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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

In budget-feasible mechanism design, a buyer wishes to procure a set of items of maximum value from self-interested rational players. We are given an item-set U and a nonnegative valuation function v: 2^U ↦ ℝ_+. Each item e is held by a player who incurs a private cost c_e for supplying item e. The goal is to devise a truthful mechanism such that the total payment made to the players is at most some given budget B, and the value of the set returned is a good approximation to OPT: = max {v(S): c(S) ≤ B, S ⊆ U}. We call such a mechanism a budget-feasible mechanism. More generally, there may be additional side constraints requiring that the set returned lies in some downwards-monotone family ℐ ⊆ 2^U. Budget-feasible mechanisms have been widely studied, but there are still significant gaps in our understanding of these mechanisms, both in terms of what kind of oracle access to the valuation is required to obtain good approximation ratios, and the best approximation ratio that can be achieved.
We substantially advance the state of the art of budget-feasible mechanisms by devising mechanisms that are simpler, and also better, both in terms of requiring weaker oracle access and the approximation factors they obtain. For XOS valuations, we devise the first polytime O(1)-approximation budget-feasible mechanism using only demand oracles, and also significantly improve the approximation factor. For subadditive valuations, we give the first explicit construction of an O(1)-approximation mechanism, where previously only an existential result was known.
We also introduce a fairly rich class of mechanism-design problems that we dub using the umbrella term generalized budget-feasible mechanism design, which allow one to capture payment constraints that are much-more nuanced than a single constraint on the total payment doled out. We demonstrate the versatility of our ideas by showing that our constructions can be adapted to yield approximation guarantees in such general settings as well.
A prominent insight to emerge from our work is the usefulness of a property called nobossiness, which allows us to nicely decouple the truthfulness + approximation, and budget-feasibility requirements. Some of our constructions can be viewed as reductions showing that an O(1)-approximation budget-feasible mechanism can be obtained provided we have a (randomized) truthful mechanism satisfying nobossiness that returns a (random) feasible set having (expected) value Ω(OPT).

Rian Neogi, Kanstantsin Pashkovich, and Chaitanya Swamy. Budget-Feasible Mechanism Design: Simpler, Better Mechanisms and General Payment Constraints. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 84:1-84:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{neogi_et_al:LIPIcs.ITCS.2024.84, author = {Neogi, Rian and Pashkovich, Kanstantsin and Swamy, Chaitanya}, title = {{Budget-Feasible Mechanism Design: Simpler, Better Mechanisms and General Payment Constraints}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {84:1--84:22}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.84}, URN = {urn:nbn:de:0030-drops-196128}, doi = {10.4230/LIPIcs.ITCS.2024.84}, annote = {Keywords: Algorithmic mechanism design, Approximation algorithms, Budget-feasible mechanisms} }

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Complete Volume

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume

Amit Chakrabarti and Chaitanya Swamy. LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 1-1064, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@Proceedings{chakrabarti_et_al:LIPIcs.APPROX/RANDOM.2022, title = {{LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {1--1064}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022}, URN = {urn:nbn:de:0030-drops-171211}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022}, annote = {Keywords: LIPIcs, Volume 245, APPROX/RANDOM 2022, Complete Volume} }

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Front Matter

**Published in:** LIPIcs, Volume 245, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)

Front Matter, Table of Contents, Preface, Conference Organization

Amit Chakrabarti and Chaitanya Swamy. Front Matter, Table of Contents, Preface, Conference Organization. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 245, pp. 0:i-0:xx, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022)

