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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

We study the problem of maximizing a non-monotone submodular function subject to a cardinality constraint in the streaming model. Our main contributions are two single-pass (semi-)streaming algorithms that use Õ(k)⋅poly(1/ε) memory, where k is the size constraint. At the end of the stream, both our algorithms post-process their data structures using any offline algorithm for submodular maximization, and obtain a solution whose approximation guarantee is α/(1+α)-ε, where α is the approximation of the offline algorithm. If we use an exact (exponential time) post-processing algorithm, this leads to 1/2-ε approximation (which is nearly optimal). If we post-process with the algorithm of [Niv Buchbinder and Moran Feldman, 2019], that achieves the state-of-the-art offline approximation guarantee of α = 0.385, we obtain 0.2779-approximation in polynomial time, improving over the previously best polynomial-time approximation of 0.1715 due to [Feldman et al., 2018]. One of our algorithms is combinatorial and enjoys fast update and overall running times. Our other algorithm is based on the multilinear extension, enjoys an improved space complexity, and can be made deterministic in some settings of interest.

Naor Alaluf, Alina Ene, Moran Feldman, Huy L. Nguyen, and Andrew Suh. Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 6:1-6:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{alaluf_et_al:LIPIcs.ICALP.2020.6, author = {Alaluf, Naor and Ene, Alina and Feldman, Moran and Nguyen, Huy L. and Suh, Andrew}, title = {{Optimal Streaming Algorithms for Submodular Maximization with Cardinality Constraints}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {6:1--6:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.6}, URN = {urn:nbn:de:0030-drops-124137}, doi = {10.4230/LIPIcs.ICALP.2020.6}, annote = {Keywords: Submodular maximization, streaming algorithms, cardinality constraint} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider the problem of maximizing a monotone submodular function subject to a knapsack constraint. Our main contribution is an algorithm that achieves a nearly-optimal, 1 - 1/e - epsilon approximation, using (1/epsilon)^{O(1/epsilon^4)} n log^2{n} function evaluations and arithmetic operations. Our algorithm is impractical but theoretically interesting, since it overcomes a fundamental running time bottleneck of the multilinear extension relaxation framework. This is the main approach for obtaining nearly-optimal approximation guarantees for important classes of constraints but it leads to Omega(n^2) running times, since evaluating the multilinear extension is expensive. Our algorithm maintains a fractional solution with only a constant number of entries that are strictly fractional, which allows us to overcome this obstacle.

Alina Ene and Huy L. Nguyen. A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 53:1-53:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ene_et_al:LIPIcs.ICALP.2019.53, author = {Ene, Alina and Nguyen, Huy L.}, title = {{A Nearly-Linear Time Algorithm for Submodular Maximization with a Knapsack Constraint}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {53:1--53:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.53}, URN = {urn:nbn:de:0030-drops-106290}, doi = {10.4230/LIPIcs.ICALP.2019.53}, annote = {Keywords: submodular maximization, knapsack constraint, fast algorithms} }

Document

Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 132, 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)

We consider fast algorithms for monotone submodular maximization subject to a matroid constraint. We assume that the matroid is given as input in an explicit form, and the goal is to obtain the best possible running times for important matroids. We develop a new algorithm for a general matroid constraint with a 1 - 1/e - epsilon approximation that achieves a fast running time provided we have a fast data structure for maintaining an approximately maximum weight base in the matroid through a sequence of decrease weight operations. We construct such data structures for graphic matroids and partition matroids, and we obtain the first algorithms for these classes of matroids that achieve a nearly-optimal, 1 - 1/e - epsilon approximation, using a nearly-linear number of function evaluations and arithmetic operations.

Alina Ene and Huy L. Nguyen. Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid Constraint. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 54:1-54:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{ene_et_al:LIPIcs.ICALP.2019.54, author = {Ene, Alina and Nguyen, Huy L.}, title = {{Towards Nearly-Linear Time Algorithms for Submodular Maximization with a Matroid Constraint}}, booktitle = {46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)}, pages = {54:1--54:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-109-2}, ISSN = {1868-8969}, year = {2019}, volume = {132}, editor = {Baier, Christel and Chatzigiannakis, Ioannis and Flocchini, Paola and Leonardi, Stefano}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2019.54}, URN = {urn:nbn:de:0030-drops-106303}, doi = {10.4230/LIPIcs.ICALP.2019.54}, annote = {Keywords: submodular maximization, matroid constraints, fast running times} }

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**Published in:** LIPIcs, Volume 93, 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)

This paper studies the stochastic variant of the classical k-TSP problem where rewards at the vertices are independent random variables which are instantiated upon the tour's visit. The objective is to minimize the expected length of a tour that collects reward at least k. The solution is a policy describing the tour which may (adaptive) or may not (non-adaptive) depend on the observed rewards.
Our work presents an adaptive O(log k)-approximation algorithm for Stochastic k-TSP, along with a non-adaptive O(log^2 k)-approximation algorithm which also upper bounds the adaptivity gap by O(log^2 k). We also show that the adaptivity gap of Stochastic k-TSP is at least e, even in the special case of stochastic knapsack cover.

