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### Approximation Algorithms for Mixed Fractional Packing and Covering Problems [chapter]

Klaus Jansen
2005 Lecture Notes in Computer Science
The main contribution is that the algorithm solves the general mixed fractional packing and covering problem (in contrast to pure fractional packing and covering problems and to the special mixed packing  ...  We propose an approximation algorithm based on the Lagrangian or pricedirective decomposition method to compute an¯-approximate solution of the mixed fractional packing and covering problem: find Ü ¾ such  ...  Our approximation algorithm In this section we describe the approximation algorithms for the mixed fractional packing and covering problem.  ...

### Approximation Algorithms for Mixed Fractional Packing and Covering Problems [chapter]

Klaus Jansen
IFIP International Federation for Information Processing
The main contribution is that the algorithm solves the general mixed fractional packing and covering problem (in contrast to pure fractional packing and covering problems and to the special mixed packing  ...  We propose an approximation algorithm based on the Lagrangian or pricedirective decomposition method to compute an¯-approximate solution of the mixed fractional packing and covering problem: find Ü ¾ such  ...  Our approximation algorithm In this section we describe the approximation algorithms for the mixed fractional packing and covering problem.  ...

### Packing Interdiction and Partial Covering Problems [chapter]

Michael Dinitz, Anupam Gupta
2013 Lecture Notes in Computer Science
There is a polynomial time O(log q · min{q, log k})-approximation algorithm for the (k, q)-Packing Interdiction problem.  ...  we consider approximation algorithms for it.  ...  There is a polynomial time α-approximation algorithm for (q, k)-Packing Interdiction if and only if there is a polynomial time α-approximation algorithm for (k, q)-Partial Covering. Proof.  ...

### Distributed algorithms for covering, packing and maximum weighted matching

Christos Koufogiannakis, Neal E. Young
2011 Distributed computing
This paper gives poly-logarithmic-round, distributed D-approximation algorithms for covering problems with submodular cost and monotone covering constraints (Submodular-cost Covering).  ...  Via duality, the paper also gives poly-logarithmic-round, distributed D-approximation algorithms for Fractional Packing linear programs (where D is the maximum number of constraints in which any variable  ...  Acknowledgements Thanks to two anonymous referees for their helpful comments.  ...

### Randomized Rounding without Solving the Linear Program [article]

Neal E. Young
2002 arXiv   pre-print
This paper explores how this bottleneck can be avoided for randomized rounding algorithms for packing and covering problems (linear programs, or mixed integer linear programs, having no negative coefficients  ...  This approach can also be used to understand Lagrangian-relaxation algorithms for packing/covering linear programs: such algorithms can be viewed as as (derandomized) randomized-rounding schemes.  ...  To demonstrate this, we give approximation algorithms for general packing and covering problems corresponding to integer and noninteger linear programs of small width, including a parallel algorithm for  ...

### Approximating Optimal Transport With Linear Programs

Kent Quanrud, Michael Wagner
2018 ACM-SIAM Symposium on Discrete Algorithms
faster additive approximation algorithms for optimal transport.  ...  In the regime of bounded transportation costs, additive approximations for the optimal transport problem are reduced (rather simply) to relative approximations for positive linear programs, resulting in  ...  Acknowledgements We thank Jason Altschuler and Chandra Chekuri for insightful discussions and helpful feedback.  ...

### Approximation of a batch consolidation problem

Junho Chang, Soo Y. Chang, Sung-Pil Hong, Yun-Hong Min, Myoung-Ju Park
2010 Networks
The problem is NP-hard and cannot be approximated within 1.0021 of the optimum under the premise, P = NP. However, the problem admits a 3 2 -approximation.  ...  We consider the batch consolidation problem of minimizing the number of batches of a finite set of items.  ...  Acknowledgments The authors are grateful to the anonymous referees for their helpful comments.  ...

### Approximating optimal transport with linear programs [article]

Kent Quanrud
2018 arXiv   pre-print
faster additive approximation algorithms for optimal transport.  ...  In the regime of bounded transportation costs, additive approximations for the optimal transport problem are reduced (rather simply) to relative approximations for positive linear programs, resulting in  ...  We thank Jason Altschuler and Chandra Chekuri for insightful discussions and helpful feedback.  ...