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@InProceedings{chakrabarti_et_al:LIPIcs.APPROX/RANDOM.2022.0, author = {Chakrabarti, Amit and Swamy, Chaitanya}, title = {{Front Matter, Table of Contents, Preface, Conference Organization}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2022)}, pages = {0:i--0:xx}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-249-5}, ISSN = {1868-8969}, year = {2022}, volume = {245}, editor = {Chakrabarti, Amit and Swamy, Chaitanya}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2022.0}, URN = {urn:nbn:de:0030-drops-171229}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2022.0}, annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We investigate a genre of vehicle-routing problems (VRPs), that we call max-reward VRPs, wherein nodes located in a metric space have associated rewards that depend on their visiting times, and we seek a path that earns maximum reward. A prominent problem in this genre is deadline TSP, where nodes have deadlines and we seek a path that visits all nodes by their deadlines and earns maximum reward. Our main result is a constant-factor approximation for deadline TSP running in time O(n^O(log(nΔ))) in metric spaces with integer distances at most Δ. This is the first improvement over the approximation factor of O(log n) due to Bansal et al. [N. Bansal et al., 2004] in over 15 years (but is achieved in super-polynomial time). Our result provides the first concrete indication that log n is unlikely to be a real inapproximability barrier for deadline TSP, and raises the exciting possibility that deadline TSP might admit a polytime constant-factor approximation.
At a high level, we obtain our result by carefully guessing an appropriate sequence of O(log (nΔ)) nodes appearing on the optimal path, and finding suitable paths between any two consecutive guessed nodes. We argue that the problem of finding a path between two consecutive guessed nodes can be relaxed to an instance of a special case of deadline TSP called point-to-point (P2P) orienteering. Any approximation algorithm for P2P orienteering can then be utilized in conjunction with either a greedy approach, or an LP-rounding approach, to find a good set of paths overall between every pair of guessed nodes. While concatenating these paths does not immediately yield a feasible solution, we argue that it can be covered by a constant number of feasible solutions. Overall our result therefore provides a novel reduction showing that any α-approximation for P2P orienteering can be leveraged to obtain an O(α)-approximation for deadline TSP in O(n^O(log nΔ)) time.
Our results extend to yield the same guarantees (in approximation ratio and running time) for a substantial generalization of deadline TSP, where the reward obtained by a client is given by an arbitrary non-increasing function (specified by a value oracle) of its visiting time. Finally, we discuss applications of our results to variants of deadline TSP, including settings where both end-nodes are specified, nodes have release dates, and orienteering with time windows.

Zachary Friggstad and Chaitanya Swamy. Constant-Factor Approximation to Deadline TSP and Related Problems in (Almost) Quasi-Polytime. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 67:1-67:21, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{friggstad_et_al:LIPIcs.ICALP.2021.67, author = {Friggstad, Zachary and Swamy, Chaitanya}, title = {{Constant-Factor Approximation to Deadline TSP and Related Problems in (Almost) Quasi-Polytime}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {67:1--67:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.67}, URN = {urn:nbn:de:0030-drops-141369}, doi = {10.4230/LIPIcs.ICALP.2021.67}, annote = {Keywords: Approximation algorithms, Vehicle routing problems, Deadline TSP, Orienteering} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 198, 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)

We consider the minimum-norm load-balancing (MinNormLB) problem, wherein there are n jobs, each of which needs to be assigned to one of m machines, and we are given the processing times {p_{ij}} of the jobs on the machines. We also have a monotone, symmetric norm f:ℝ^m → ℝ_{≥ 0}. We seek an assignment σ of jobs to machines that minimizes the f-norm of the induced load vector load->_σ ∈ ℝ_{≥ 0}^m, where load_σ(i) = ∑_{j:σ(j) = i}p_{ij}. This problem was introduced by [Deeparnab Chakrabarty and Chaitanya Swamy, 2019], and the current-best result for MinNormLB is a (4+ε)-approximation [Deeparnab Chakrabarty and Chaitanya Swamy, 2019]. In the stochastic version of MinNormLB, the job processing times are given by nonnegative random variables X_{ij}, and jobs are independent; the goal is to find an assignment σ that minimizes the expected f-norm of the induced random load vector.
We obtain results that (essentially) match the best-known guarantees for deterministic makespan minimization (MinNormLB with 𝓁_∞ norm). For MinNormLB, we obtain a (2+ε)-approximation for unrelated machines, and a PTAS for identical machines. For stochastic MinNormLB, we consider the setting where the X_{ij}s are Poisson random variables, denoted PoisNormLB. Our main result here is a novel and powerful reduction showing that, for any machine environment (e.g., unrelated/identical machines), any α-approximation algorithm for MinNormLB in that machine environment yields a randomized α(1+ε)-approximation for PoisNormLB in that machine environment. Combining this with our results for MinNormLB, we immediately obtain a (2+ε)-approximation for PoisNormLB on unrelated machines, and a PTAS for PoisNormLB on identical machines. The latter result substantially generalizes a PTAS for makespan minimization with Poisson jobs obtained recently by [Anindya De et al., 2020].

