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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)

One of the most basic graph problems, All-Pairs Shortest Paths (APSP) is known to be solvable in n^{3-o(1)} time, and it is widely open whether it has an O(n^{3-ε}) time algorithm for ε > 0. To better understand APSP, one often strives to obtain subcubic time algorithms for structured instances of APSP and problems equivalent to it, such as the Min-Plus matrix product.
A natural structured version of Min-Plus product is Monotone Min-Plus product which has been studied in the context of the Batch Range Mode [SODA'20] and Dynamic Range Mode [ICALP'20] problems. This paper improves the known algorithms for Monotone Min-Plus Product and for Batch and Dynamic Range Mode, and establishes a connection between Monotone Min-Plus Product and the Single Source Replacement Paths (SSRP) problem on an n-vertex graph with potentially negative edge weights in {-M, …, M}.
SSRP with positive integer edge weights bounded by M can be solved in Õ(Mn^ω) time, whereas the prior fastest algorithm for graphs with possibly negative weights [FOCS'12] runs in O(M^{0.7519} n^{2.5286}) time, the current best running time for directed APSP with small integer weights. Using Monotone Min-Plus Product, we obtain an improved O(M^{0.8043} n^{2.4957}) time SSRP algorithm, showing that SSRP with constant negative integer weights is likely easier than directed unweighted APSP, a problem that is believed to require n^{2.5-o(1)} time.
Complementing our algorithm for SSRP, we give a reduction from the Bounded-Difference Min-Plus Product problem studied by Bringmann et al. [FOCS'16] to negative weight SSRP. This reduction shows that it might be difficult to obtain an Õ(M n^{ω}) time algorithm for SSRP with negative weight edges, thus separating the problem from SSRP with only positive weight edges.

Yuzhou Gu, Adam Polak, Virginia Vassilevska Williams, and Yinzhan Xu. Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths. In 48th International Colloquium on Automata, Languages, and Programming (ICALP 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 198, pp. 75:1-75:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{gu_et_al:LIPIcs.ICALP.2021.75, author = {Gu, Yuzhou and Polak, Adam and Vassilevska Williams, Virginia and Xu, Yinzhan}, title = {{Faster Monotone Min-Plus Product, Range Mode, and Single Source Replacement Paths}}, booktitle = {48th International Colloquium on Automata, Languages, and Programming (ICALP 2021)}, pages = {75:1--75:20}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2021.75}, URN = {urn:nbn:de:0030-drops-141440}, doi = {10.4230/LIPIcs.ICALP.2021.75}, annote = {Keywords: APSP, Min-Plus Product, Range Mode, Single-Source Replacement Paths} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

We introduce a simple logical inference structure we call a spanoid (generalizing the notion of a matroid), which captures well-studied problems in several areas. These include combinatorial geometry (point-line incidences), algebra (arrangements of hypersurfaces and ideals), statistical physics (bootstrap percolation), network theory (gossip / infection processes) and coding theory. We initiate a thorough investigation of spanoids, from computational and structural viewpoints, focusing on parameters relevant to the applications areas above and, in particular, to questions regarding Locally Correctable Codes (LCCs).
One central parameter we study is the rank of a spanoid, extending the rank of a matroid and related to the dimension of codes. This leads to one main application of our work, establishing the first known barrier to improving the nearly 20-year old bound of Katz-Trevisan (KT) on the dimension of LCCs. On the one hand, we prove that the KT bound (and its more recent refinements) holds for the much more general setting of spanoid rank. On the other hand we show that there exist (random) spanoids whose rank matches these bounds. Thus, to significantly improve the known bounds one must step out of the spanoid framework.
Another parameter we explore is the functional rank of a spanoid, which captures the possibility of turning a given spanoid into an actual code. The question of the relationship between rank and functional rank is one of the main questions we raise as it may reveal new avenues for constructing new LCCs (perhaps even matching the KT bound). As a first step, we develop an entropy relaxation of functional rank to create a small constant gap and amplify it by tensoring to construct a spanoid whose functional rank is smaller than rank by a polynomial factor. This is evidence that the entropy method we develop can prove polynomially better bounds than KT-type methods on the dimension of LCCs.
To facilitate the above results we also develop some basic structural results on spanoids including an equivalent formulation of spanoids as set systems and properties of spanoid products. We feel that given these initial findings and their motivations, the abstract study of spanoids merits further investigation. We leave plenty of concrete open problems and directions.

