A Static 2-Approximation Algorithm for Vertex Connectivity and Incremental Approximation Algorithms for Edge and Vertex Connectivity

1997 ◽  
Vol 24 (1) ◽  
pp. 194-220 ◽  
Author(s):  
Monika Rauch Henzinger
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Vertex Connectivity

Two-stage submodular maximization problem beyond nonnegative and monotone

2021 ◽  
pp. 1-16
Author(s):  
Zhicheng Liu ◽  
Hong Chang ◽  
Ran Ma ◽  
Donglei Du ◽  
Xiaoyan Zhang
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Objective Function ◽  
Modular Function ◽  
Submodular Function ◽  
Maximization Problem ◽  
Cardinality Constraint ◽  
Two Stage ◽  
Time Efficiency ◽  
Submodular Maximization

Abstract We consider a two-stage submodular maximization problem subject to a cardinality constraint and k matroid constraints, where the objective function is the expected difference of a nonnegative monotone submodular function and a nonnegative monotone modular function. We give two bi-factor approximation algorithms for this problem. The first is a deterministic $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right),1} \right)$ -approximation algorithm, and the second is a randomized $\left( {{1 \over {k + 1}}\left( {1 - {1 \over {{e^{k + 1}}}}} \right) - \varepsilon ,1} \right)$ -approximation algorithm with improved time efficiency.

scholarly journals MAXIMIZING THE AREA OF OVERLAP OF TWO UNIONS OF DISKS UNDER RIGID MOTION

2009 ◽  
Vol 19 (06) ◽  
pp. 533-556 ◽  
Author(s):  
SERGIO CABELLO ◽  
MARK DE BERG ◽  
PANOS GIANNOPOULOS ◽  
CHRISTIAN KNAUER ◽  
RENÉ VAN OOSTRUM ◽  
...  
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
High Probability ◽  
Rigid Motion ◽  
Maximum Area ◽  
Empty Intersection ◽  
Rigid Motions ◽  
Unit Disks

Let A and B be two sets of n resp. m disjoint unit disks in the plane, with m ≥ n. We consider the problem of finding a translation or rigid motion of A that maximizes the total area of overlap with B. The function describing the area of overlap is quite complex, even for combinatorially equivalent translations and, hence, we turn our attention to approximation algorithms. We give deterministic (1 - ∊)-approximation algorithms for translations and for rigid motions, which run in O((nm/∊2) log (m/∊)) and O((n2m2/∊3) log m)) time, respectively. For rigid motions, we can also compute a (1 - ∊)-approximation in O((m2n4/3Δ1/3/∊3) log n log m) time, where Δ is the diameter of set A. Under the condition that the maximum area of overlap is at least a constant fraction of the area of A, we give a probabilistic (1 - ∊)-approximation algorithm for rigid motions that runs in O((m2/∊4) log 2(m/∊) log m) time and succeeds with high probability. Our results generalize to the case where A and B consist of possibly intersecting disks of different radii, provided that (i) the ratio of the radii of any two disks in A ∪ B is bounded, and (ii) within each set, the maximum number of disks with a non-empty intersection is bounded.

APPROXIMATION ALGORITHMS FOR SCHEDULING MALLEABLE TASKS UNDER PRECEDENCE CONSTRAINTS

2002 ◽  
Vol 13 (04) ◽  
pp. 613-627 ◽  
Author(s):  
RENAUD LEPÈRE ◽  
DENIS TRYSTRAM ◽  
GERHARD J. WOEGINGER
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Polynomial Time ◽  
Precedence Constraints ◽  
Performance Guarantee ◽  
Time Approximation ◽  
Polynomial Time Approximation Algorithm ◽  
Special Cases ◽  
Special Case ◽  
Polynomial Time Approximation

This work presents approximation algorithms for scheduling the tasks of a parallel application that are subject to precedence constraints. The considered tasks are malleable which means that they may be executed on a varying number of processors in parallel. The considered objective criterion is the makespan, i.e., the largest task completion time. We demonstrate a close relationship between this scheduling problem and one of its subproblems, the allotment problem. By exploiting this relationship, we design a polynomial time approximation algorithm with performance guarantee arbitrarily close to [Formula: see text] for the special case of series parallel precedence constraints and for the special case of precedence constraints of bounded width. These special cases cover the important situation of tree structured precedence constraints. For arbitrary precedence constraints, we give a polynomial time approximation algorithm with performance guarantee [Formula: see text].