Sharat Ibrahimpur and Chaitanya Swamy. Minimum-Norm Load Balancing Is (Almost) as Easy as Minimizing Makespan. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 81:1-81:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ibrahimpur_et_al:LIPIcs.ICALP.2021.81, author = {Ibrahimpur, Sharat and Swamy, Chaitanya}, title = {{Minimum-Norm Load Balancing Is (Almost) as Easy as Minimizing Makespan}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {81:1--81:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-195-5}, ISSN = {1868-8969}, year = {2021}, volume = {198}, editor = {Bansal, Nikhil and Merelli, Emanuela and Worrell, James}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.81}, URN = {urn:nbn:de:0030-drops-141504}, doi = {10.4230/LIPIcs.ICALP.2021.81}, annote = {Keywords: Approximation algorithms, Load balancing, Minimum-norm optimization, LP rounding} }

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**Published in:** LIPIcs, Volume 173, 28th Annual European Symposium on Algorithms (ESA 2020)

We consider the directed minimum latency problem (DirLat), wherein we seek a path P visiting all points (or clients) in a given asymmetric metric starting at a given root node r, so as to minimize the sum of the client waiting times, where the waiting time of a client v is the length of the r-v portion of P. We give the first constant-factor approximation guarantee for DirLat, but in quasi-polynomial time. Previously, a polynomial-time O(log n)-approximation was known [Z. Friggstad et al., 2013], and no better approximation guarantees were known even in quasi-polynomial time.
A key ingredient of our result, and our chief technical contribution, is an extension of a recent result of [A. Köhne et al., 2019] showing that the integrality gap of the natural Held-Karp relaxation for asymmetric TSP-Path (ATSPP) is at most a constant, which itself builds on the breakthrough similar result established for asymmetric TSP (ATSP) by Svensson et al. [O. Svensson et al., 2018]. We show that the integrality gap of the Held-Karp relaxation for ATSPP is bounded by a constant even if the cut requirements of the LP relaxation are relaxed from x(δ^{in}(S)) ≥ 1 to x(δ^{in}(S)) ≥ ρ for some constant 1/2 < ρ ≤ 1.
We also give a better approximation guarantee for the minimum total-regret problem, where the goal is to find a path P that minimizes the total time that nodes spend in excess of their shortest-path distances from r, which can be cast as a special case of DirLat involving so-called regret metrics.

Zachary Friggstad and Chaitanya Swamy. A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time. In 28th Annual European Symposium on Algorithms (ESA 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 173, pp. 52:1-52:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020)

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@InProceedings{friggstad_et_al:LIPIcs.ESA.2020.52, author = {Friggstad, Zachary and Swamy, Chaitanya}, title = {{A Constant-Factor Approximation for Directed Latency in Quasi-Polynomial Time}}, booktitle = {28th Annual European Symposium on Algorithms (ESA 2020)}, pages = {52:1--52:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-162-7}, ISSN = {1868-8969}, year = {2020}, volume = {173}, editor = {Grandoni, Fabrizio and Herman, Grzegorz and Sanders, Peter}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2020.52}, URN = {urn:nbn:de:0030-drops-129183}, doi = {10.4230/LIPIcs.ESA.2020.52}, annote = {Keywords: Approximation Algorithms, Directed Latency, TSP} }

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**Published in:** LIPIcs, Volume 144, 27th Annual European Symposium on Algorithms (ESA 2019)

Recently, Chakrabarty and Swamy (STOC 2019) introduced the minimum-norm load-balancing problem on unrelated machines, wherein we are given a set J of jobs that need to be scheduled on a set of m unrelated machines, and a monotone, symmetric norm; We seek an assignment sigma: J -> [m] that minimizes the norm of the resulting load vector load_{sigma} in R_+^m, where load_{sigma}(i) is the load on machine i under the assignment sigma. Besides capturing all l_p norms, symmetric norms also capture other norms of interest including top-l norms, and ordered norms. Chakrabarty and Swamy (STOC 2019) give a (38+epsilon)-approximation algorithm for this problem via a general framework they develop for minimum-norm optimization that proceeds by first carefully reducing this problem (in a series of steps) to a problem called min-max ordered load balancing, and then devising a so-called deterministic oblivious LP-rounding algorithm for ordered load balancing.
We give a direct, and simple 4+epsilon-approximation algorithm for the minimum-norm load balancing based on rounding a (near-optimal) solution to a novel convex-programming relaxation for the problem. Whereas the natural convex program encoding minimum-norm load balancing problem has a large non-constant integrality gap, we show that this issue can be remedied by including a key constraint that bounds the "norm of the job-cost vector." Our techniques also yield a (essentially) 4-approximation for: (a) multi-norm load balancing, wherein we are given multiple monotone symmetric norms, and we seek an assignment respecting a given budget for each norm; (b) the best simultaneous approximation factor achievable for all symmetric norms for a given instance.

