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  1. null (Ed.)
    Can linear systems be solved faster than matrix multiplication? While there has been remarkable progress for the special cases of graph structured linear systems, in the general setting, the bit complexity of solving an $n \times n$ linear system $Ax=b$ is $\tilde{O}(n^\omega)$, where $\omega < 2.372864$ is the matrix multiplication exponent. Improving on this has been an open problem even for sparse linear systems with poly$(n)$ condition number. In this paper, we present an algorithm that solves linear systems in sparse matrices asymptotically faster than matrix multiplication for any $\omega > 2$. This speedup holds for any input matrix $A$ with $o(n^{\omega -1}/\log(\kappa(A)))$ non-zeros, where $\kappa(A)$ is the condition number of $A$. For poly$(n)$-conditioned matrices with $\tilde{O}(n)$ nonzeros, and the current value of $\omega$, the bit complexity of our algorithm to solve to within any $1/\text{poly}(n)$ error is $O(n^{2.331645})$. Our algorithm can be viewed as an efficient, randomized implementation of the block Krylov method via recursive low displacement rank factorizations. It is inspired by the algorithm of [Eberly et al. ISSAC `06 `07] for inverting matrices over finite fields. In our analysis of numerical stability, we develop matrix anti-concentration techniques to bound the smallest eigenvalue and the smallest gap in eigenvalues of semi-random matrices. 
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  2. null (Ed.)
    Graph compression or sparsification is a basic information-theoretic and computational question. A major open problem in this research area is whether $(1+\epsilon)$-approximate cut-preserving vertex sparsifiers with size close to the number of terminals exist. As a step towards this goal, we initiate the study of a thresholded version of the problem: for a given parameter $c$, find a smaller graph, which we call \emph{connectivity-$c$ mimicking network}, which preserves connectivity among $k$ terminals exactly up to the value of $c$. We show that connectivity-$c$ mimicking networks of size $O(kc^4)$ exist and can be found in time $m(c\log n)^{O(c)}$. We also give a separate algorithm that constructs such graphs of size $k \cdot O(c)^{2c}$ in time $mc^{O(c)}\log^{O(1)}n$. These results lead to the first offline data structures for answering fully dynamic $c$-edge-connectivity queries for $c \ge 4$ in polylogarithmic time per query as well as more efficient algorithms for survivable network design on bounded treewidth graphs. 
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  3. 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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