59,504 research outputs found
Covering Problems via Structural Approaches
The minimum set cover problem is, without question, among the most ubiquitous and well-studied problems in computer science. Its theoretical hardness has been fully characterized--logarithmic approximability has been established, and no sublogarithmic approximation exists unless P=NP. However, the gap between real-world instances and the theoretical worst case is often immense--many covering problems of practical relevance admit much better approximations, or even solvability in polynomial time. Simple combinatorial or geometric structure can often be exploited to obtain improved algorithms on a problem-by-problem basis, but there is no general method of determining the extent to which this is possible.
In this thesis, we aim to shed light on the relationship between the structure and the hardness of covering problems. We discuss several measures of structural complexity of set cover instances and prove new algorithmic and hardness results linking the approximability of a set cover problem to its underlying structure. In particular, we provide:
- An APX-hardness proof for a wide family of problems that encode a simple covering problem known as Special-3SC.
- A class of polynomial dynamic programming algorithms for a group of weighted geometric set cover problems having simple structure.
- A simplified quasi-uniform sampling algorithm that yields improved approximations for weighted covering problems having low cell complexity or geometric union complexity.
- Applications of the above to various capacitated covering problems via linear programming strengthening and rounding.
In total, we obtain new results for dozens of covering problems exhibiting geometric or combinatorial structure. We tabulate these problems and classify them according to their approximability
Algorithms for covering multiple submodular constraints and applications
We consider the problem of covering multiple submodular constraints. Given a finite ground set N, a weight function , r monotone submodular functions over N and requirements the goal is to find a minimum weight subset such that for . We refer to this problem as Multi-Submod-Cover and it was recently considered by Har-Peled and Jones (Few cuts meet many point sets. CoRR. arxiv:abs1808.03260Har-Peled and Jones 2018) who were motivated by an application in geometry. Even with Multi-Submod-Cover generalizes the well-known Submodular Set Cover problem (Submod-SC), and it can also be easily reduced to Submod-SC. A simple greedy algorithm gives an approximation where and this ratio cannot be improved in the general case. In this paper, motivated by several concrete applications, we consider two ways to improve upon the approximation given by the greedy algorithm. First, we give a bicriteria approximation algorithm for Multi-Submod-Cover that covers each constraint to within a factor of while incurring an approximation of in the cost. Second, we consider the special case when each is a obtained from a truncated coverage function and obtain an algorithm that generalizes previous work on partial set cover (Partial-SC), covering integer programs (CIPs) and multiple vertex cover constraints Bera et al. (Theoret Comput Sci 555:2–8 Bera et al. 2014). Both these algorithms are based on mathematical programming relaxations that avoid the limitations of the greedy algorithm. We demonstrate the implications of our algorithms and related ideas to several applications ranging from geometric covering problems to clustering with outliers. Our work highlights the utility of the high-level model and the lens of submodularity in addressing this class of covering problems.publishedVersio
Approximation Algorithms for Polynomial-Expansion and Low-Density Graphs
We study the family of intersection graphs of low density objects in low
dimensional Euclidean space. This family is quite general, and includes planar
graphs. We prove that such graphs have small separators. Next, we present
efficient -approximation algorithms for these graphs, for
Independent Set, Set Cover, and Dominating Set problems, among others. We also
prove corresponding hardness of approximation for some of these optimization
problems, providing a characterization of their intractability in terms of
density
The Geometry of Scheduling
We consider the following general scheduling problem: The input consists of n
jobs, each with an arbitrary release time, size, and a monotone function
specifying the cost incurred when the job is completed at a particular time.
The objective is to find a preemptive schedule of minimum aggregate cost. This
problem formulation is general enough to include many natural scheduling
objectives, such as weighted flow, weighted tardiness, and sum of flow squared.
Our main result is a randomized polynomial-time algorithm with an approximation
ratio O(log log nP), where P is the maximum job size. We also give an O(1)
approximation in the special case when all jobs have identical release times.
The main idea is to reduce this scheduling problem to a particular geometric
set-cover problem which is then solved using the local ratio technique and
Varadarajan's quasi-uniform sampling technique. This general algorithmic
approach improves the best known approximation ratios by at least an
exponential factor (and much more in some cases) for essentially all of the
nontrivial common special cases of this problem. Our geometric interpretation
of scheduling may be of independent interest.Comment: Conference version in FOCS 201
Coresets-Methods and History: A Theoreticians Design Pattern for Approximation and Streaming Algorithms
We present a technical survey on the state of the art approaches in data reduction and the coreset framework. These include geometric decompositions, gradient methods, random sampling, sketching and random projections. We further outline their importance for the design of streaming algorithms and give a brief overview on lower bounding techniques
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