23,944 research outputs found
Weighted Min-Cut: Sequential, Cut-Query and Streaming Algorithms
Consider the following 2-respecting min-cut problem. Given a weighted graph
and its spanning tree , find the minimum cut among the cuts that contain
at most two edges in . This problem is an important subroutine in Karger's
celebrated randomized near-linear-time min-cut algorithm [STOC'96]. We present
a new approach for this problem which can be easily implemented in many
settings, leading to the following randomized min-cut algorithms for weighted
graphs.
* An -time sequential algorithm:
This improves Karger's and bounds when the input graph is not extremely
sparse or dense. Improvements over Karger's bounds were previously known only
under a rather strong assumption that the input graph is simple [Henzinger et
al. SODA'17; Ghaffari et al. SODA'20]. For unweighted graphs with parallel
edges, our bound can be improved to .
* An algorithm requiring cut queries to compute the min-cut of
a weighted graph: This answers an open problem by Rubinstein et al. ITCS'18,
who obtained a similar bound for simple graphs.
* A streaming algorithm that requires space and
passes to compute the min-cut: The only previous non-trivial exact min-cut
algorithm in this setting is the 2-pass -space algorithm on simple
graphs [Rubinstein et al., ITCS'18] (observed by Assadi et al. STOC'19).
In contrast to Karger's 2-respecting min-cut algorithm which deploys
sophisticated dynamic programming techniques, our approach exploits some cute
structural properties so that it only needs to compute the values of cuts corresponding to removing pairs of tree edges, an
operation that can be done quickly in many settings.Comment: Updates on this version: (1) Minor corrections in Section 5.1, 5.2;
(2) Reference to newer results by GMW SOSA21 (arXiv:2008.02060v2), DEMN
STOC21 (arXiv:2004.09129v2) and LMN 21 (arXiv:2102.06565v1
The First Proven Performance Guarantees for the Non-Dominated Sorting Genetic Algorithm II (NSGA-II) on a Combinatorial Optimization Problem
The Non-dominated Sorting Genetic Algorithm-II (NSGA-II) is one of the most
prominent algorithms to solve multi-objective optimization problems. Recently,
the first mathematical runtime guarantees have been obtained for this
algorithm, however only for synthetic benchmark problems.
In this work, we give the first proven performance guarantees for a classic
optimization problem, the NP-complete bi-objective minimum spanning tree
problem. More specifically, we show that the NSGA-II with population size computes all extremal points of the Pareto front in
an expected number of iterations, where
is the number of vertices, the number of edges, and is the
maximum edge weight in the problem instance. This result confirms, via
mathematical means, the good performance of the NSGA-II observed empirically.
It also shows that mathematical analyses of this algorithm are not only
possible for synthetic benchmark problems, but also for more complex
combinatorial optimization problems.
As a side result, we also obtain a new analysis of the performance of the
global SEMO algorithm on the bi-objective minimum spanning tree problem, which
improves the previous best result by a factor of , the number of extremal
points of the Pareto front, a set that can be as large as . The
main reason for this improvement is our observation that both multi-objective
evolutionary algorithms find the different extremal points in parallel rather
than sequentially, as assumed in the previous proofs.Comment: Author-generated version of a paper appearing in the proceedings of
IJCAI 202
Bicriteria Network Design Problems
We study a general class of bicriteria network design problems. A generic
problem in this class is as follows: Given an undirected graph and two
minimization objectives (under different cost functions), with a budget
specified on the first, find a <subgraph \from a given subgraph-class that
minimizes the second objective subject to the budget on the first. We consider
three different criteria - the total edge cost, the diameter and the maximum
degree of the network. Here, we present the first polynomial-time approximation
algorithms for a large class of bicriteria network design problems for the
above mentioned criteria. The following general types of results are presented.
First, we develop a framework for bicriteria problems and their
approximations. Second, when the two criteria are the same %(note that the cost
functions continue to be different) we present a ``black box'' parametric
search technique. This black box takes in as input an (approximation) algorithm
for the unicriterion situation and generates an approximation algorithm for the
bicriteria case with only a constant factor loss in the performance guarantee.
Third, when the two criteria are the diameter and the total edge costs we use a
cluster-based approach to devise a approximation algorithms --- the solutions
output violate both the criteria by a logarithmic factor. Finally, for the
class of treewidth-bounded graphs, we provide pseudopolynomial-time algorithms
for a number of bicriteria problems using dynamic programming. We show how
these pseudopolynomial-time algorithms can be converted to fully
polynomial-time approximation schemes using a scaling technique.Comment: 24 pages 1 figur
Parallel Graph Connectivity in Log Diameter Rounds
We study graph connectivity problem in MPC model. On an undirected graph with
nodes and edges, round connectivity algorithms have been
known for over 35 years. However, no algorithms with better complexity bounds
were known. In this work, we give fully scalable, faster algorithms for the
connectivity problem, by parameterizing the time complexity as a function of
the diameter of the graph. Our main result is a
time connectivity algorithm for diameter- graphs, using total
memory. If our algorithm can use more memory, it can terminate in fewer rounds,
and there is no lower bound on the memory per processor.
We extend our results to related graph problems such as spanning forest,
finding a DFS sequence, exact/approximate minimum spanning forest, and
bottleneck spanning forest. We also show that achieving similar bounds for
reachability in directed graphs would imply faster boolean matrix
multiplication algorithms.
We introduce several new algorithmic ideas. We describe a general technique
called double exponential speed problem size reduction which roughly means that
if we can use total memory to reduce a problem from size to , for
in one phase, then we can solve the problem in
phases. In order to achieve this fast reduction for graph
connectivity, we use a multistep algorithm. One key step is a carefully
constructed truncated broadcasting scheme where each node broadcasts neighbor
sets to its neighbors in a way that limits the size of the resulting neighbor
sets. Another key step is random leader contraction, where we choose a smaller
set of leaders than many previous works do
Minimum Cuts in Near-Linear Time
We significantly improve known time bounds for solving the minimum cut
problem on undirected graphs. We use a ``semi-duality'' between minimum cuts
and maximum spanning tree packings combined with our previously developed
random sampling techniques. We give a randomized algorithm that finds a minimum
cut in an m-edge, n-vertex graph with high probability in O(m log^3 n) time. We
also give a simpler randomized algorithm that finds all minimum cuts with high
probability in O(n^2 log n) time. This variant has an optimal RNC
parallelization. Both variants improve on the previous best time bound of O(n^2
log^3 n). Other applications of the tree-packing approach are new, nearly tight
bounds on the number of near minimum cuts a graph may have and a new data
structure for representing them in a space-efficient manner
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