Deeparnab Chakrabarty and Chaitanya Swamy. Simpler and Better Algorithms for Minimum-Norm Load Balancing. In 27th Annual European Symposium on Algorithms (ESA 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 144, pp. 27:1-27:12, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2019)

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@InProceedings{chakrabarty_et_al:LIPIcs.ESA.2019.27, author = {Chakrabarty, Deeparnab and Swamy, Chaitanya}, title = {{Simpler and Better Algorithms for Minimum-Norm Load Balancing}}, booktitle = {27th Annual European Symposium on Algorithms (ESA 2019)}, pages = {27:1--27:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-124-5}, ISSN = {1868-8969}, year = {2019}, volume = {144}, editor = {Bender, Michael A. and Svensson, Ola and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2019.27}, URN = {urn:nbn:de:0030-drops-111488}, doi = {10.4230/LIPIcs.ESA.2019.27}, annote = {Keywords: Approximation Algorithms} }

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**Published in:** LIPIcs, Volume 112, 26th Annual European Symposium on Algorithms (ESA 2018)

We study inverse optimization problems, wherein the goal is to map given solutions to an underlying optimization problem to a cost vector for which the given solutions are the (unique) optimal solutions. Inverse optimization problems find diverse applications and have been widely studied. A prominent problem in this field is the inverse shortest path (ISP) problem [D. Burton and Ph.L. Toint, 1992; W. Ben-Ameur and E. Gourdin, 2004; A. Bley, 2007], which finds applications in shortest-path routing protocols used in telecommunications. Here we seek a cost vector that is positive, integral, induces a set of given paths as the unique shortest paths, and has minimum l_infty norm. Despite being extensively studied, very few algorithmic results are known for inverse optimization problems involving integrality constraints on the desired cost vector whose norm has to be minimized.
Motivated by ISP, we initiate a systematic study of such integral inverse optimization problems from the perspective of designing polynomial time approximation algorithms. For ISP, our main result is an additive 1-approximation algorithm for multicommodity ISP with node-disjoint commodities, which we show is tight assuming P!=NP. We then consider the integral-cost inverse versions of various other fundamental combinatorial optimization problems, including min-cost flow, max/min-cost bipartite matching, and max/min-cost basis in a matroid, and obtain tight or nearly-tight approximation guarantees for these. Our guarantees for the first two problems are based on results for a broad generalization, namely integral inverse polyhedral optimization, for which we also give approximation guarantees. Our techniques also give similar results for variants, including l_p-norm minimization of the integral cost vector, and distance-minimization from an initial cost vector.

Sara Ahmadian, Umang Bhaskar, Laura Sanità, and Chaitanya Swamy. Algorithms for Inverse Optimization Problems. In 26th Annual European Symposium on Algorithms (ESA 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 112, pp. 1:1-1:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ahmadian_et_al:LIPIcs.ESA.2018.1, author = {Ahmadian, Sara and Bhaskar, Umang and Sanit\`{a}, Laura and Swamy, Chaitanya}, title = {{Algorithms for Inverse Optimization Problems}}, booktitle = {26th Annual European Symposium on Algorithms (ESA 2018)}, pages = {1:1--1:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-081-1}, ISSN = {1868-8969}, year = {2018}, volume = {112}, editor = {Azar, Yossi and Bast, Hannah and Herman, Grzegorz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ESA.2018.1}, URN = {urn:nbn:de:0030-drops-94646}, doi = {10.4230/LIPIcs.ESA.2018.1}, annote = {Keywords: Inverse optimization, Shortest paths, Approximation algorithms, Linear programming, Polyhedral theory, Combinatorial optimization} }

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**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We consider a generalization of k-median and k-center, called the ordered k-median problem. In this problem, we are given a metric space (D,{c_{ij}}) with n=|D| points, and a non-increasing weight vector w in R_+^n, and the goal is to open k centers and assign each point j in D to a center so as to minimize w_1 *(largest assignment cost)+w_2 *(second-largest assignment cost)+...+w_n *(n-th largest assignment cost). We give an (18+epsilon)-approximation algorithm for this problem. Our algorithms utilize Lagrangian relaxation and the primal-dual schema, combined with an enumeration procedure of Aouad and Segev. For the special case of {0,1}-weights, which models the problem of minimizing the l largest assignment costs that is interesting in and of by itself, we provide a novel reduction to the (standard) k-median problem, showing that LP-relative guarantees for k-median translate to guarantees for the ordered k-median problem; this yields a nice and clean (8.5+epsilon)-approximation algorithm for {0,1} weights.

