57 research outputs found
Algorithms to Approximate Column-Sparse Packing Problems
Column-sparse packing problems arise in several contexts in both
deterministic and stochastic discrete optimization. We present two unifying
ideas, (non-uniform) attenuation and multiple-chance algorithms, to obtain
improved approximation algorithms for some well-known families of such
problems. As three main examples, we attain the integrality gap, up to
lower-order terms, for known LP relaxations for k-column sparse packing integer
programs (Bansal et al., Theory of Computing, 2012) and stochastic k-set
packing (Bansal et al., Algorithmica, 2012), and go "half the remaining
distance" to optimal for a major integrality-gap conjecture of Furedi, Kahn and
Seymour on hypergraph matching (Combinatorica, 1993).Comment: Extended abstract appeared in SODA 2018. Full version in ACM
Transactions of Algorithm
Distributed Dominating Set Approximations beyond Planar Graphs
The Minimum Dominating Set (MDS) problem is one of the most fundamental and
challenging problems in distributed computing. While it is well-known that
minimum dominating sets cannot be approximated locally on general graphs, over
the last years, there has been much progress on computing local approximations
on sparse graphs, and in particular planar graphs.
In this paper we study distributed and deterministic MDS approximation
algorithms for graph classes beyond planar graphs. In particular, we show that
existing approximation bounds for planar graphs can be lifted to bounded genus
graphs, and present (1) a local constant-time, constant-factor MDS
approximation algorithm and (2) a local -time
approximation scheme. Our main technical contribution is a new analysis of a
slightly modified variant of an existing algorithm by Lenzen et al.
Interestingly, unlike existing proofs for planar graphs, our analysis does not
rely on direct topological arguments.Comment: arXiv admin note: substantial text overlap with arXiv:1602.0299
Inapproximability of maximal strip recovery
In comparative genomic, the first step of sequence analysis is usually to
decompose two or more genomes into syntenic blocks that are segments of
homologous chromosomes. For the reliable recovery of syntenic blocks, noise and
ambiguities in the genomic maps need to be removed first. Maximal Strip
Recovery (MSR) is an optimization problem proposed by Zheng, Zhu, and Sankoff
for reliably recovering syntenic blocks from genomic maps in the midst of noise
and ambiguities. Given genomic maps as sequences of gene markers, the
objective of \msr{d} is to find subsequences, one subsequence of each
genomic map, such that the total length of syntenic blocks in these
subsequences is maximized. For any constant , a polynomial-time
2d-approximation for \msr{d} was previously known. In this paper, we show that
for any , \msr{d} is APX-hard, even for the most basic version of the
problem in which all gene markers are distinct and appear in positive
orientation in each genomic map. Moreover, we provide the first explicit lower
bounds on approximating \msr{d} for all . In particular, we show that
\msr{d} is NP-hard to approximate within . From the other
direction, we show that the previous 2d-approximation for \msr{d} can be
optimized into a polynomial-time algorithm even if is not a constant but is
part of the input. We then extend our inapproximability results to several
related problems including \cmsr{d}, \gapmsr{\delta}{d}, and
\gapcmsr{\delta}{d}.Comment: A preliminary version of this paper appeared in two parts in the
Proceedings of the 20th International Symposium on Algorithms and Computation
(ISAAC 2009) and the Proceedings of the 4th International Frontiers of
Algorithmics Workshop (FAW 2010
Adaptive Out-Orientations with Applications
We give simple algorithms for maintaining edge-orientations of a
fully-dynamic graph, such that the out-degree of each vertex is bounded. On one
hand, we show how to orient the edges such that the out-degree of each vertex
is proportional to the arboricity of the graph, in a worst-case update
time of . On the other hand, motivated by applications
in dynamic maximal matching, we obtain a different trade-off, namely the
improved worst case update time of for the problem of
maintaining an edge-orientation with at most out-edges per
vertex. Since our algorithms have update times with worst-case guarantees, the
number of changes to the solution (i.e. the recourse) is naturally limited.
