10,881 research outputs found
Reoptimization of Some Maximum Weight Induced Hereditary Subgraph Problems
The reoptimization issue studied in this paper can be described as follows: given an instance I of some problem Î , an optimal solution OPT for Î in I and an instance IâČ resulting from a local perturbation of I that consists of insertions or removals of a small number of data, we wish to use OPT in order to solve Î in I', either optimally or by guaranteeing an approximation ratio better than that guaranteed by an ex nihilo computation and with running time better than that needed for such a computation. We use this setting in order to study weighted versions of several representatives of a broad class of problems known in the literature as maximum induced hereditary subgraph problems. The main problems studied are max independent set, max k-colorable subgraph and max split subgraph under vertex insertions and deletion
From Gap-ETH to FPT-Inapproximability: Clique, Dominating Set, and More
We consider questions that arise from the intersection between the areas of
polynomial-time approximation algorithms, subexponential-time algorithms, and
fixed-parameter tractable algorithms. The questions, which have been asked
several times (e.g., [Marx08, FGMS12, DF13]), are whether there is a
non-trivial FPT-approximation algorithm for the Maximum Clique (Clique) and
Minimum Dominating Set (DomSet) problems parameterized by the size of the
optimal solution. In particular, letting be the optimum and be
the size of the input, is there an algorithm that runs in
time and outputs a solution of size
, for any functions and that are independent of (for
Clique, we want )?
In this paper, we show that both Clique and DomSet admit no non-trivial
FPT-approximation algorithm, i.e., there is no
-FPT-approximation algorithm for Clique and no
-FPT-approximation algorithm for DomSet, for any function
(e.g., this holds even if is the Ackermann function). In fact, our results
imply something even stronger: The best way to solve Clique and DomSet, even
approximately, is to essentially enumerate all possibilities. Our results hold
under the Gap Exponential Time Hypothesis (Gap-ETH) [Dinur16, MR16], which
states that no -time algorithm can distinguish between a satisfiable
3SAT formula and one which is not even -satisfiable for some
constant .
Besides Clique and DomSet, we also rule out non-trivial FPT-approximation for
Maximum Balanced Biclique, Maximum Subgraphs with Hereditary Properties, and
Maximum Induced Matching in bipartite graphs. Additionally, we rule out
-FPT-approximation algorithm for Densest -Subgraph although this
ratio does not yet match the trivial -approximation algorithm.Comment: 43 pages. To appear in FOCS'1
Near-Optimal Approximate Shortest Paths and Transshipment in Distributed and Streaming Models
We present a method for solving the transshipment problem - also known as
uncapacitated minimum cost flow - up to a multiplicative error of in undirected graphs with non-negative edge weights using a
tailored gradient descent algorithm. Using to hide
polylogarithmic factors in (the number of nodes in the graph), our gradient
descent algorithm takes iterations, and in each
iteration it solves an instance of the transshipment problem up to a
multiplicative error of . In particular, this allows
us to perform a single iteration by computing a solution on a sparse spanner of
logarithmic stretch. Using a randomized rounding scheme, we can further extend
the method to finding approximate solutions for the single-source shortest
paths (SSSP) problem. As a consequence, we improve upon prior work by obtaining
the following results: (1) Broadcast CONGEST model: -approximate SSSP using rounds, where is the (hop) diameter of the network.
(2) Broadcast congested clique model: -approximate
transshipment and SSSP using rounds. (3)
Multipass streaming model: -approximate transshipment and
SSSP using space and passes. The
previously fastest SSSP algorithms for these models leverage sparse hop sets.
We bypass the hop set construction; computing a spanner is sufficient with our
method. The above bounds assume non-negative edge weights that are polynomially
bounded in ; for general non-negative weights, running times scale with the
logarithm of the maximum ratio between non-zero weights.Comment: Accepted to SIAM Journal on Computing. Preliminary version in DISC
2017. Abstract shortened to fit arXiv's limitation to 1920 character
A cutting-plane approach to the edge-weighted maximal clique problem
We investigated the computational performance of a cutting-plane algorithm for the problem of determining a maximal subclique in an edge-weighted complete graph. Our numerical results are contrasted with reports on closely related problems for which cutting-plane approaches perform well in instances of moderate size. Somewhat surprisingly, we find that our approach already in the case of n = 15 or N = 25 nodes in the underlying graph typically neither produces an integral solution nor yields a good approximation to the true optimal objective function value. This result seems to shed some doubt on the universal applicability of cuttingplane approaches as an efficient means to solve linear (0, 1)-programming problems of moderate size
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
Super-Fast Distributed Algorithms for Metric Facility Location
This paper presents a distributed O(1)-approximation algorithm, with
expected- running time, in the model for
the metric facility location problem on a size- clique network. Though
metric facility location has been considered by a number of researchers in
low-diameter settings, this is the first sub-logarithmic-round algorithm for
the problem that yields an O(1)-approximation in the setting of non-uniform
facility opening costs. In order to obtain this result, our paper makes three
main technical contributions. First, we show a new lower bound for metric
facility location, extending the lower bound of B\u{a}doiu et al. (ICALP 2005)
that applies only to the special case of uniform facility opening costs. Next,
we demonstrate a reduction of the distributed metric facility location problem
to the problem of computing an O(1)-ruling set of an appropriate spanning
subgraph. Finally, we present a sub-logarithmic-round (in expectation)
algorithm for computing a 2-ruling set in a spanning subgraph of a clique. Our
algorithm accomplishes this by using a combination of randomized and
deterministic sparsification.Comment: 15 pages, 2 figures. This is the full version of a paper that
appeared in ICALP 201
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