Deeparnab Chakrabarty and Chaitanya Swamy. Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 29:1-29:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chakrabarty_et_al:LIPIcs.ICALP.2018.29, author = {Chakrabarty, Deeparnab and Swamy, Chaitanya}, title = {{Interpolating between k-Median and k-Center: Approximation Algorithms for Ordered k-Median}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {29:1--29:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.29}, URN = {urn:nbn:de:0030-drops-90335}, doi = {10.4230/LIPIcs.ICALP.2018.29}, annote = {Keywords: Approximation algorithms, Clustering, Facility location, Primal-dual method} }

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**Published in:** LIPIcs, Volume 81, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)

In the prize-collecting Steiner forest (PCSF) problem, we are given an undirected graph G=(V,E), nonnegative edge costs {c_e} for e in E, terminal pairs {(s_i,t_i)} for i=1,...,k, and penalties {pi_i} for i=1,...,k for each terminal pair; the goal is to find a forest F to minimize c(F) + sum{ pi_i: (s_i,t_i) is not connected in F }. The Steiner forest problem can be viewed as the special case where pi_i are infinite for all i. It was widely believed that the integrality gap of the natural (and well-studied) linear-programming (LP) relaxation for PCSF (PCSF-LP) is at most 2. We dispel this belief by showing that the integrality gap of this LP is at least 9/4 even if the input instance is planar. We also show that using this LP, one cannot devise a Lagrangian-multiplier-preserving (LMP) algorithm with approximation guarantee better than 4. Our results thus show a separation between the integrality gaps of the LP-relaxations for prize-collecting and non-prize-collecting (i.e., standard) Steiner forest, as well as the approximation ratios achievable relative to the optimal LP solution by LMP- and non-LMP-approximation algorithms for PCSF. For the special case of prize-collecting Steiner tree (PCST), we prove that the natural LP relaxation admits basic feasible solutions with all coordinates of value at most 1/3 and all edge variables positive. Thus, we rule out the possibility of approximating PCST with guarantee better than 3 using a direct iterative rounding method.

Jochen Könemann, Neil Olver, Kanstantsin Pashkovich, R. Ravi, Chaitanya Swamy, and Jens Vygen. On the Integrality Gap of the Prize-Collecting Steiner Forest LP. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 81, pp. 17:1-17:13, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{konemann_et_al:LIPIcs.APPROX-RANDOM.2017.17, author = {K\"{o}nemann, Jochen and Olver, Neil and Pashkovich, Kanstantsin and Ravi, R. and Swamy, Chaitanya and Vygen, Jens}, title = {{On the Integrality Gap of the Prize-Collecting Steiner Forest LP}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2017)}, pages = {17:1--17:13}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-044-6}, ISSN = {1868-8969}, year = {2017}, volume = {81}, editor = {Jansen, Klaus and Rolim, Jos\'{e} D. P. and Williamson, David P. and Vempala, Santosh S.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2017.17}, URN = {urn:nbn:de:0030-drops-75665}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2017.17}, annote = {Keywords: Integrality gap, Steiner tree, Steiner forest, prize-collecting, Lagrangianmultiplier- preserving} }

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**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We consider the MST-interdiction problem: given a multigraph G = (V, E), edge weights {w_e >= 0}_{e in E}, interdiction costs {c_e >= 0}_{e in E}, and an interdiction budget B >= 0, the goal is to remove a subset R of edges of total interdiction cost at most B so as to maximize the w-weight of an MST of G-R:=(V,E-R).
Our main result is a 4-approximation algorithm for this problem. This improves upon the previous-best 14-approximation [Zenklusen, FOCS 2015]. Notably, our analysis is also significantly simpler and cleaner than the one in [Zenklusen, FOCS 2015]. Whereas Zenklusen uses a greedy algorithm with an involved analysis to extract a good interdiction set from an over-budget set, we utilize a generalization of knapsack called the tree knapsack problem that nicely captures the key combinatorial aspects of this "extraction problem." We prove a simple, yet strong, LP-relative approximation bound for tree knapsack, which leads to our improved guarantees for MST interdiction. Our algorithm and analysis are nearly tight, as we show that one cannot achieve an approximation ratio better than 3 relative to the upper bound used in our analysis (and the one in [Zenklusen, FOCS 2015]).
Our guarantee for MST-interdiction yields an 8-approximation for metric-TSP interdiction (improving over the 28-approximation in [Zenklusen, FOCS 2015]). We also show that maximum-spanning-tree interdiction is at least as hard to approximate as the minimization version of densest-k-subgraph.

