12 research outputs found

    Linear-Sized Spectral Sparsifiers and the Kadison-Singer Problem

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    The Kadison-Singer Conjecture, as proved by Marcus, Spielman, and Srivastava (MSS) [Ann. Math. 182, 327-350 (2015)], has been informally thought of as a strengthening of Batson, Spielman, and Srivastava's theorem that every undirected graph has a linear-sized spectral sparsifier [SICOMP 41, 1704-1721 (2012)]. We formalize this intuition by using a corollary of the MSS result to derive the existence of spectral sparsifiers with a number of edges linear in their number of vertices for all undirected, weighted graphs. The proof consists of two steps. First, following a suggestion of Srivastava [Asia Pac. Math. Newsl. 3, 15-20 (2013)], we show the result in the special case of graphs with bounded leverage scores by repeatedly applying the MSS corollary to partition the graph, while maintaining an appropriate bound on the leverage scores of each subgraph. Then, we extend to the general case by constructing a recursive algorithm that repeatedly (i) divides edges with high leverage scores into multiple parallel edges and (ii) uses the bounded leverage score case to sparsify the resulting graph

    Space Hardness of Solving Structured Linear Systems

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    Space-efficient Laplacian solvers are closely related to derandomization of space-bound randomized computations. We show that if the probabilistic logarithmic-space solver or the deterministic nearly logarithmic-space solver for undirected Laplacian matrices can be extended to solve slightly larger subclasses of linear systems, then they can be used to solve all linear systems with similar space complexity. Previously Kyng and Zhang [Rasmus Kyng and Peng Zhang, 2017] proved such results in the time complexity setting using reductions between approximate solvers. We prove that their reductions can be implemented using constant-depth, polynomial-size threshold circuits

    Error Reduction for Weighted PRGs Against Read Once Branching Programs

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    Pseudodistributions That Beat All Pseudorandom Generators (Extended Abstract)

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    New PRGs for Unbounded-Width/Adaptive-Order Read-Once Branching Programs

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    Pseudorandom Generators for Unbounded-Width Permutation Branching Programs

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    Better Pseudodistributions and Derandomization for Space-Bounded Computation

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    Spectral Sparsification via Bounded-Independence Sampling

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    We give a deterministic, nearly logarithmic-space algorithm for mild spectral sparsification of undirected graphs. Given a weighted, undirected graph GG on nn vertices described by a binary string of length NN, an integer klognk\leq \log n, and an error parameter ϵ>0\epsilon > 0, our algorithm runs in space O~(klog(Nwmax/wmin))\tilde{O}(k\log (N\cdot w_{\mathrm{max}}/w_{\mathrm{min}})) where wmaxw_{\mathrm{max}} and wminw_{\mathrm{min}} are the maximum and minimum edge weights in GG, and produces a weighted graph HH with O~(n1+2/k/ϵ2)\tilde{O}(n^{1+2/k}/\epsilon^2) edges that spectrally approximates GG, in the sense of Spielmen and Teng [ST04], up to an error of ϵ\epsilon. Our algorithm is based on a new bounded-independence analysis of Spielman and Srivastava's effective resistance based edge sampling algorithm [SS08] and uses results from recent work on space-bounded Laplacian solvers [MRSV17]. In particular, we demonstrate an inherent tradeoff (via upper and lower bounds) between the amount of (bounded) independence used in the edge sampling algorithm, denoted by kk above, and the resulting sparsity that can be achieved.Comment: 37 page

    Recursive Error Reduction for Regular Branching Programs

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    In a recent work, Chen, Hoza, Lyu, Tal and Wu (FOCS 2023) showed an improved error reduction framework for the derandomization of regular read-once branching programs (ROBPs). Their result is based on a clever modification to the inverse Laplacian perspective of space-bounded derandomization, which was originally introduced by Ahmadinejad, Kelner, Murtagh, Peebles, Sidford and Vadhan (FOCS 2020). In this work, we give an alternative error reduction framework for regular ROBPs. Our new framework is based on a binary recursive formula from the work of Chattopadhyay and Liao (CCC 2020), that they used to construct weighted pseudorandom generators (WPRGs) for general ROBPs. Based on our new error reduction framework, we give alternative proofs to the following results for regular ROBPs of length nn and width ww, both of which were proved in the work of Chen et al. using their error reduction: \bullet There is a WPRG with error ε\varepsilon that has seed length O~(log(n)(log(1/ε)+log(w))+log(1/ε)).\tilde{O}(\log(n)(\sqrt{\log(1/\varepsilon)}+\log(w))+\log(1/\varepsilon)). \bullet There is a (non-black-box) deterministic algorithm which estimates the expectation of any such program within error ±ε\pm\varepsilon with space complexity O~(log(nw)loglog(1/ε)).\tilde{O}(\log(nw)\cdot\log\log(1/\varepsilon)). (This was first proved in the work of Ahmadinejad et al., but the proof by Chen et al. is simpler.) Because of the binary recursive nature of our new framework, both of our proofs are based on a straightforward induction that is arguably simpler than the Laplacian-based proof in the work of Chen et al

    Hitting Sets Give Two-Sided Derandomization of Small Space

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