92 research outputs found
Improving the integrality gap for multiway cut
In the multiway cut problem, we are given an undirected graph with
non-negative edge weights and a collection of terminal nodes, and the goal
is to partition the node set of the graph into non-empty parts each
containing exactly one terminal so that the total weight of the edges crossing
the partition is minimized. The multiway cut problem for is APX-hard.
For arbitrary , the best-known approximation factor is due to
[Sharma and Vondr\'{a}k, 2014] while the best known inapproximability factor is
due to [Angelidakis, Makarychev and Manurangsi, 2017]. In this work, we
improve on the lower bound to by constructing an integrality gap
instance for the CKR relaxation.
A technical challenge in improving the gap has been the lack of geometric
tools to understand higher-dimensional simplices. Our instance is a non-trivial
-dimensional instance that overcomes this technical challenge. We analyze
the gap of the instance by viewing it as a convex combination of
-dimensional instances and a uniform 3-dimensional instance. We believe that
this technique could be exploited further to construct instances with larger
integrality gap. One of the ingredients of our proof technique is a
generalization of a result on \emph{Sperner admissible labelings} due to
[Mirzakhani and Vondr\'{a}k, 2015] that might be of independent combinatorial
interest.Comment: 28 page
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
Local Guarantees in Graph Cuts and Clustering
Correlation Clustering is an elegant model that captures fundamental graph
cut problems such as Min Cut, Multiway Cut, and Multicut, extensively
studied in combinatorial optimization. Here, we are given a graph with edges
labeled or and the goal is to produce a clustering that agrees with the
labels as much as possible: edges within clusters and edges across
clusters. The classical approach towards Correlation Clustering (and other
graph cut problems) is to optimize a global objective. We depart from this and
study local objectives: minimizing the maximum number of disagreements for
edges incident on a single node, and the analogous max min agreements
objective. This naturally gives rise to a family of basic min-max graph cut
problems. A prototypical representative is Min Max Cut: find an cut
minimizing the largest number of cut edges incident on any node. We present the
following results: an -approximation for the problem of
minimizing the maximum total weight of disagreement edges incident on any node
(thus providing the first known approximation for the above family of min-max
graph cut problems), a remarkably simple -approximation for minimizing
local disagreements in complete graphs (improving upon the previous best known
approximation of ), and a -approximation for
maximizing the minimum total weight of agreement edges incident on any node,
hence improving upon the -approximation that follows from
the study of approximate pure Nash equilibria in cut and party affiliation
games
Half-integrality, LP-branching and FPT Algorithms
A recent trend in parameterized algorithms is the application of polytope
tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011;
Narayanaswamy et al., 2012). However, although interesting results have been
achieved, the methods require the underlying polytope to have very restrictive
properties (half-integrality and persistence), which are known only for few
problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node
Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we
view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a
relaxation of the search space from to such that
the new problem admits a polynomial-time exact solution. Using tools from CSP
(in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such
relaxations, we provide a much broader class of half-integral polytopes with
the required properties, unifying and extending previously known cases.
In addition to the insight into problems with half-integral relaxations, our
results yield a range of new and improved FPT algorithms, including an
-time algorithm for node-deletion Unique Label Cover with
label set and an -time algorithm for Group Feedback Vertex
Set, including the setting where the group is only given by oracle access. All
these significantly improve on previous results. The latter result also implies
the first single-exponential time FPT algorithm for Subset Feedback Vertex Set,
answering an open question of Cygan et al. (2012).
Additionally, we propose a network flow-based approach to solve some cases of
the relaxation problem. This gives the first linear-time FPT algorithm to
edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA
paper
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
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