André Linhares and Chaitanya Swamy. Improved Algorithms for MST and Metric-TSP Interdiction. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 32:1-32:14, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017)

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@InProceedings{linhares_et_al:LIPIcs.ICALP.2017.32, author = {Linhares, Andr\'{e} and Swamy, Chaitanya}, title = {{Improved Algorithms for MST and Metric-TSP Interdiction}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {32:1--32:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.32}, URN = {urn:nbn:de:0030-drops-74552}, doi = {10.4230/LIPIcs.ICALP.2017.32}, annote = {Keywords: Approximation algorithms, interdiction problems, LP-rounding algorithms, iterative rounding, tree-knapsack problem, supermodular functions} }

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**Published in:** LIPIcs, Volume 55, 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)

We consider clustering problems with non-uniform lower bounds and outliers, and obtain the first approximation guarantees for these problems. We have a set F of facilities with lower bounds {L_i}_{i in F} and a set D of clients located in a common metric space {c(i,j)}_{i,j in F union D}, and bounds k, m. A feasible solution is a pair (S subseteq F, sigma: D -> S union {out}), where sigma specifies the client assignments, such that |S| <=k, |sigma^{-1}(i)| >= L_i for all i in S, and |sigma^{-1}(out)| <= m. In the lower-bounded min-sum-of-radii with outliers P (LBkSRO) problem, the objective is to minimize sum_{i in S} max_{j in sigma^{-1})i)}, and in the lower-bounded k-supplier with outliers (LBkSupO) problem, the objective is to minimize max_{i in S} max_{j in sigma^{-1})i)} c(i,j).
We obtain an approximation factor of 12.365 for LBkSRO, which improves to 3.83 for the non-outlier version (i.e., m = 0). These also constitute the first approximation bounds for the min-sum-of-radii objective when we consider lower bounds and outliers separately. We apply the primal-dual method to the relaxation where we Lagrangify the |S| <= k constraint. The chief technical contribution and novelty of our algorithm is that, departing from the standard paradigm used for such constrained problems, we obtain an O(1)-approximation despite the fact that we do not obtain a Lagrangian-multiplier-preserving algorithm for the Lagrangian relaxation. We believe that our ideas have broader applicability to other clustering problems with outliers as well.
We obtain approximation factors of 5 and 3 respectively for LBkSupO and its non-outlier version. These are the first approximation results for k-supplier with non-uniform lower bounds.

Sara Ahmadian and Chaitanya Swamy. Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 69:1-69:15, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ahmadian_et_al:LIPIcs.ICALP.2016.69, author = {Ahmadian, Sara and Swamy, Chaitanya}, title = {{Approximation Algorithms for Clustering Problems with Lower Bounds and Outliers}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {69:1--69:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-013-2}, ISSN = {1868-8969}, year = {2016}, volume = {55}, editor = {Chatzigiannakis, Ioannis and Mitzenmacher, Michael and Rabani, Yuval and Sangiorgi, Davide}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2016.69}, URN = {urn:nbn:de:0030-drops-62153}, doi = {10.4230/LIPIcs.ICALP.2016.69}, annote = {Keywords: Approximation algorithms, facililty-location problems, primal-dual method, Lagrangian relaxation, k-center problems, minimizing sum of radii} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

We consider a facility-location problem that abstracts settings where the cost of serving the clients assigned to a facility is incurred by the facility. Formally, we consider the minimum-load k-facility location (MLkFL) problem, which is defined as follows. We have a set F of facilities, a set C of clients, and an integer k > 0. Assigning client j to a facility f incurs a connection cost d(f, j). The goal is to open a set F' of k facilities, and assign each client j to a facility f(j) in F' so as to minimize maximum, over all facilities in F', of the sum of distances of clients j assigned to F' to F'. We call
this sum the load of facility f. This problem was studied under the name of min-max star cover in [6, 2], who (among other results) gave bicriteria approximation algorithms for MLkFL for when F = C. MLkFL is rather poorly understood, and only an O(k)-approximation is currently known for MLkFL, even for line metrics. Our main result is the first polynomial time approximation scheme (PTAS) for MLkFL on line metrics (note that no non-trivial true approximation of any kind was known for this metric). Complementing this, we prove that MLkFL is strongly NP-hard on line metrics. We also devise a quasi-PTAS for MLkFL on tree metrics. MLkFL turns out to be surprisingly challenging even on line metrics, and resilient to attack by the variety of techniques that have been successfully applied to facility-location problems. For instance, we show that: (a) even a configuration-style LP-relaxation has a bad integrality gap; and (b) a multi-swap k-median style local-search heuristic has a bad locality gap. Thus, we need to devise various novel techniques to attack MLkFL. Our PTAS for line metrics consists of two main ingredients. First, we prove that there always exists a near-optimal solution possessing some nice structural properties. A novel aspect of this proof is that we first move to a mixed-integer LP (MILP) encoding the problem, and argue that a MILP-solution minimizing a certain potential function possesses the desired structure, and then use a rounding algorithm for the generalized-assignment problem to "transfer" this structure to the rounded integer solution. Complementing this, we show that these structural properties enable one to find such a structured solution via dynamic programming.

