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**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

Inspired by applications on search engines and web servers, we consider a load balancing problem with a general convex objective function. In this problem, we are given a bipartite graph on a set of sources S and a set of workers W and the goal is to distribute the load from each source among its neighboring workers such that the total load of workers are as balanced as possible. We present a new distributed algorithm that works with any symmetric non-decreasing convex function for evaluating the balancedness of the workers' load. Our algorithm computes a nearly optimal allocation of loads in O(log n log² d/ε³) rounds where n is the number of nodes, d is the maximum degree, and ε is the desired precision. If the objective is to minimize the maximum load, we modify the algorithm to obtain a nearly optimal solution in O(log n log d/ε²) rounds. This improves a line of algorithms that require a polynomial number of rounds in n and d and appear to encounter a fundamental barrier that prevents them from obtaining poly-logarithmic runtime [Berenbrink et al., 2005; Berenbrink et al., 2009; Subramanian and Scherson, 1994; Rabani et al., 1998]. In our paper, we introduce a novel primal-dual approach with multiplicative weight updates that allows us to circumvent this barrier. Our algorithm is inspired by [Agrawal et al., 2018] and other distributed algorithms for optimizing linear objectives but introduces several new twists to deal with general convex objectives.

Sara Ahmadian, Allen Liu, Binghui Peng, and Morteza Zadimoghaddam. Distributed Load Balancing: A New Framework and Improved Guarantees. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 79:1-79:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ahmadian_et_al:LIPIcs.ITCS.2021.79, author = {Ahmadian, Sara and Liu, Allen and Peng, Binghui and Zadimoghaddam, Morteza}, title = {{Distributed Load Balancing: A New Framework and Improved Guarantees}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {79:1--79:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.79}, URN = {urn:nbn:de:0030-drops-136186}, doi = {10.4230/LIPIcs.ITCS.2021.79}, annote = {Keywords: Load balancing, Distributed 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 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We pursue a study of the Generalized Demand Matching problem, a common generalization of the b-Matching and Knapsack problems. Here, we are given a graph with vertex capacities, edge profits, and asymmetric demands on the edges. The goal is to find a maximum-profit subset of edges so the demands of chosen edges do not violate the vertex capacities. This problem is APX-hard and constant-factor approximations are already known.
Our main results fall into two categories. First, using iterated relaxation and various filtering strategies, we show with an efficient rounding algorithm that if an additional matroid structure M is given and we further only allow sets that are independent in M, the natural LP relaxation has an integrality gap of at most 25/3. This can be further improved
in various special cases, for example we improve over the 15-approximation for the previously- studied Coupled Placement problem [Korupolu et al. 2014] by giving a 7-approximation.
Using similar techniques, we show the problem of computing a minimum-cost base in M satisfying vertex capacities admits a (1,3)-bicriteria approximation: the cost is at most the optimum and the capacities are violated by a factor of at most 3. This improves over the previous (1,4)-approximation in the special case that M is the graphic matroid over the given graph [Fukanaga and Nagamochi, 2009].
Second, we show Demand Matching admits a polynomial-time approximation scheme in graphs that exclude a fixed minor. If all demands are polynomially-bounded integers, this is somewhat easy using dynamic programming in bounded-treewidth graphs. Our main technical contribution is a sparsification lemma that allows us to scale the demands of some items to be used in a more intricate dynamic programming algorithm, followed by some randomized rounding to filter our scaled-demand solution to one whose original demands satisfy all constraints.

Sara Ahmadian and Zachary Friggstad. Further Approximations for Demand Matching: Matroid Constraints and Minor-Closed Graphs. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 55:1-55:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{ahmadian_et_al:LIPIcs.ICALP.2017.55, author = {Ahmadian, Sara and Friggstad, Zachary}, title = {{Further Approximations for Demand Matching: Matroid Constraints and Minor-Closed Graphs}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {55:1--55:13}, 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.2017.55}, URN = {urn:nbn:de:0030-drops-74600}, doi = {10.4230/LIPIcs.ICALP.2017.55}, annote = {Keywords: Approximation Algorithms, Column-Restricted Packing, Demand Matching, Matroids, Planar Graphs} }

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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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