Improved Approximation Guarantees for Packing and Covering Integer Programs
SIAM Journal on Computing
Strengthening integrality gaps for capacitated network design and covering problems
SODA '00 Proceedings of the eleventh annual ACM-SIAM symposium on Discrete algorithms
New approaches to covering and packing problems
SODA '01 Proceedings of the twelfth annual ACM-SIAM symposium on Discrete algorithms
Proceedings of the thirty-fifth annual ACM symposium on Theory of computing
Approximation algorithms for covering/packing integer programs
Journal of Computer and System Sciences
A general approach to online network optimization problems
ACM Transactions on Algorithms (TALG)
An Extension of the Lova´sz Local Lemma, and its Applications to Integer Programming
SIAM Journal on Computing
A Primal-Dual Randomized Algorithm for Weighted Paging
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Randomized competitive algorithms for generalized caching
STOC '08 Proceedings of the fortieth annual ACM symposium on Theory of computing
Online Primal-Dual Algorithms for Covering and Packing
Mathematics of Operations Research
The Design of Competitive Online Algorithms via a Primal: Dual Approach
Foundations and Trends® in Theoretical Computer Science
ICALP '09 Proceedings of the 36th International Colloquium on Automata, Languages and Programming: Part I
Approximability of Sparse Integer Programs
Algorithmica - Special Issue: European Symposium on Algorithms, Design and Analysis
On column-restricted and priority covering integer programs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
On k-column sparse packing programs
IPCO'10 Proceedings of the 14th international conference on Integer Programming and Combinatorial Optimization
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A covering integer program (CIP) is a mathematical program of the form: $$\begin{aligned} \min \{ c^\top \mathbf{x} \mid A\mathbf{x} \geq \mathbf{1},\; \mathbf{0} \leq \mathbf{x} \leq \mathbf{u},\; \mathbf{x} \in {\ensuremath{\mathbb{Z}}}^n\},\nonumber \end{aligned}$$ where $A \in R_{\geq 0}^{m \times n}, c, u \in {\ensuremath{\mathbb{R}}}_{\geq 0}^n$. In the online setting, the constraints (i.e., the rows of the constraint matrix A) arrive over time, and the algorithm can only increase the coordinates of x to maintain feasibility. As an intermediate step, we consider solving the covering linear program (CLP) online, where the requirement x∈ℤn is replaced by x∈ℝn. Our main results are (a) an O(logk)-competitive online algorithm for solving the CLP, and (b) an O(logk ·logℓ)-competitive randomized online algorithm for solving the CIP. Here k≤n and ℓ≤m respectively denote the maximum number of non-zero entries in any row and column of the constraint matrix A. By a result of Feige and Korman, this is the best possible for polynomial-time online algorithms, even in the special case of set cover (where A∈{0,1}m ×n and c, u∈{0,1}n). The novel ingredient of our approach is to allow the dual variables to increase and decrease throughout the course of the algorithm. We show that the previous approaches, which either only raise dual variables, or lower duals only within a guess-and-double framework, cannot give a performance better than O(logn), even when each constraint only has a single variable (i.e., k=1).