Sara Ahmadian, Babak Behsaz, Zachary Friggstad, Amin Jorati, Mohammad R. Salavatipour, and Chaitanya Swamy. Approximation Algorithms for Minimum-Load k-Facility Location. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 17-33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{ahmadian_et_al:LIPIcs.APPROX-RANDOM.2014.17, author = {Ahmadian, Sara and Behsaz, Babak and Friggstad, Zachary and Jorati, Amin and Salavatipour, Mohammad R. and Swamy, Chaitanya}, title = {{Approximation Algorithms for Minimum-Load k-Facility Location}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {17--33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.17}, URN = {urn:nbn:de:0030-drops-47154}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.17}, annote = {Keywords: approximation algorithms, min-max star cover, facility location, line metrics} }

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**Published in:** LIPIcs, Volume 28, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)

We consider the matroid median problem, wherein we are given a set of facilities with opening costs and a matroid on the facility-set, and clients with demands and connection costs, and we seek to open an independent set of facilities and assign clients to open facilities so as to minimize the sum of the facility-opening and client-connection costs. We give a simple 8-approximation algorithm for this problem based on LP-rounding, which improves upon the 16-approximation by Krishnaswamy et al. We illustrate the power and versatility of our techniques by deriving: (a) an 8-approximation for the two-matroid median problem, a generalization of matroid median that we introduce involving two matroids; and (b) a 24-approximation algorithm for matroid median with penalties, which is a vast improvement over the 360-approximation obtained by Krishnaswamy et al. We show that a variety of seemingly disparate facility-location problems considered in the literature -- data placement problem, mobile facility location, k-median forest, metric uniform minimum-latency UFL -- in fact reduce to the matroid median or two-matroid median problems, and thus obtain improved approximation guarantees for all these problems. Our techniques also yield an improvement for the knapsack median problem.

Chaitanya Swamy. Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 403-418, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2014)

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@InProceedings{swamy:LIPIcs.APPROX-RANDOM.2014.403, author = {Swamy, Chaitanya}, title = {{Improved Approximation Algorithms for Matroid and Knapsack Median Problems and Applications}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {403--418}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-74-3}, ISSN = {1868-8969}, year = {2014}, volume = {28}, editor = {Jansen, Klaus and Rolim, Jos\'{e} and Devanur, Nikhil R. and Moore, Cristopher}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2014.403}, URN = {urn:nbn:de:0030-drops-47125}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.403}, annote = {Keywords: Approximation algorithms, LP rounding, facility location, matroid and submodular polyhedra, knapsack constraints} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 7391, Probabilistic Methods in the Design and Analysis of Algorithms (2007)

Stochastic optimization problems provide a means to model uncertainty in the input data where the uncertainty is modeled by a probability distribution over the possible realizations of the data. We consider a broad class of these problems, called {it multi-stage stochastic programming problems with recourse}, where the uncertainty evolves through a series of stages and one take decisions in each stage in response to the new information learned. These problems are often computationally quite difficult with even very specialized (sub)problems being $#P$-complete.
We obtain the first fully polynomial randomized approximation scheme (FPRAS) for a broad class of multi-stage stochastic linear programming problems with any constant number of stages, without placing any restrictions on the underlying probability distribution or on the cost structure of the input. For any fixed $k$, for a rich class of $k$-stage stochastic linear programs (LPs), we show that, for any probability distribution, for any $epsilon>0$, one can compute, with high probability, a solution with expected cost at most $(1+e)$ times the optimal expected cost, in time polynomial in the input size, $frac{1}{epsilon}$, and a parameter $lambda$ that is an upper bound on the cost-inflation over successive stages. Moreover, the algorithm analyzed is a simple and intuitive algorithm that is often used in practice, the {it sample average approximation} (SAA) method. In this method, one draws certain samples from the underlying distribution, constructs an approximate distribution from these samples, and solves the stochastic problem given by this approximate distribution. This is the first result establishing that the SAA method yields near-optimal solutions for (a class of) multi-stage programs with a polynomial number of samples.
As a corollary of this FPRAS, by adapting a generic rounding technique of Shmoys and Swamy, we also obtain the first approximation algorithms for the analogous class of multi-stage stochastic integer programs, which includes the multi-stage versions of the set cover, vertex cover, multicut on trees, facility location, and multicommodity flow problems.

