1,337 research outputs found
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
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
Directed Multicut with linearly ordered terminals
Motivated by an application in network security, we investigate the following
"linear" case of Directed Mutlicut. Let be a directed graph which includes
some distinguished vertices . What is the size of the
smallest edge cut which eliminates all paths from to for all ? We show that this problem is fixed-parameter tractable when parametrized in
the cutset size via an algorithm running in time.Comment: 12 pages, 1 figur
Improved Hardness for Cut, Interdiction, and Firefighter Problems
We study variants of the classic s-t cut problem and prove the following improved hardness results assuming the Unique Games Conjecture (UGC).
* For Length-Bounded Cut and Shortest Path Interdiction, we show that both problems are hard to approximate within any constant factor, even if we allow bicriteria approximation. If we want to cut vertices or the graph is directed, our hardness ratio for Length-Bounded Cut matches the best approximation ratio up to a constant. Previously, the best hardness ratio was 1.1377 for Length-Bounded Cut and 2 for Shortest Path Interdiction.
* For any constant k >= 2 and epsilon > 0, we show that Directed Multicut with k source-sink pairs is hard to approximate within a factor k - epsilon. This matches the trivial k-approximation algorithm. By a simple reduction, our result for k = 2 implies that Directed Multiway Cut with two terminals (also known as s-t Bicut} is hard to approximate within a factor 2 - epsilon, matching the trivial 2-approximation algorithm.
* Assuming a variant of the UGC (implied by another variant of Bansal and Khot), we prove that it is hard to approximate Resource Minimization Fire Containment within any constant factor. Previously, the best hardness ratio was 2. For directed layered graphs with b layers, our hardness ratio Omega(log b) matches the best approximation algorithm.
Our results are based on a general method of converting an integrality gap instance to a length-control dictatorship test for variants of the s-t cut problem, which may be useful for other problems
Bilu-Linial Stable Instances of Max Cut and Minimum Multiway Cut
We investigate the notion of stability proposed by Bilu and Linial. We obtain
an exact polynomial-time algorithm for -stable Max Cut instances with
for some absolute constant . Our
algorithm is robust: it never returns an incorrect answer; if the instance is
-stable, it finds the maximum cut, otherwise, it either finds the
maximum cut or certifies that the instance is not -stable. We prove
that there is no robust polynomial-time algorithm for -stable instances
of Max Cut when , where is the best
approximation factor for Sparsest Cut with non-uniform demands.
Our algorithm is based on semidefinite programming. We show that the standard
SDP relaxation for Max Cut (with triangle inequalities) is integral
if , where
is the least distortion with which every point metric space of negative
type embeds into . On the negative side, we show that the SDP
relaxation is not integral when .
Moreover, there is no tractable convex relaxation for -stable instances
of Max Cut when . That suggests that solving
-stable instances with might be difficult or
impossible.
Our results significantly improve previously known results. The best
previously known algorithm for -stable instances of Max Cut required
that (for some ) [Bilu, Daniely, Linial, and
Saks]. No hardness results were known for the problem. Additionally, we present
an algorithm for 4-stable instances of Minimum Multiway Cut. We also study a
relaxed notion of weak stability.Comment: 24 page
Fixed-Parameter Tractability of Directed Multiway Cut Parameterized by the Size of the Cutset
Given a directed graph , a set of terminals and an integer , the
\textsc{Directed Vertex Multiway Cut} problem asks if there is a set of at
most (nonterminal) vertices whose removal disconnects each terminal from
all other terminals. \textsc{Directed Edge Multiway Cut} is the analogous
problem where is a set of at most edges. These two problems indeed are
known to be equivalent. A natural generalization of the multiway cut is the
\emph{multicut} problem, in which we want to disconnect only a set of given
pairs instead of all pairs. Marx (Theor. Comp. Sci. 2006) showed that in
undirected graphs multiway cut is fixed-parameter tractable (FPT) parameterized
by . Marx and Razgon (STOC 2011) showed that undirected multicut is FPT and
directed multicut is W[1]-hard parameterized by . We complete the picture
here by our main result which is that both \textsc{Directed Vertex Multiway
Cut} and \textsc{Directed Edge Multiway Cut} can be solved in time
, i.e., FPT parameterized by size of the cutset of
the solution. This answers an open question raised by Marx (Theor. Comp. Sci.
2006) and Marx and Razgon (STOC 2011). It follows from our result that
\textsc{Directed Multicut} is FPT for the case of terminal pairs, which
answers another open problem raised in Marx and Razgon (STOC 2011)
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