12 research outputs found
Linear-Sized Spectral Sparsifiers and the Kadison-Singer Problem
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
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
Spectral Sparsification via Bounded-Independence Sampling
We give a deterministic, nearly logarithmic-space algorithm for mild spectral
sparsification of undirected graphs. Given a weighted, undirected graph on
vertices described by a binary string of length , an integer , and an error parameter , our algorithm runs in space
where
and are the maximum and minimum edge
weights in , and produces a weighted graph with
edges that spectrally approximates , in
the sense of Spielmen and Teng [ST04], up to an error of .
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 above, and the resulting sparsity that can be
achieved.Comment: 37 page
Recursive Error Reduction for Regular Branching Programs
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 and width , both of which
were proved in the work of Chen et al. using their error reduction:
There is a WPRG with error that has seed length
There is a (non-black-box) deterministic algorithm which estimates
the expectation of any such program within error with space
complexity (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