Chaitanya Swamy and David Shmoys. Sampling-based Approximation Algorithms for Multi-stage Stochastic Optimization. In Probabilistic Methods in the Design and Analysis of Algorithms. Dagstuhl Seminar Proceedings, Volume 7391, pp. 1-24, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2007)

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@InProceedings{swamy_et_al:DagSemProc.07391.2, author = {Swamy, Chaitanya and Shmoys, David}, title = {{Sampling-based Approximation Algorithms for Multi-stage Stochastic Optimization}}, booktitle = {Probabilistic Methods in the Design and Analysis of Algorithms}, pages = {1--24}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2007}, volume = {7391}, editor = {Martin Dietzfelbinger and Shang-Hua Teng and Eli Upfal and Berthold V\"{o}cking}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.07391.2}, URN = {urn:nbn:de:0030-drops-12906}, doi = {10.4230/DagSemProc.07391.2}, annote = {Keywords: Stochastic optimization, approximation algorithms, randomized algorithms, linear programming} }

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**Published in:** Dagstuhl Seminar Proceedings, Volume 5031, Algorithms for Optimization with Incomplete Information (2005)

Stochastic optimization problems attempt to model uncertainty in the data by assuming that (part of) the input is specified by a probability distribution. We consider the well-studied paradigm of stochastic recourse models, where the uncertainty evolves through a series of stages and one can take decisions in each stage in response to the new information learned. We obtain the first approximation algorithms for a variety of 2-stage and k-stage stochastic integer optimization problems where the underlying random data is given by a "black box" and no restrictions are placed on the costs of the two stages: one can merely sample data from this distribution, but no direct information about the distributions is given. Our results are based on two principal components. First, we show that for a broad class of 2-stage and k-stage linear programs, where k is not part of the input, given only a "black box" to draw independent samples from the distribution, one can, for any \epsilon>0, compute a solution of cost guaranteed to be within a (1+\epsilon) factor of the optimum, in time polynomial in 1/\epsilon, the size of the input, and a parameter \lambda that is the ratio of the cost of the same action in successive stages which is a lower bound on the sample complexity in the "black-box" model. This is based on reformulating the stochastic linear program, which has both an exponential number of variables and an exponential number of constraints, as a compact convex program, and adapting tools from convex optimization to solve the resulting program to near optimality. In doing so, a significant difficulty that we must overcome is that even evaluating the objective function of this convex program at a given point may be quite difficult and provably hard. To the best of our knowledge, this is the first such result for multi-stage stochastic programs. Second, we give a rounding approach for stochastic integer programs that shows that approximation algorithms for a deterministic analogue yields, with a small constant-factor loss, provably near-optimal solutions for the stochastic generalization. Thus we obtain approximation algorithms for several stochastic problems, including the stochastic versions of the set cover, vertex cover, facility location, multicut (on trees) and multicommodity flow problems.

Chaitanya Swamy and David Shmoys. Approximation Algorithms for 2-stage and Multi-stage Stochastic Optimization. In Algorithms for Optimization with Incomplete Information. Dagstuhl Seminar Proceedings, Volume 5031, pp. 1-5, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2005)

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@InProceedings{swamy_et_al:DagSemProc.05031.5, author = {Swamy, Chaitanya and Shmoys, David}, title = {{Approximation Algorithms for 2-stage and Multi-stage Stochastic Optimization}}, booktitle = {Algorithms for Optimization with Incomplete Information}, pages = {1--5}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2005}, volume = {5031}, editor = {Susanne Albers and Rolf H. M\"{o}hring and Georg Ch. Pflug and R\"{u}diger Schultz}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.05031.5}, URN = {urn:nbn:de:0030-drops-723}, doi = {10.4230/DagSemProc.05031.5}, annote = {Keywords: Algorithms, Approximation Algorithms, Optimization, Convex Optimization, Stochastic Optimization} }

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