308 research outputs found
On Weighted Graph Sparsification by Linear Sketching
A seminal work of [Ahn-Guha-McGregor, PODS'12] showed that one can compute a
cut sparsifier of an unweighted undirected graph by taking a near-linear number
of linear measurements on the graph. Subsequent works also studied computing
other graph sparsifiers using linear sketching, and obtained near-linear upper
bounds for spectral sparsifiers [Kapralov-Lee-Musco-Musco-Sidford, FOCS'14] and
first non-trivial upper bounds for spanners [Filtser-Kapralov-Nouri, SODA'21].
All these linear sketching algorithms, however, only work on unweighted graphs.
In this paper, we initiate the study of weighted graph sparsification by
linear sketching by investigating a natural class of linear sketches that we
call incidence sketches, in which each measurement is a linear combination of
the weights of edges incident on a single vertex. Our results are:
1. Weighted cut sparsification: We give an algorithm that computes a -cut sparsifier using linear
measurements, which is nearly optimal.
2. Weighted spectral sparsification: We give an algorithm that computes a -spectral sparsifier using
linear measurements. Complementing our algorithm, we then prove a superlinear
lower bound of measurements for computing some
-spectral sparsifier using incidence sketches.
3. Weighted spanner computation: We focus on graphs whose largest/smallest
edge weights differ by an factor, and prove that, for incidence
sketches, the upper bounds obtained by~[Filtser-Kapralov-Nouri, SODA'21] are
optimal up to an factor
Sketching Cuts in Graphs and Hypergraphs
Sketching and streaming algorithms are in the forefront of current research
directions for cut problems in graphs. In the streaming model, we show that
-approximation for Max-Cut must use space;
moreover, beating -approximation requires polynomial space. For the
sketching model, we show that -uniform hypergraphs admit a
-cut-sparsifier (i.e., a weighted subhypergraph that
approximately preserves all the cuts) with
edges. We also make first steps towards sketching general CSPs (Constraint
Satisfaction Problems)
- …