Zeev Dvir, Sivakanth Gopi, Yuzhou Gu, and Avi Wigderson. Spanoids - An Abstraction of Spanning Structures, and a Barrier for LCCs. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{dvir_et_al:LIPIcs.ITCS.2019.32, author = {Dvir, Zeev and Gopi, Sivakanth and Gu, Yuzhou and Wigderson, Avi}, title = {{Spanoids - An Abstraction of Spanning Structures, and a Barrier for LCCs}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {32:1--32:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.32}, URN = {urn:nbn:de:0030-drops-101258}, doi = {10.4230/LIPIcs.ITCS.2019.32}, annote = {Keywords: Locally correctable codes, spanoids, entropy, bootstrap percolation, gossip spreading, matroid, union-closed family} }

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**Published in:** LIPIcs, Volume 101, 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)

Since the introduction of retroactive data structures at SODA 2004, a major unsolved problem has been to bound the gap between the best partially retroactive data structure (where changes can be made to the past, but only the present can be queried) and the best fully retroactive data structure (where the past can also be queried) for any problem. It was proved in 2004 that any partially retroactive data structure with operation time T_{op}(n,m) can be transformed into a fully retroactive data structure with operation time O(sqrt{m} * T_{op}(n,m)), where n is the size of the data structure and m is the number of operations in the timeline [Demaine et al., 2004]. But it has been open for 14 years whether such a gap is necessary.
In this paper, we prove nearly matching upper and lower bounds on this gap for all n and m. We improve the upper bound for n << sqrt m by showing a new transformation with multiplicative overhead n log m. We then prove a lower bound of min {n log m, sqrt m}^{1-o(1)} assuming any of the following conjectures:
- Conjecture I: Circuit SAT requires 2^{n - o(n)} time on n-input circuits of size 2^{o(n)}. This conjecture is far weaker than the well-believed SETH conjecture from complexity theory, which asserts that CNF SAT with n variables and O(n) clauses already requires 2^{n-o(n)} time.
- Conjecture II: Online (min,+) product between an integer n x n matrix and n vectors requires n^{3 - o(1)} time. This conjecture is weaker than the APSP conjectures widely used in fine-grained complexity.
- Conjecture III (3-SUM Conjecture): Given three sets A,B,C of integers, each of size n, deciding whether there exist a in A, b in B, c in C such that a + b + c = 0 requires n^{2 - o(1)} time. This 1995 conjecture [Anka Gajentaan and Mark H. Overmars, 1995] was the first conjecture in fine-grained complexity.
Our lower bound construction illustrates an interesting power of fully retroactive queries: they can be used to quickly solve batched pair evaluation. We believe this technique can prove useful for other data structure lower bounds, especially dynamic ones.

Lijie Chen, Erik D. Demaine, Yuzhou Gu, Virginia Vassilevska Williams, Yinzhan Xu, and Yuancheng Yu. Nearly Optimal Separation Between Partially and Fully Retroactive Data Structures. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 33:1-33:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{chen_et_al:LIPIcs.SWAT.2018.33, author = {Chen, Lijie and Demaine, Erik D. and Gu, Yuzhou and Williams, Virginia Vassilevska and Xu, Yinzhan and Yu, Yuancheng}, title = {{Nearly Optimal Separation Between Partially and Fully Retroactive Data Structures}}, booktitle = {16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018)}, pages = {33:1--33:12}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-068-2}, ISSN = {1868-8969}, year = {2018}, volume = {101}, editor = {Eppstein, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.SWAT.2018.33}, URN = {urn:nbn:de:0030-drops-88593}, doi = {10.4230/LIPIcs.SWAT.2018.33}, annote = {Keywords: retroactive data structure, conditional lower bound} }

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