10 research outputs found
Multiway Cut, Pairwise Realizable Distributions, and Descending Thresholds
We design new approximation algorithms for the Multiway Cut problem,
improving the previously known factor of 1.32388 [Buchbinder et al., 2013].
We proceed in three steps. First, we analyze the rounding scheme of
Buchbinder et al., 2013 and design a modification that improves the
approximation to (3+sqrt(5))/4 (approximately 1.309017). We also present a
tight example showing that this is the best approximation one can achieve with
the type of cuts considered by Buchbinder et al., 2013: (1) partitioning by
exponential clocks, and (2) single-coordinate cuts with equal thresholds.
Then, we prove that this factor can be improved by introducing a new rounding
scheme: (3) single-coordinate cuts with descending thresholds. By combining
these three schemes, we design an algorithm that achieves a factor of (10 + 4
sqrt(3))/13 (approximately 1.30217). This is the best approximation factor that
we are able to verify by hand.
Finally, we show that by combining these three rounding schemes with the
scheme of independent thresholds from Karger et al., 2004, the approximation
factor can be further improved to 1.2965. This approximation factor has been
verified only by computer.Comment: This is an updated version and is the full version of STOC 2014 pape
Rounding Algorithms for a Geometric Embedding of Minimum Multiway Cut
The multiway-cut problem is, given a weighted graph and k >= 2 terminal
nodes, to find a minimum-weight set of edges whose removal separates all the
terminals. The problem is NP-hard, and even NP-hard to approximate within
1+delta for some small delta > 0.
Calinescu, Karloff, and Rabani (1998) gave an algorithm with performance
guarantee 3/2-1/k, based on a geometric relaxation of the problem. In this
paper, we give improved randomized rounding schemes for their relaxation,
yielding a 12/11-approximation algorithm for k=3 and a 1.3438-approximation
algorithm in general.
Our approach hinges on the observation that the problem of designing a
randomized rounding scheme for a geometric relaxation is itself a linear
programming problem. The paper explores computational solutions to this
problem, and gives a proof that for a general class of geometric relaxations,
there are always randomized rounding schemes that match the integrality gap.Comment: Conference version in ACM Symposium on Theory of Computing (1999). To
appear in Mathematics of Operations Researc
Generating partitions of a graph into a fixed number of minimum weight cuts
AbstractIn this paper, we present an algorithm for the generation of all partitions of a graph G with positive edge weights into k mincuts. The algorithm is an enumeration procedure based on the cactus representation of the mincuts of G. We report computational results demonstrating the efficiency of the algorithm in practice and describe in more detail a specific application for generating cuts in branch-and-cut algorithms for the traveling salesman problem
Attacking Shortest Paths by Cutting Edges
Identifying shortest paths between nodes in a network is a common graph
analysis problem that is important for many applications involving routing of
resources. An adversary that can manipulate the graph structure could alter
traffic patterns to gain some benefit (e.g., make more money by directing
traffic to a toll road). This paper presents the Force Path Cut problem, in
which an adversary removes edges from a graph to make a particular path the
shortest between its terminal nodes. We prove that this problem is APX-hard,
but introduce PATHATTACK, a polynomial-time approximation algorithm that
guarantees a solution within a logarithmic factor of the optimal value. In
addition, we introduce the Force Edge Cut and Force Node Cut problems, in which
the adversary targets a particular edge or node, respectively, rather than an
entire path. We derive a nonconvex optimization formulation for these problems,
and derive a heuristic algorithm that uses PATHATTACK as a subroutine. We
demonstrate all of these algorithms on a diverse set of real and synthetic
networks, illustrating the network types that benefit most from the proposed
algorithms.Comment: 37 pages, 11 figures; Extended version of arXiv:2104.0376
Local Search Approximation Algorithms for Clustering Problems
In this research we study the use of local search in the design of approximation algorithms for NP-hard optimization problems. For our study we have selected several well-known clustering problems: k-facility location problem, minimum mutliway cut problem, and constrained maximum k-cut problem.
We show that by careful use of the local optimality property of the solutions produced by local search algorithms it is possible to bound the ratio between solutions produced by local search approximation algorithms and optimum solutions. This ratio is known as the locality gap.
The locality gap of our algorithm for the k-uncapacitated facility location problem is 2+sqrt(3) +epsilon for any constant epsilon \u3e0. This matches the approximation ratio of the best known algorithm for the problem, proposed by Zhang but our algorithm is simpler. For the minimum multiway cut problem our algorithm has locality gap 2-2/k, which matches the approximation ratio of the isolation heuristic of Dahlhaus et al; however, our experimental results show that in practice our local search algorithm greatly outperforms the isolation heuristic, and furthermore it has comparable performance as that of the three currently best algorithms for the minimum multiway cut problem. For the constrained maximum k-cut problem on hypergraphs we proposed a local search based approximation algorithm with locality gap 1-1/k for a variety of constraints imposed on the k-cuts. The locality gap of our algorithm matches the approximation ratio of the best known algorithm for the max k-cut problem on graphs designed by Vazirani, but our algorithm is more general
Optimal 3-Terminal Cuts and Linear Programming
. Given an undirected graph G = (V; E) and three specified terminal nodes t1 ; t2 ; t3 , a 3-cut is a subset A of E such that no two terminals are in the same component of GnA. If a non-negative edge weight ce is specified for each e 2 E, the optimal 3-cut problem is to find a 3-cut of minimum total weight. This problem is NP-hard, and in fact, is max-SNP-hard. An approximation algorithm having performance guarantee 7 6 has recently been given by Calinescu, Karloff, and Rabani. It is based on a certain linear programming relaxation, for which it is shown that the optimal 3-cut has weight at most 7 6 times the optimal LP value. It is proved here that 7 6 can be improved to 12 11 , and that this is best possible. As a consequence, we obtain an approximation algorithm for the optimal 3-cut problem having performance guarantee 12 11 . 1 Introduction Given an undirected graph G = (V; E) and k specified terminal nodes t 1 ; : : : ; t k , a k-cut is a subset A of E such that no two term..