Approximability of Sparse Integer Programs

Pritchard, David; Chakrabarty, Deeparnab

doi:10.1007/978-3-642-04128-0_8

Cited by 19 publications

(16 citation statements)

References 26 publications

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“…They gave a deterministic 3.5-approximation algorithm and randomized 3.264-approximation algorithm for demand matching. Chakrabarty and Pritchard [62] recently gave a deterministic 4-approximation algorithm and randomized 3.764-approximation algorithm for the more general 2-CS-PIP problem. Shepherd and Vetta also established a lower bound of 3 on the integrality gap of the natural LP for demand matching.…”

Section: Related Workmentioning

confidence: 99%

“…Pritchard initiated the improvements with an iterative rounding based 2 k k 2 -approximation [61], which was improved to an O(k 2 )-approximation by Chekuri, Ene, and Korula (see [62] and [6]) and Chakrabarty and Pritchard [62]. Most recently, Bansal et al [6] devised a deterministic 8k-approximation and a randomized (ek + o(k))-approximation.…”

Section: Related Workmentioning

confidence: 99%

See 1 more Smart Citation

LDRD final report : combinatorial optimization with demands.

Parekh¹,

Carr²,

Pritchard

2012

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This report summarizes the research conducted and results obtained under the LaboratoryDirected Research and Development (LDRD) project, "Combinatorial Optimization with Demands," from October 2010 to September 2012. Complex resource allocation problems are ubiquitous in mission-driven applications. We investigated resource allocation problems endowed with additional parameters called demands, which allow for richer modeling. Demandendowed resource allocation problems are considerably harder to solve than their traditional counterparts, and we developed a conceptually simple but powerful new framework called iterative packing which allowed us to address the added complexity of demands. By leveraging iterative packing, we designed approximation algorithms to solve a large class of hard resource allocation problems with demands. Approximation algorithms are efficient heuristics which also offer a performance guarantee on the worst case deviation from the cost of an optimal solution. Our algorithms offer better performance guarantees over previously known algorithms; moreover, we are able to show that our algorithms provide performance guarantees that are likely close to best possible. 3 AcknowledgmentsWe thank Cindy Phillips for insightful and entertaining discussions. We thank Brian Barrett and Rich Field for maintaining the L A T E X template on which this report is based.4

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Section: Related Workmentioning

confidence: 99%

Section: Related Workmentioning

confidence: 99%

LDRD final report : combinatorial optimization with demands.

Parekh¹,

Carr²,

Pritchard

2012

Self Cite

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“…The algorithm is roughly as follows. problem approximation ratio method where comment VERTEX COVER 2 − ln ln ∆/ ln ∆ local ratio [30] see also [34,7,50,28,29,31,24,40] SET COVER ∆ LP; greedy [33,34]; [6] ∆ = max i |{j | A ij > 0}| CIP-01 w/A ij ∈ Z ≥0 max i j A ij primal-dual [10,27] quadratic time CIP-UB ∆ ellipsoid [16,54,55] KC-ineq., high-degree-poly time SUBMOD-COST COVER ∆ greedy [our §2] min{c(x) | x ∈ S (∀S ∈ C)} new SET/VERTEX COVER ∆ greedy [ [39,47] PAGING k = ∆ potential function [59,56] e.g. LRU, FIFO, FWF, Harmonic CONNECTION CACHING O(k) reduction to paging [20,1] WEIGHTED CACHING k primal-dual [63,64,56] e.g.…”

Section: Definition 1 (Submodular-cost Covering) An Instance Is a Trmentioning

confidence: 99%

“…from to CMIP-UB). (The standard KC inequalities suffice for O(log( ∆))-approximation of CMIP-UB [41], but may require some modification to give ∆-approximation of CMIP-UB [54,55].) The primal-dual analysis in Section 6 uses a new linear program (LP) relaxation for LINEAR-COST COVERING that may help better understand how to extend the KC inequalities.…”

Section: Definition 1 (Submodular-cost Covering) An Instance Is a Trmentioning

confidence: 99%

Greedy ${\ensuremath{\Delta}}$ -Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

Koufogiannakis

Young

2009

Automata, Languages and Programming

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This paper describes a simple greedy ∆-approximation algorithm for any covering problem whose objective function is submodular and non-decreasing, and whose feasible region can be expressed as the intersection of arbitrary (closed upwards) covering constraints, each of which constrains at most ∆ variables of the problem. (A simple example is VERTEX COVER, with ∆ = 2.) The algorithm generalizes previous approximation algorithms for fundamental covering problems and online paging and caching problems.

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“…Analyzing performance guarantees for covering/packing integer programs in terms of row (k) and column (ℓ) sparsity has received much attention in the offline setting, e.g. [15,17,11,14,6]. This paper obtains tight bounds in terms of these parameters for online covering integer programs.…”

Section: Introductionmentioning

confidence: 99%

Approximating Sparse Covering Integer Programs Online

Gupta

Nagarajan

2012

Automata, Languages, and Programming

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A covering integer program (CIP) is a mathematical program of the form:where A ∈ R m×n ≥0 , c, u ∈ R n ≥0 . 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 ∈ Z n is replaced by x ∈ R n .Our main results are (a) an O(log k)-competitive online algorithm for solving the CLP, and (b) an O(log k · log ℓ)-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(log n), even when each constraint only has a single variable (i.e., k = 1).

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Approximability of Sparse Integer Programs

Cited by 19 publications

References 26 publications

LDRD final report : combinatorial optimization with demands.

LDRD final report : combinatorial optimization with demands.

Greedy ${\ensuremath{\Delta}}$ -Approximation Algorithm for Covering with Arbitrary Constraints and Submodular Cost

Approximating Sparse Covering Integer Programs Online

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