65 research outputs found
A Cut-Matching Game for Constant-Hop Expanders
This paper provides a cut-strategy that produces constant-hop expanders in
the well-known cut-matching game framework.
Constant-hop expanders strengthen expanders with constant conductance by
guaranteeing that any demand can be (obliviously) routed along constant-hop
paths - in contrast to the -hop routes in expanders.
Cut-matching games for expanders are key tools for obtaining
close-to-linear-time approximation algorithms for many hard problems, including
finding (balanced or approximately-largest) sparse cuts, certifying the
expansion of a graph by embedding an (explicit) expander, as well as computing
expander decompositions, hierarchical cut decompositions, oblivious routings,
multi-cuts, and multicommodity flows. The cut-matching game provided in this
paper is crucial in extending this versatile and powerful machinery to
constant-hop expanders. It is also a key ingredient towards close-to-linear
time algorithms for computing a constant approximation of multicommodity-flows
and multi-cuts - the approximation factor being a constant relies on the
expanders being constant-hop
Expander Decomposition with Fewer Inter-Cluster Edges Using a Spectral Cut Player
A -expander-decomposition of a graph is a partition of
into clusters with conductance ,
such that there are at most inter-cluster edges. We consider the
problem of computing such a decomposition for a given and , in near
linear time, while minimizing as a function of . Saranurak and
Wang [SW19] gave a randomized algorithm for
computing a -expander decomposition. There are graphs that
do not admit an expander decomposition with less than inter-cluster edges, so the number of inter-cluster edges in [SW19] is
within two logarithmic factors of optimal. As a main building block, [SW19] use
an adaptation of the algorithm of R\"{a}cke et al. [RST14] for computing an
approximate balanced sparse cut. Both algorithms rely on the cut-matching game
of Khandekar et al. [KRV09].
Orecchia et al. [OSVV08], using spectral analysis, improved upon [KRV09] by
giving a fast algorithm that computes a sparse cut with better approximation
guarantee. Using the technique of [OSVV08] for computing expander
decompositions or balanced cuts [RST14, SW19], encounters many hurdles. In
particular, in [RST14, SW19] the relevant part of the graph constantly changes,
making it difficult to perform spectral analysis.
In this paper, we manage to exploit the technique of [OSVV08] to compute an
expander decomposition, improving the result by Saranurak and Wang [SW19].
Specifically, we give a randomized algorithm for computing a
-expander decomposition of a graph, in
time. Our new result is achieved by using a
novel combination of a symmetric version of the potential functions of [OSVV08,
RST14, SW19] with a new variation of Cheeger's inequality for the notion of
near-expansion.Comment: 43 pages; Revised the introduction, fixed typo
Locality-Aware Qubit Routing for the Grid Architecture
Due to the short decohorence time of qubits available in the NISQ-era, it is essential to pack (minimize the size and or the depth of) a logical quantum circuit as efficiently as possible given a sparsely coupled physical architecture. In this work we introduce a locality-aware qubit routing algorithm based on a graph theoretic framework. Our algorithm is designed for the grid and certain \u27grid-like\u27 architectures. We experimentally show the competitiveness of algorithm by comparing it against the approximate token swapping algorithm, which is used as a primitive in many state-of-the-art quantum trans pilers. Our algorithm produces circuits of comparable depth (better on random permutations) while being an order of magnitude faster than a typical implementation of the approximate token swapping algorithm
Dynamic Maxflow via Dynamic Interior Point Methods
In this paper we provide an algorithm for maintaining a
-approximate maximum flow in a dynamic, capacitated graph
undergoing edge additions. Over a sequence of -additions to an -node
graph where every edge has capacity our algorithm runs in
time . To obtain this result we
design dynamic data structures for the more general problem of detecting when
the value of the minimum cost circulation in a dynamic graph undergoing edge
additions obtains value at most (exactly) for a given threshold . Over a
sequence -additions to an -node graph where every edge has capacity
and cost we solve this thresholded
minimum cost flow problem in . Both of our algorithms
succeed with high probability against an adaptive adversary. We obtain these
results by dynamizing the recent interior point method used to obtain an almost
linear time algorithm for minimum cost flow (Chen, Kyng, Liu, Peng, Probst
Gutenberg, Sachdeva 2022), and introducing a new dynamic data structure for
maintaining minimum ratio cycles in an undirected graph that succeeds with high
probability against adaptive adversaries.Comment: 30 page
Undergraduate Catalog, 2022-2023
https://scholar.valpo.edu/undergradcatalogs/1097/thumbnail.jp
Undergraduate Catalog, 2021-2022
https://scholar.valpo.edu/undergradcatalogs/1096/thumbnail.jp
Faster Parallel Algorithm for Approximate Shortest Path
We present the first work, time
algorithm in the PRAM model that computes -approximate
single-source shortest paths on weighted, undirected graphs. This improves upon
the breakthrough result of Cohen~[JACM'00] that achieves
work and time. While most previous approaches, including
Cohen's, leveraged the power of hopsets, our algorithm builds upon the recent
developments in \emph{continuous optimization}, studying the shortest path
problem from the lens of the closely-related \emph{minimum transshipment}
problem. To obtain our algorithm, we demonstrate a series of near-linear work,
polylogarithmic-time reductions between the problems of approximate shortest
path, approximate transshipment, and -embeddings, and establish a
recursive algorithm that cycles through the three problems and reduces the
graph size on each cycle. As a consequence, we also obtain faster parallel
algorithms for approximate transshipment and -embeddings with
polylogarithmic distortion. The minimum transshipment algorithm in particular
improves upon the previous best work sequential algorithm of
Sherman~[SODA'17].
To improve readability, the paper is almost entirely self-contained, save for
several staple theorems in algorithms and combinatorics.Comment: 53 pages, STOC 202
Undergraduate Catalog, 2020-2021
https://scholar.valpo.edu/undergradcatalogs/1095/thumbnail.jp
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