3,262 research outputs found
The Fast Heuristic Algorithms and Post-Processing Techniques to Design Large and Low-Cost Communication Networks
It is challenging to design large and low-cost communication networks. In
this paper, we formulate this challenge as the prize-collecting Steiner Tree
Problem (PCSTP). The objective is to minimize the costs of transmission routes
and the disconnected monetary or informational profits. Initially, we note that
the PCSTP is MAX SNP-hard. Then, we propose some post-processing techniques to
improve suboptimal solutions to PCSTP. Based on these techniques, we propose
two fast heuristic algorithms: the first one is a quasilinear time heuristic
algorithm that is faster and consumes less memory than other algorithms; and
the second one is an improvement of a stateof-the-art polynomial time heuristic
algorithm that can find high-quality solutions at a speed that is only inferior
to the first one. We demonstrate the competitiveness of our heuristic
algorithms by comparing them with the state-of-the-art ones on the largest
existing benchmark instances (169 800 vertices and 338 551 edges). Moreover, we
generate new instances that are even larger (1 000 000 vertices and 10 000 000
edges) to further demonstrate their advantages in large networks. The
state-ofthe-art algorithms are too slow to find high-quality solutions for
instances of this size, whereas our new heuristic algorithms can do this in
around 6 to 45s on a personal computer. Ultimately, we apply our
post-processing techniques to update the bestknown solution for a notoriously
difficult benchmark instance to show that they can improve near-optimal
solutions to PCSTP. In conclusion, we demonstrate the usefulness of our
heuristic algorithms and post-processing techniques for designing large and
low-cost communication networks
Fast and Deterministic Approximations for k-Cut
In an undirected graph, a k-cut is a set of edges whose removal breaks the graph into at least k connected components. The minimum weight k-cut can be computed in n^O(k) time, but when k is treated as part of the input, computing the minimum weight k-cut is NP-Hard [Goldschmidt and Hochbaum, 1994]. For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi, 2017]. Saran and Vazirani [1995] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed via O(k) minimum cuts, which implies a O~(km) randomized running time via the nearly linear time randomized min-cut algorithm of Karger [2000]. Nagamochi and Kamidoi [2007] showed that a (2 - 2/k)-approximately minimum weight k-cut can be computed deterministically in O(mn + n^2 log n) time. These results prompt two basic questions. The first concerns the role of randomization. Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of O~(km)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut. Can 2-approximate k-cuts be computed as fast as the minimum cut - in O~(m) randomized time?
We give a deterministic approximation algorithm that computes (2 + eps)-minimum k-cuts in O(m log^3 n / eps^2) time, via a (1 + eps)-approximation for an LP relaxation of k-cut
Colored Non-Crossing Euclidean Steiner Forest
Given a set of -colored points in the plane, we consider the problem of
finding trees such that each tree connects all points of one color class,
no two trees cross, and the total edge length of the trees is minimized. For
, this is the well-known Euclidean Steiner tree problem. For general ,
a -approximation algorithm is known, where is the
Steiner ratio.
We present a PTAS for , a -approximation algorithm
for , and two approximation algorithms for general~, with ratios
and
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
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