Our algorithms make choices based entirely on local information, which makes
them automatically adaptive to the current arboricity of the graph. In other
words, they are arboricity-oblivious, while they are arboricity-sensitive. This
both simplifies and improves upon previous work, by having fewer assumptions or
better asymptotic guarantees.
As a consequence, one obtains an algorithm with improved efficiency for
maintaining a approximation of the maximum subgraph density,
and an algorithm for dynamic maximal matching whose worst-case update time is
guaranteed to be upper bounded by , where
is the arboricity at the time of the update
An overview of the lowest outdegree orientation problem
Στο πρόβλημα εύρεσης προσανατολισμού ελάχιστου βαθμού, διερευνούμε πώς να βρούμε έναν προσανατολισμό ενός δεδομένου γραφήματος, τέτοιο ώστε να ελαχιστοποιεί τον έξω- βαθμό του γραφήματος. Το πρόβλημα σχετίζεται στενά με μια σειρά από άλλα προβλήματα, όπως η δημιουργία συστημάτων επισήμανσης (labeling schemes) και η προσέγγιση των μέτρων πυκνότητας ενός γραφήματος.In the lowest outdegree orientation problem, we investigate how to find an orientation of a given graph that minimizes the outdegree, or maximum number of edges leaving a vertex. The problem is closely related to a number of other problems, such as creating labeling schemes and approximating graph density measures
Distributed CONGEST Approximation of Weighted Vertex Covers and Matchings
We provide CONGEST model algorithms for approximating minimum weighted vertex
cover and the maximum weighted matching. For bipartite graphs, we show that a
-approximate weighted vertex cover can be computed
deterministically in polylogarithmic time. This generalizes a corresponding
result for the unweighted vertex cover problem shown in [Faour, Kuhn; OPODIS
'20]. Moreover, we show that in general weighted graph families that are closed
under taking subgraphs and in which we can compute an independent set of weight
at least a -fraction of the total weight, one can compute a
-approximate weighted vertex cover in
polylogarithmic time in the CONGEST model. Our result in particular implies
that in graphs of arboricity , one can compute a
-approximate weighted vertex cover.
For maximum weighted matchings, we show that a -approximate
solution can be computed deterministically in polylogarithmic CONGEST rounds
(for constant ). We also provide a more efficient randomized
algorithm. Our algorithm generalizes results of [Lotker, Patt-Shamir, Pettie;
SPAA '08] and [Bar-Yehuda, Hillel, Ghaffari, Schwartzman; PODC '17] for the
unweighted case.
Finally, we show that even in the LOCAL model and in bipartite graphs of
degree , if for some constant
, then computing a -approximation for the
unweighted minimum vertex cover problem requires rounds. This generalizes aresult of [G\"o\"os, Suomela;
DISC '12], who showed that computing a -approximation in
such graphs requires rounds
Local Multicoloring Algorithms: Computing a Nearly-Optimal TDMA Schedule in Constant Time
The described multicoloring problem has direct applications in the context of
wireless ad hoc and sensor networks. In order to coordinate the access to the
shared wireless medium, the nodes of such a network need to employ some medium
access control (MAC) protocol. Typical MAC protocols control the access to the
shared channel by time (TDMA), frequency (FDMA), or code division multiple
access (CDMA) schemes. Many channel access schemes assign a fixed set of time
slots, frequencies, or (orthogonal) codes to the nodes of a network such that
nodes that interfere with each other receive disjoint sets of time slots,
frequencies, or code sets. Finding a valid assignment of time slots,
frequencies, or codes hence directly corresponds to computing a multicoloring
of a graph . The scarcity of bandwidth, energy, and computing resources in
ad hoc and sensor networks, as well as the often highly dynamic nature of these
networks require that the multicoloring can be computed based on as little and
as local information as possible
- …