64 research outputs found
A Survey on Approximation in Parameterized Complexity: Hardness and Algorithms
Parameterization and approximation are two popular ways of coping with
NP-hard problems. More recently, the two have also been combined to derive many
interesting results. We survey developments in the area both from the
algorithmic and hardness perspectives, with emphasis on new techniques and
potential future research directions
Diameter Minimization by Shortcutting with Degree Constraints
We consider the problem of adding a fixed number of new edges to an
undirected graph in order to minimize the diameter of the augmented graph, and
under the constraint that the number of edges added for each vertex is bounded
by an integer. The problem is motivated by network-design applications, where
we want to minimize the worst case communication in the network without
excessively increasing the degree of any single vertex, so as to avoid
additional overload. We present three algorithms for this task, each with their
own merits. The special case of a matching augmentation, when every vertex can
be incident to at most one new edge, is of particular interest, for which we
show an inapproximability result, and provide bounds on the smallest achievable
diameter when these edges are added to a path. Finally, we empirically evaluate
and compare our algorithms on several real-life networks of varying types.Comment: A shorter version of this work has been accepted at the IEEE ICDM
2022 conferenc
Some results on more flexible versions of Graph Motif
The problems studied in this paper originate from Graph Motif, a problem
introduced in 2006 in the context of biological networks. Informally speaking,
it consists in deciding if a multiset of colors occurs in a connected subgraph
of a vertex-colored graph. Due to the high rate of noise in the biological
data, more flexible definitions of the problem have been outlined. We present
in this paper two inapproximability results for two different optimization
variants of Graph Motif: one where the size of the solution is maximized, the
other when the number of substitutions of colors to obtain the motif from the
solution is minimized. We also study a decision version of Graph Motif where
the connectivity constraint is replaced by the well known notion of graph
modularity. While the problem remains NP-complete, it allows algorithms in FPT
for biologically relevant parameterizations
On Approximating Four Covering and Packing Problems
In this paper, we consider approximability issues of the following four
problems: triangle packing, full sibling reconstruction, maximum profit
coverage and 2-coverage. All of them are generalized or specialized versions of
set-cover and have applications in biology ranging from full-sibling
reconstructions in wild populations to biomolecular clusterings; however, as
this paper shows, their approximability properties differ considerably. Our
inapproximability constant for the triangle packing problem improves upon the
previous results; this is done by directly transforming the inapproximability
gap of Haastad for the problem of maximizing the number of satisfied equations
for a set of equations over GF(2) and is interesting in its own right. Our
approximability results on the full siblings reconstruction problems answers
questions originally posed by Berger-Wolf et al. and our results on the maximum
profit coverage problem provides almost matching upper and lower bounds on the
approximation ratio, answering a question posed by Hassin and Or.Comment: 25 page
Approximability of (Simultaneous) Class Cover for Boxes
Bereg et al. (2012) introduced the Boxes Class Cover problem, which has its
roots in classification and clustering applications: Given a set of n points in
the plane, each colored red or blue, find the smallest cardinality set of
axis-aligned boxes whose union covers the red points without covering any blue
point. In this paper we give an alternative proof of APX-hardness for this
problem, which also yields an explicit lower bound on its approximability. Our
proof also directly applies when restricted to sets of points in general
position and to the case where so-called half-strips are considered instead of
boxes, which is a new result.
We also introduce a symmetric variant of this problem, which we call
Simultaneous Boxes Class Cover and can be stated as follows: Given a set S of n
points in the plane, each colored red or blue, find the smallest cardinality
set of axis-aligned boxes which together cover S such that all boxes cover only
points of the same color and no box covering a red point intersects a box
covering a blue point. We show that this problem is also APX-hard and give a
polynomial-time constant-factor approximation algorithm
Overlapping and Robust Edge-Colored Clustering in Hypergraphs
A recent trend in data mining has explored (hyper)graph clustering algorithms
for data with categorical relationship types. Such algorithms have applications
in the analysis of social, co-authorship, and protein interaction networks, to
name a few. Many such applications naturally have some overlap between
clusters, a nuance which is missing from current combinatorial models.
Additionally, existing models lack a mechanism for handling noise in datasets.
We address these concerns by generalizing Edge-Colored Clustering, a recent
framework for categorical clustering of hypergraphs. Our generalizations allow
for a budgeted number of either (a) overlapping cluster assignments or (b) node
deletions. For each new model we present a greedy algorithm which approximately
minimizes an edge mistake objective, as well as bicriteria approximations where
the second approximation factor is on the budget. Additionally, we address the
parameterized complexity of each problem, providing FPT algorithms and hardness
results
Minimum d-dimensional arrangement with fixed points
In the Minimum -Dimensional Arrangement Problem (d-dimAP) we are given a
graph with edge weights, and the goal is to find a 1-1 map of the vertices into
(for some fixed dimension ) minimizing the total
weighted stretch of the edges. This problem arises in VLSI placement and chip
design.
Motivated by these applications, we consider a generalization of d-dimAP,
where the positions of some of the vertices (pins) is fixed and specified as
part of the input. We are asked to extend this partial map to a map of all the
vertices, again minimizing the weighted stretch of edges. This generalization,
which we refer to as d-dimAP+, arises naturally in these application domains
(since it can capture blocked-off parts of the board, or the requirement of
power-carrying pins to be in certain locations, etc.). Perhaps surprisingly,
very little is known about this problem from an approximation viewpoint.
For dimension , we obtain an -approximation
algorithm, based on a strengthening of the spreading-metric LP for 2-dimAP. The
integrality gap for this LP is shown to be . We also show that
it is NP-hard to approximate 2-dimAP+ within a factor better than
\Omega(k^{1/4-\eps}). We also consider a (conceptually harder, but
practically even more interesting) variant of 2-dimAP+, where the target space
is the grid , instead of
the entire integer lattice . For this problem, we obtain a -approximation using the same LP relaxation. We complement
this upper bound by showing an integrality gap of , and an
\Omega(k^{1/2-\eps})-inapproximability result.
Our results naturally extend to the case of arbitrary fixed target dimension
Matroid and Knapsack Center Problems
In the classic -center problem, we are given a metric graph, and the
objective is to open nodes as centers such that the maximum distance from
any vertex to its closest center is minimized. In this paper, we consider two
important generalizations of -center, the matroid center problem and the
knapsack center problem. Both problems are motivated by recent content
distribution network applications. Our contributions can be summarized as
follows:
1. We consider the matroid center problem in which the centers are required
to form an independent set of a given matroid. We show this problem is NP-hard
even on a line. We present a 3-approximation algorithm for the problem on
general metrics. We also consider the outlier version of the problem where a
given number of vertices can be excluded as the outliers from the solution. We
present a 7-approximation for the outlier version.
2. We consider the (multi-)knapsack center problem in which the centers are
required to satisfy one (or more) knapsack constraint(s). It is known that the
knapsack center problem with a single knapsack constraint admits a
3-approximation. However, when there are at least two knapsack constraints, we
show this problem is not approximable at all. To complement the hardness
result, we present a polynomial time algorithm that gives a 3-approximate
solution such that one knapsack constraint is satisfied and the others may be
violated by at most a factor of . We also obtain a 3-approximation
for the outlier version that may violate the knapsack constraint by
.Comment: A preliminary version of this paper is accepted to IPCO 201
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