphs obtained from the Stanford Network Analysis Project (SNAP) database. Our results show that both our approximate and exact counting algorithms exhibit improvements in terms of round complexity and approximation ratio, respectively, compared to two previous widely used algorithms for these problems. 

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5. Feasible Q-Learning for Average Reward Reinforcement Learning

   Ying, Jin; Ramki, Gummadi; Zhengyuan, Zhou; Blanchet, Jose (May 2024, Proceedings of The 27th International Conference on Artificial Intelligence and Statistics, PMLR)

   Dasgupta, Sanjoy; Mandt, Stephan; Li, Yingzhen (Ed.)

   Average reward reinforcement learning (RL) provides a suitable framework for capturing the objective (i.e. long-run average reward) for continuing tasks, where there is often no natural way to identify a discount factor. However, existing average reward RL algorithms with sample complexity guarantees are not feasible, as they take as input the (unknown) mixing time of the Markov decision process (MDP). In this paper, we make initial progress towards addressing this open problem. We design a feasible average-reward $$Q$$-learning framework that requires no knowledge of any problem parameter as input. Our framework is based on discounted $$Q$$-learning, while we dynamically adapt the discount factor (and hence the effective horizon) to progressively approximate the average reward. In the synchronous setting, we solve three tasks: (i) learn a policy that is $$\epsilon$$-close to optimal, (ii) estimate optimal average reward with $$\epsilon$$-accuracy, and (iii) estimate the bias function (similar to $$Q$$-function in discounted case) with $$\epsilon$$-accuracy. We show that with carefully designed adaptation schemes, (i) can be achieved with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^{8}}{\epsilon^{8}})$$ samples, (ii) with $$\tilde{O}(\frac{SA t_{\mathrm{mix}}^5}{\epsilon^5})$$ samples, and (iii) with $$\tilde{O}(\frac{SA B}{\epsilon^9})$$ samples, where $$t_\mathrm{mix}$$ is the mixing time, and $B > 0$ is an MDP-dependent constant. To our knowledge, we provide the first finite-sample guarantees that are polynomial in $$S, A, t_{\mathrm{mix}}, \epsilon$$ for a feasible variant of $$Q$$-learning. That said, the sample complexity bounds have tremendous room for improvement, which we leave for the community’s best minds. Preliminary simulations verify that our framework is effective without prior knowledge of parameters as input. 

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