283 research outputs found
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
Tight Bounds for Gomory-Hu-like Cut Counting
By a classical result of Gomory and Hu (1961), in every edge-weighted graph
, the minimum -cut values, when ranging over all ,
take at most distinct values. That is, these instances
exhibit redundancy factor . They further showed how to construct
from a tree that stores all minimum -cut values. Motivated
by this result, we obtain tight bounds for the redundancy factor of several
generalizations of the minimum -cut problem.
1. Group-Cut: Consider the minimum -cut, ranging over all subsets
of given sizes and . The redundancy
factor is .
2. Multiway-Cut: Consider the minimum cut separating every two vertices of
, ranging over all subsets of a given size . The
redundancy factor is .
3. Multicut: Consider the minimum cut separating every demand-pair in
, ranging over collections of demand pairs. The
redundancy factor is . This result is a bit surprising, as
the redundancy factor is much larger than in the first two problems.
A natural application of these bounds is to construct small data structures
that stores all relevant cut values, like the Gomory-Hu tree. We initiate this
direction by giving some upper and lower bounds.Comment: This version contains additional references to previous work (which
have some overlap with our results), see Bibliographic Update 1.
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
Discrete Convex Functions on Graphs and Their Algorithmic Applications
The present article is an exposition of a theory of discrete convex functions
on certain graph structures, developed by the author in recent years. This
theory is a spin-off of discrete convex analysis by Murota, and is motivated by
combinatorial dualities in multiflow problems and the complexity classification
of facility location problems on graphs. We outline the theory and algorithmic
applications in combinatorial optimization problems
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