### Knapsack Cover Subject to a Matroid Constraint

Venkatesan T. Chakaravarthy, Anamitra Roy Choudhury, Sivaramakrishnan R. Natarajan, Sambuddha Roy, Marc Herbstritt
2013 Foundations of Software Technology and Theoretical Computer Science
Our main result proves a 2-factor approximation for this problem. The problem described above falls in the realm of mixed packing covering problems.  ...  We also consider packing extensions of certain other covering problems and prove that in such cases it is not possible to derive any constant factor approximations.  ...  The submission version contained only a 2-factor approximation for the KCM problem; a reviewer noted that a slight modification  ...

### A Tight Approximation for Submodular Maximization with Mixed Packing and Covering Constraints

Eyal Mizrachi, Roy Schwartz, Joachim Spoerhase, Sumedha Uniyal, Michael Wagner
2019 International Colloquium on Automata, Languages and Programming
Motivated by applications in machine learning, such as subset selection and data summarization, we consider the problem of maximizing a monotone submodular function subject to mixed packing and covering  ...  Our algorithm is based on a novel enumeration method, which unlike previously known enumeration techniques, can handle both packing and covering constraints.  ...  Acknowledgements Joachim Spoerhase and Sumedha Uniyal thank an anonymous reviewer for pointing them to the fact that Theorem 6 also applies to polytopes that are not down-closed, which makes it possible  ...

### Fast and Deterministic Approximations for k-Cut [article]

Kent Quanrud
2018 arXiv   pre-print
For poly(m,n,k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis [Manurangsi 2017].  ...  Is there a deterministic algorithm for 2-approximate k-cuts matching the randomized running time of Õ(mk)? The second question qualitatively compares minimum cut to 2-approximate minimum k-cut.  ...  Acknowledgements The author thanks Chandra Chekuri for introducing him to the problem and providing helpful feedback, including pointers to the literature for rounding the LP.  ...

### Online Weighted Degree-Bounded Steiner Networks via Novel Online Mixed Packing/Covering [article]

Sina Dehghani, Soheil Ehsani, MohammadTaghi Hajiaghayi, Vahid Liaghat, Harald Racke, Saeed Seddighin
2017 arXiv   pre-print
We design an online deterministic integral algorithm with competitive ratio of O(k log m) for the mixed packing/covering integer programs.  ...  We then design a generic integral algorithm for solving this restricted family of IPs. We demonstrate a new technique for solving mixed packing/covering integer programs.  ...   generalize this method for the fractional mixed packing and covering LPs.  ...

### Fast and Deterministic Approximations for k-Cut

Kent Quanrud, Michael Wagner
2019 International Workshop on Approximation Algorithms for Combinatorial Optimization
For poly(m, n, k)-time algorithms, the best possible approximation factor is essentially 2 under the small set expansion hypothesis  .  ...  We give a deterministic approximation algorithm that computes (2 + )-minimum k-cuts in O m log 3 n/ 2 time, via a (1 + )-approximation for an LP relaxation of k-cut.  ...  The standard LP (L) is difficult because it is a mixed packing and covering problem, and fast approximation algorithms for mixed packing and covering problems lead to bicriteria approximations that we  ...

### Online Mixed Packing and Covering [chapter]

Yossi Azar, Umang Bhaskar, Lisa Fleischer, Debmalya Panigrahi
2013 Proceedings of the Twenty-Fourth Annual ACM-SIAM Symposium on Discrete Algorithms
We use ideas generated from our result for mixed packing and covering to obtain polylogarithmic-competitive algorithms for these problems.  ...  We extend this framework in a fundamental way by demonstrating that it can be used to solve mixed packing and covering LPs online, where packing constraints are given offline and covering constraints are  ...  They are used in both offline approximation algorithms for packing and covering problems [11, 18, 21, 24, 25, 26, 30, 32, 33, 39, 41, 42] , and online algorithms for problems with only packing or only  ...

### Optimizing over Consecutive 1's and Circular 1's Constraints

Dorit S. Hochbaum, Asaf Levin
2006 SIAM Journal on Optimization
For the "mixed" case with both covering and packing consecutive 1's constraints we present an O(mn) time algorithm.  ...  Finally, we show an O(n min{mn, n 2 log n + m log 2 n}) time algorithm for the most general problem of mixed covering and packing case where the constraints are circular.  ...  Tamir for discussing an earlier version of this paper and to anonymous referees whose comments and suggestions improved and simplified the presentation of the results in this paper.  ...
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