103 research outputs found

    Optimal Bounds for the kk-cut Problem

    Full text link
    In the kk-cut problem, we want to find the smallest set of edges whose deletion breaks a given (multi)graph into kk connected components. Algorithms of Karger & Stein and Thorup showed how to find such a minimum kk-cut in time approximately O(n2k)O(n^{2k}). The best lower bounds come from conjectures about the solvability of the kk-clique problem, and show that solving kk-cut is likely to require time Ω(nk)\Omega(n^k). Recent results of Gupta, Lee & Li have given special-purpose algorithms that solve the problem in time n1.98k+O(1)n^{1.98k + O(1)}, and ones that have better performance for special classes of graphs (e.g., for small integer weights). In this work, we resolve the problem for general graphs, by showing that the Contraction Algorithm of Karger outputs any fixed kk-cut of weight αλk\alpha \lambda_k with probability Ωk(n−αk)\Omega_k(n^{-\alpha k}), where λk\lambda_k denotes the minimum kk-cut size. This also gives an extremal bound of Ok(nk)O_k(n^k) on the number of minimum kk-cuts and an algorithm to compute a minimum kk-cut in similar runtime. Both are tight up to lower-order factors, with the algorithmic lower bound assuming hardness of max-weight kk-clique. The first main ingredient in our result is a fine-grained analysis of how the graph shrinks -- and how the average degree evolves -- in the Karger process. The second ingredient is an extremal bound on the number of cuts of size less than 2λk/k2 \lambda_k/k, using the Sunflower lemma.Comment: Final version of arXiv:1911.09165 with new and more general proof

    Matrix probing and its conditioning

    Full text link
    When a matrix A with n columns is known to be well approximated by a linear combination of basis matrices B_1,..., B_p, we can apply A to a random vector and solve a linear system to recover this linear combination. The same technique can be used to recover an approximation to A^-1. A basic question is whether this linear system is invertible and well-conditioned. In this paper, we show that if the Gram matrix of the B_j's is sufficiently well-conditioned and each B_j has a high numerical rank, then n {proportional} p log^2 n will ensure that the linear system is well-conditioned with high probability. Our main application is probing linear operators with smooth pseudodifferential symbols such as the wave equation Hessian in seismic imaging. We demonstrate numerically that matrix probing can also produce good preconditioners for inverting elliptic operators in variable media

    Practical Minimum Cut Algorithms

    Full text link
    The minimum cut problem for an undirected edge-weighted graph asks us to divide its set of nodes into two blocks while minimizing the weight sum of the cut edges. Here, we introduce a linear-time algorithm to compute near-minimum cuts. Our algorithm is based on cluster contraction using label propagation and Padberg and Rinaldi's contraction heuristics [SIAM Review, 1991]. We give both sequential and shared-memory parallel implementations of our algorithm. Extensive experiments on both real-world and generated instances show that our algorithm finds the optimal cut on nearly all instances significantly faster than other state-of-the-art algorithms while our error rate is lower than that of other heuristic algorithms. In addition, our parallel algorithm shows good scalability

    Learning and Testing Variable Partitions

    Get PDF
    Let FF be a multivariate function from a product set Σn\Sigma^n to an Abelian group GG. A kk-partition of FF with cost δ\delta is a partition of the set of variables V\mathbf{V} into kk non-empty subsets (X1,…,Xk)(\mathbf{X}_1, \dots, \mathbf{X}_k) such that F(V)F(\mathbf{V}) is δ\delta-close to F1(X1)+⋯+Fk(Xk)F_1(\mathbf{X}_1)+\dots+F_k(\mathbf{X}_k) for some F1,…,FkF_1, \dots, F_k with respect to a given error metric. We study algorithms for agnostically learning kk partitions and testing kk-partitionability over various groups and error metrics given query access to FF. In particular we show that 1.1. Given a function that has a kk-partition of cost δ\delta, a partition of cost O(kn2)(δ+ϵ)\mathcal{O}(k n^2)(\delta + \epsilon) can be learned in time O~(n2poly(1/ϵ))\tilde{\mathcal{O}}(n^2 \mathrm{poly} (1/\epsilon)) for any ϵ>0\epsilon > 0. In contrast, for k=2k = 2 and n=3n = 3 learning a partition of cost δ+ϵ\delta + \epsilon is NP-hard. 2.2. When FF is real-valued and the error metric is the 2-norm, a 2-partition of cost δ2+ϵ\sqrt{\delta^2 + \epsilon} can be learned in time O~(n5/ϵ2)\tilde{\mathcal{O}}(n^5/\epsilon^2). 3.3. When FF is Zq\mathbb{Z}_q-valued and the error metric is Hamming weight, kk-partitionability is testable with one-sided error and O(kn3/ϵ)\mathcal{O}(kn^3/\epsilon) non-adaptive queries. We also show that even two-sided testers require Ω(n)\Omega(n) queries when k=2k = 2. This work was motivated by reinforcement learning control tasks in which the set of control variables can be partitioned. The partitioning reduces the task into multiple lower-dimensional ones that are relatively easier to learn. Our second algorithm empirically increases the scores attained over previous heuristic partitioning methods applied in this context.Comment: Innovations in Theoretical Computer Science (ITCS) 202

    Fast and Deterministic Approximations for k-Cut

    Get PDF
    In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time? We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut

    Computing the Girth of a Planar Graph in Linear Time

    Full text link
    The girth of a graph is the minimum weight of all simple cycles of the graph. We study the problem of determining the girth of an n-node unweighted undirected planar graph. The first non-trivial algorithm for the problem, given by Djidjev, runs in O(n^{5/4} log n) time. Chalermsook, Fakcharoenphol, and Nanongkai reduced the running time to O(n log^2 n). Weimann and Yuster further reduced the running time to O(n log n). In this paper, we solve the problem in O(n) time.Comment: 20 pages, 7 figures, accepted to SIAM Journal on Computin
    • …
    corecore