Alina Ene, Viswanath Nagarajan, and Rishi Saket. Approximation Algorithms for Stochastic k-TSP. In 37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 93, pp. 27:1-27:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{ene_et_al:LIPIcs.FSTTCS.2017.27, author = {Ene, Alina and Nagarajan, Viswanath and Saket, Rishi}, title = {{Approximation Algorithms for Stochastic k-TSP}}, booktitle = {37th IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2017)}, pages = {27:1--27:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-055-2}, ISSN = {1868-8969}, year = {2018}, volume = {93}, editor = {Lokam, Satya and Ramanujam, R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2017.27}, URN = {urn:nbn:de:0030-drops-83910}, doi = {10.4230/LIPIcs.FSTTCS.2017.27}, annote = {Keywords: Stochastic TSP, algorithms, approximation, adaptivity gap} }

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

We study the problem of routing symmetric demand pairs in planar digraphs. The input consists of a directed planar graph G = (V, E) and a collection of k source-destination pairs M = {s_1t_1, ..., s_kt_k}. The goal is to maximize the number of pairs that are routed along disjoint paths. A pair s_it_i is routed in the symmetric setting if there is a directed path connecting s_i to t_i and a directed path connecting t_i to s_i. In this paper we obtain a randomized poly-logarithmic approximation with constant congestion for this problem in planar digraphs. The main technical contribution is to show that a planar digraph with directed treewidth h contains a constant congestion crossbar of size Omega(h/polylog(h)).

Chandra Chekuri, Alina Ene, and Marcin Pilipczuk. Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs. In 43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 55, pp. 7:1-7:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{chekuri_et_al:LIPIcs.ICALP.2016.7, author = {Chekuri, Chandra and Ene, Alina and Pilipczuk, Marcin}, title = {{Constant Congestion Routing of Symmetric Demands in Planar Directed Graphs}}, booktitle = {43rd International Colloquium on Automata, Languages, and Programming (ICALP 2016)}, pages = {7:1--7:14}, 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.2016.7}, URN = {urn:nbn:de:0030-drops-62737}, doi = {10.4230/LIPIcs.ICALP.2016.7}, annote = {Keywords: Disjoint paths, symmetric demands, planar directed graph} }

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**Published in:** LIPIcs, Volume 53, 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)

We study the problem of routing on disjoint paths in bounded treewidth graphs with both edge and node capacities. The input consists of a capacitated graph G and a collection of k source-destination pairs M = (s_1, t_1), ..., (s_k, t_k). The goal is to maximize the number of pairs that can be routed subject to the capacities in the graph. A routing of a subset M' of the pairs is a collection P of paths such that, for each pair (s_i, t_i) in M', there is a path in P connecting s_i to t_i. In the Maximum Edge Disjoint Paths (MaxEDP) problem, the graph G has capacities cap(e) on the edges and a routing P is feasible if each edge e is in at most cap(e) of the paths of P. The Maximum Node Disjoint Paths (MaxNDP) problem is the node-capacitated counterpart of MaxEDP.
In this paper we obtain an O(r^3) approximation for MaxEDP on graphs of treewidth at most r and a matching approximation for MaxNDP on graphs of pathwidth at most r. Our results build on and significantly improve the work by Chekuri et al. [ICALP 2013] who obtained an O(r * 3^r) approximation for MaxEDP.

Alina Ene, Matthias Mnich, Marcin Pilipczuk, and Andrej Risteski. On Routing Disjoint Paths in Bounded Treewidth Graphs. In 15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 53, pp. 15:1-15:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{ene_et_al:LIPIcs.SWAT.2016.15, author = {Ene, Alina and Mnich, Matthias and Pilipczuk, Marcin and Risteski, Andrej}, title = {{On Routing Disjoint Paths in Bounded Treewidth Graphs}}, booktitle = {15th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2016)}, pages = {15:1--15:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-011-8}, ISSN = {1868-8969}, year = {2016}, volume = {53}, editor = {Pagh, Rasmus}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2016.15}, URN = {urn:nbn:de:0030-drops-60378}, doi = {10.4230/LIPIcs.SWAT.2016.15}, annote = {Keywords: Algorithms and data structures, disjoint paths, treewidth} }

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

We consider the Minimum Submodular Cost Allocation (MSCA) problem.
In this problem, we are given k submodular cost functions f_1, ... ,
f_k: 2^V -> R_+ and the goal is to partition V into k sets A_1, ...,
A_k so as to minimize the total cost sum_{i = 1}^k f_i(A_i). We show
that MSCA is inapproximable within any multiplicative factor even in
very restricted settings; prior to our work, only Set Cover hardness
was known. In light of this negative result, we turn our attention
to special cases of the problem. We consider the setting in which
each function f_i satisfies f_i = g_i + h, where each g_i is monotone
submodular and h is (possibly non-monotone) submodular. We give an
O(k log |V|) approximation for this problem. We provide some evidence
that a factor of k may be necessary, even in the special case of
HyperLabel. In particular, we formulate a simplex-coloring
conjecture that implies a Unique-Games-hardness of (k - 1 - epsilon)
for k-uniform HyperLabel and label set [k]. We provide a proof of the
simplex-coloring conjecture for k=3.

Alina Ene and Jan Vondrák. Hardness of Submodular Cost Allocation: Lattice Matching and a Simplex Coloring Conjecture. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014). Leibniz International Proceedings in Informatics (LIPIcs), Volume 28, pp. 144-159, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2014)

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@InProceedings{ene_et_al:LIPIcs.APPROX-RANDOM.2014.144, author = {Ene, Alina and Vondr\'{a}k, Jan}, title = {{Hardness of Submodular Cost Allocation: Lattice Matching and a Simplex Coloring Conjecture}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2014)}, pages = {144--159}, 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.2014.144}, URN = {urn:nbn:de:0030-drops-46943}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2014.144}, annote = {Keywords: Minimum Cost Submodular Allocation, Submodular Optimization, Hypergraph Labeling} }