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Documents authored by Zhang, Jialin

Document
DOI: 10.4230/LIPIcs.ESA.2023.98
Simple Deterministic Approximation for Submodular Multiple Knapsack Problem

Authors: Xiaoming Sun, Jialin Zhang, and Zhijie Zhang

Published in: LIPIcs, Volume 274, 31st Annual European Symposium on Algorithms (ESA 2023)

Abstract
Submodular maximization has been a central topic in theoretical computer science and combinatorial optimization over the last decades. Plenty of well-performed approximation algorithms have been designed for the problem over a variety of constraints. In this paper, we consider the submodular multiple knapsack problem (SMKP). In SMKP, the profits of each subset of elements are specified by a monotone submodular function. The goal is to find a feasible packing of elements over multiple bins (knapsacks) to maximize the profit. Recently, Fairstein et al. [ESA20] proposed a nearly optimal (1-e^{-1}-ε)-approximation algorithm for SMKP. Their algorithm is obtained by combining configuration LP, a grouping technique for bin packing, and the continuous greedy algorithm for submodular maximization. As a result, the algorithm is somewhat sophisticated and inherently randomized. In this paper, we present an arguably simple deterministic combinatorial algorithm for SMKP, which achieves a (1-e^{-1}-ε)-approximation ratio. Our algorithm is based on very different ideas compared with Fairstein et al. [ESA20].
Cite as

Xiaoming Sun, Jialin Zhang, and Zhijie Zhang. Simple Deterministic Approximation for Submodular Multiple Knapsack Problem. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 98:1-98:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{sun_et_al:LIPIcs.ESA.2023.98,
  author =	{Sun, Xiaoming and Zhang, Jialin and Zhang, Zhijie},
  title =	{{Simple Deterministic Approximation for Submodular Multiple Knapsack Problem}},
  booktitle =	{31st Annual European Symposium on Algorithms (ESA 2023)},
  pages =	{98:1--98:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-295-2},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{274},
  editor =	{G{\o}rtz, Inge Li and Farach-Colton, Martin and Puglisi, Simon J. 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.2023.98},
  URN =		{urn:nbn:de:0030-drops-187517},
  doi =		{10.4230/LIPIcs.ESA.2023.98},
  annote =	{Keywords: Submodular maximization, knapsack problem, deterministic algorithm}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2019.94
Querying a Matrix Through Matrix-Vector Products

Authors: Xiaoming Sun, David P. Woodruff, Guang Yang, and Jialin Zhang

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

Abstract
We consider algorithms with access to an unknown matrix M in F^{n x d} via matrix-vector products, namely, the algorithm chooses vectors v^1, ..., v^q, and observes Mv^1, ..., Mv^q. Here the v^i can be randomized as well as chosen adaptively as a function of Mv^1, ..., Mv^{i-1}. Motivated by applications of sketching in distributed computation, linear algebra, and streaming models, as well as connections to areas such as communication complexity and property testing, we initiate the study of the number q of queries needed to solve various fundamental problems. We study problems in three broad categories, including linear algebra, statistics problems, and graph problems. For example, we consider the number of queries required to approximate the rank, trace, maximum eigenvalue, and norms of a matrix M; to compute the AND/OR/Parity of each column or row of M, to decide whether there are identical columns or rows in M or whether M is symmetric, diagonal, or unitary; or to compute whether a graph defined by M is connected or triangle-free. We also show separations for algorithms that are allowed to obtain matrix-vector products only by querying vectors on the right, versus algorithms that can query vectors on both the left and the right. We also show separations depending on the underlying field the matrix-vector product occurs in. For graph problems, we show separations depending on the form of the matrix (bipartite adjacency versus signed edge-vertex incidence matrix) to represent the graph. Surprisingly, this fundamental model does not appear to have been studied on its own, and we believe a thorough investigation of problems in this model would be beneficial to a number of different application areas.
Cite as

Xiaoming Sun, David P. Woodruff, Guang Yang, and Jialin Zhang. Querying a Matrix Through Matrix-Vector Products. In 46th International Colloquium on Automata, Languages, and Programming (ICALP 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 132, pp. 94:1-94:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{sun_et_al:LIPIcs.ICALP.2019.94,
  author =	{Sun, Xiaoming and Woodruff, David P. and Yang, Guang and Zhang, Jialin},
  title =	{{Querying a Matrix Through Matrix-Vector Products}},
  booktitle =	{46th International Colloquium on Automata, Languages, and Programming (ICALP 2019)},
  pages =	{94:1--94:16},
  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.2019.94},
  URN =		{urn:nbn:de:0030-drops-106709},
  doi =		{10.4230/LIPIcs.ICALP.2019.94},
  annote =	{Keywords: Communication complexity, linear algebra, sketching}
}
Document
DOI: 10.4230/LIPIcs.ISAAC.2016.51
On the Optimality of Tape Merge of Two Lists with Similar Size

Authors: Qian Li, Xiaoming Sun, and Jialin Zhang

Published in: LIPIcs, Volume 64, 27th International Symposium on Algorithms and Computation (ISAAC 2016)

Abstract
The problem of merging sorted lists in the least number of pairwise comparisons has been solved completely only for a few special cases. Graham and Karp [TAOCP, 1999] independently discovered that the tape merge algorithm is optimal in the worst case when the two lists have the same size. Stockmeyer and Yao [SICOMP, 1980], Murphy and Paull [Inform. Control, 1979], and Christen [1978] independently showed when the lists to be merged are of size m and n satisfying m leq n leq floor(3/2 m) + 1, the tape merge algorithm is optimal in the worst case. This paper extends this result by showing that the tape merge algorithm is optimal in the worst case whenever the size of one list is no larger than 1.52 times the size of the other. The main tool we used to prove lower bounds is Knuth’s adversary methods [TAOCP, 1999]. In addition, we show that the lower bound cannot be improved to 1.8 via Knuth's adversary methods. We also develop a new inequality about Knuth's adversary methods, which might be interesting in its own right. Moreover, we design a simple procedure to achieve constant improvement of the upper bounds for 2m - 2 leq n leq 3m.
Cite as

Qian Li, Xiaoming Sun, and Jialin Zhang. On the Optimality of Tape Merge of Two Lists with Similar Size. In 27th International Symposium on Algorithms and Computation (ISAAC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 64, pp. 51:1-51:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{li_et_al:LIPIcs.ISAAC.2016.51,
  author =	{Li, Qian and Sun, Xiaoming and Zhang, Jialin},
  title =	{{On the Optimality of Tape Merge of Two Lists with Similar Size}},
  booktitle =	{27th International Symposium on Algorithms and Computation (ISAAC 2016)},
  pages =	{51:1--51:17},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-026-2},
  ISSN =	{1868-8969},
  year =	{2016},
  volume =	{64},
  editor =	{Hong, Seok-Hee},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2016.51},
  URN =		{urn:nbn:de:0030-drops-68219},
  doi =		{10.4230/LIPIcs.ISAAC.2016.51},
  annote =	{Keywords: comparison-based sorting, tape merge, optimal sort, adversary method}
}
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