CUTTING OUT POLYGONS WITH LINES AND RAYS

2006 ◽  
Vol 16 (02n03) ◽  
pp. 227-248 ◽  
Author(s):  
OVIDIU DAESCU ◽  
JUN LUO
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Convex Polygon ◽  
Linear Time ◽  
Constant Factor ◽  
Open Problems ◽  
Cutting Out ◽  
Cutting Sequence

We present approximation algorithms for cutting out a polygon P with n vertices from another convex polygon Q with m vertices by line cuts and ray cuts. For line cuts we require both P and Q are convex while for ray cuts we require Q is convex and P is ray cuttable. Our results answer a number of open problems and are either the first solutions or significantly improve over previously known solutions. For the line cutting version, we prove a key property that leads to a simple, constant factor approximation algorithm. For the ray cutting version, we prove it is possible to compute in almost linear time a cutting sequence that is an O( log 2 n)-factor approximation of an optimal cutting sequence. No algorithms were previously known for the ray cutting version.

An approximation algorithm for minimum-cost vertex-connectivity problems

Algorithmica ◽  
1997 ◽  
Vol 18 (1) ◽  
pp. 21-43 ◽  
Author(s):  
R. Ravi ◽  
D. P. Williamson
Keyword(s):  
Approximation Algorithm ◽  
Minimum Cost ◽  
Vertex Connectivity

scholarly journals Approximation Algorithms for Energy, Reliability, and Makespan Optimization Problems

2016 ◽  
Vol 26 (01) ◽  
pp. 1650001 ◽  
Author(s):  
Guillaume Aupy ◽  
Anne Benoit
Keyword(s):  
Energy Consumption ◽  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Optimization Problems ◽  
Linear Chain ◽  
Constant Factor ◽  
Task Graph ◽  
Linear Chains ◽  
Energy Reliability ◽  
Independent Tasks

We consider the problem of scheduling an application on a parallel computational platform. The application is a particular task graph, either a linear chain of tasks, or a set of independent tasks. The platform is made of identical processors, whose speed can be dynamically modified. It is also subject to failures: if a processor is slowed down to decrease the energy consumption, it has a higher chance to fail. Therefore, the scheduling problem requires us to re-execute or replicate tasks (i.e., execute twice the same task, either on the same processor, or on two distinct processors), in order to increase the reliability. It is a tri-criteria problem: the goal is to minimize the energy consumption, while enforcing a bound on the total execution time (the makespan), and a constraint on the reliability of each task. Our main contribution is to propose approximation algorithms for linear chains of tasks and independent tasks. For linear chains, we design a fully polynomial-time approximation scheme. However, we show that there exists no constant factor approximation algorithm for independent tasks, unless P=NP, and we propose in this case an approximation algorithm with a relaxation on the makespan constraint.

scholarly journals Approximation Algorithms for Multicoloring Planar Graphs and Powers of Square and Triangular Meshes

2006 ◽  
Vol Vol. 8 ◽  
Author(s):  
Mustapha Kchikech ◽  
Olivier Togni
Keyword(s):  
Approximation Algorithms ◽  
Approximation Algorithm ◽  
Cartesian Product ◽  
Weighted Graph ◽  
Clique Number ◽  
Triangular Mesh ◽  
Fourth Power ◽  
Triangular Meshes ◽  
Induced Subgraph ◽  
International Audience

International audience A multicoloring of a weighted graph G is an assignment of sets of colors to the vertices of G so that two adjacent vertices receive two disjoint sets of colors. A multicoloring problem on G is to find a multicoloring of G. In particular, we are interested in a minimum multicoloring that uses the least total number of colors. The main focus of this work is to obtain upper bounds on the weighted chromatic number of some classes of graphs in terms of the weighted clique number. We first propose an 11/6-approximation algorithm for multicoloring any weighted planar graph. We then study the multicoloring problem on powers of square and triangular meshes. Among other results, we show that the infinite triangular mesh is an induced subgraph of the fourth power of the infinite square mesh and we present 2-approximation algorithms for multicoloring a power square mesh and the second power of a triangular mesh, 3-approximation algorithms for multicoloring powers of semi-toroidal meshes and of triangular meshes and 4-approximation algorithm for multicoloring the power of a toroidal mesh. We also give similar algorithms for the Cartesian product of powers of paths and of cycles.

Sign in / Sign up

Export Citation Format

Share Document