Fast Approximation Schemes for Convex Programs with Many Blocks and Coupling Constraints

Grigoriadis, Michael D.; Khachiyan, Leonid

doi:10.1137/0804004

Cited by 110 publications

(55 citation statements)

References 13 publications

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“…This approach is very similar to the approach taken in the packing-covering literature (see, e.g. [26,12,4,7]). In each iteration t, the algorithm computes a primal iterate X (t) ∈ X such that the gradient ∇f α (u (t) ) = 5n 1 − n diag(X (t) ) .…”

Section: Notation and Definitionsmentioning

confidence: 67%

“…Subsequent research quickly reduced the dependence on to O( −2 ), and many other improvements reduced the number of iterations, the time per iteration, and expanded the techniques to broader packing and covering problems. See, e.g., [19,20,12,13,26,27,9,8] for details. A recent breakthrough by Bienstock and Iyengar [4] employed a result of Nesterov [24] to reduce the dependence on to O( −1 ).…”

Section: Introductionmentioning

confidence: 99%

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Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

Iyengar

Phillips

Stein

2005

Integer Programming and Combinatorial Optimization

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We describe the semidefinite analog of the vector packing problem, and show that the semidefinite programming relaxations for Maxcut [10] and graph coloring [16] are in this class of problems. We extend a method of Bienstock and Iyengar [4] which was based on ideas from Nesterov [24] to design an algorithm for computing -approximate solutions for this class of semidefinite programs. Our algorithm is in the spirit of Klein and Lu [17], and decreases the dependence of the run-time on from −2 to −1 . For sparse graphs, our method is faster than the best specialized interior point methods. A significant feature of our method is that it treats both the Maxcut and the graph coloring problem in a unified manner.

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Section: Notation and Definitionsmentioning

confidence: 67%

Section: Introductionmentioning

confidence: 99%

Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

Iyengar

Phillips

Stein

2005

Integer Programming and Combinatorial Optimization

View full text Add to dashboard Cite

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“…They have been improved and adapted to different models in a series of papers [12,13,17,20,22,25,27,32]. While all this research was built on the idea of rerouting existing flows, the idea of augmenting flow has led to a couple of new algorithms with improved running times [7,8,10,16,34].…”

Section: The Fptasmentioning

confidence: 99%

Multicommodity Flow Approximation Used for Exact Graph Partitioning

Sellmann

Sensen

Timajev

2003

Algorithms - ESA 2003

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Abstract. We present a fully polynomial-time approximation scheme for a multicommodity flow problem that yields lower bounds of the graph bisection problem. We compare the approximation algorithm with Lagrangian relaxation based cost-decomposition approaches and linear programming software when embedded in an exact branch&bound approach for graph bisection. It is shown that the approximation algorithm is clearly superior in this context. Furthermore, we present a new practical addition to the approximation algorithm which improves its performance distinctly. Finally, we prove the performance of the graph bisection algorithm using multicommodity flow approximation by computing formerly unknown bisection widths of some DeBruijn-and Shuffle-Exchange-Graphs.

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“…Plotkin, Shmoys, and Tardos introduce this approach in the context of fractional packing problems [27]; see also, [18]. It has since been used in boosting and in the generalized experts framework in machine learning [16], [23], [3].…”

Section: Introductionmentioning

confidence: 99%

Efficient Algorithms for Budget-Constrained Markov Decision Processes

Caramanis

Dimitrov

Morton

2014

IEEE Trans. Automat. Contr.

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Abstract-Discounted, discrete-time, discrete state-space, discrete action-space Markov decision processes (MDPs) form a classical topic in control, game theory, and learning, and as a result are widely applied, increasingly, in very large-scale applications. Many algorithms have been developed to solve large-scale MDPs. Algorithms based on value iteration are particularly popular, as they are more efficient than the generic linear programming approach, by an order of magnitude in the number of states of the MDP. Yet in the case of budget constrained MDPs, no more efficient algorithm than linear programming is known. The theoretically slower running times of linear programming may limit the scalability of constrained MDPs piratically; while, theoretically, it invites the question of whether the increase is somehow intrinsic. In this paper we show that it is not, and provide two algorithms for budgetconstrained MDPs that are as efficient as value iteration. Denoting the running time of value iteration by VI, and the magnitude of the input by U , for an MDP with m expected budget constraints our first algorithm runs in time O(poly(m, log U ) · VI). Given a pre-specified degree of precision, η, for satisfying the budget constraints, our second algorithm runs in time O(log m·poly(log U )· 1 η 2 ·VI), but may produce solutions that overutilize each of the m budgets by a multiplicative factor of 1 + η. In fact, one can substitute value iteration with any algorithm, possibly specially designed for a specific MDP, that solves the MDP quickly to achieve similar theoretical guarantees. Both algorithms restrict attention to constrained infinite-horizon MDPs under discounted costs.

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Fast Approximation Schemes for Convex Programs with Many Blocks and Coupling Constraints

Cited by 110 publications

References 13 publications

Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

Approximation Algorithms for Semidefinite Packing Problems with Applications to Maxcut and Graph Coloring

Multicommodity Flow Approximation Used for Exact Graph Partitioning

Efficient Algorithms for Budget-Constrained Markov Decision Processes

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