10 Search Results for "An, Haozhe"

Document
DOI: 10.4230/LIPIcs.ITCS.2026.102
Smoothed Analysis of Dynamic Graph Algorithms

Authors: Uri Meir and Ami Paz

Published in: LIPIcs, Volume 362, 17th Innovations in Theoretical Computer Science Conference (ITCS 2026)

Abstract
Recent years have seen significant progress in the study of dynamic graph algorithms, and most notably, the introduction of strong lower bound techniques for them (e.g., Henzinger, Krinninger, Nanongkai and Saranurak, STOC 2015; Larsen and Yu, FOCS 2023). As worst-case analysis (adversarial inputs) may lead to the necessity of high running times, a natural question arises: in which cases are high running times really necessary, and in which cases these inputs merely manifest unique pathological cases? Early attempts to tackle this question were made by Nikoletseas, Reif, Spirakis and Yung (ICALP 1995) and by Alberts and Henzinger (Algorithmica 1998), who considered models with very little adversarial control over the inputs, and showed fast algorithms exist for them. The question was then overlooked for decades, until Henzinger, Lincoln and Saha (SODA 2022) recently addressed uniformly random inputs, and presented algorithms and impossibility results for several subgraph counting problems. To tackle the above question more thoroughly, we employ smoothed analysis, a celebrated framework introduced by Spielman and Teng (J. ACM, 2004). An input is proposed by an adversary but then a noisy version of it is processed by the algorithm instead. This model of inputs is parameterized by the amount of adversarial control, and fully interpolates between worst-case inputs and a uniformly random input. Doing so, we extend impossibility results for some problems to the smoothed model with only a minor quantitative loss. That is, we show that partially-adversarial inputs suffice to impose high running times for certain problems. In contrast, we show that other problems become easy even with the slightest amount of noise. In addition, we study the interplay between the adversary and the noise, leading to three natural models of smoothed inputs, for which we show a hierarchy of increasing difficulty stretching between the average-case and the worst-case complexities.
Cite as

Uri Meir and Ami Paz. Smoothed Analysis of Dynamic Graph Algorithms. In 17th Innovations in Theoretical Computer Science Conference (ITCS 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 362, pp. 102:1-102:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026)

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@InProceedings{meir_et_al:LIPIcs.ITCS.2026.102,
  author =	{Meir, Uri and Paz, Ami},
  title =	{{Smoothed Analysis of Dynamic Graph Algorithms}},
  booktitle =	{17th Innovations in Theoretical Computer Science Conference (ITCS 2026)},
  pages =	{102:1--102:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-410-9},
  ISSN =	{1868-8969},
  year =	{2026},
  volume =	{362},
  editor =	{Saraf, Shubhangi},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2026.102},
  URN =		{urn:nbn:de:0030-drops-253896},
  doi =		{10.4230/LIPIcs.ITCS.2026.102},
  annote =	{Keywords: Dynamic graph algorithms, Smoothed analysis, Shortest paths}
}
Document
Track A: Algorithms, Complexity and Games
DOI: 10.4230/LIPIcs.ICALP.2025.78
The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs

Authors: Nick Fischer, Marvin Künnemann, Mirza Redžić, and Julian Stieß

Published in: LIPIcs, Volume 334, 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)

Abstract
Is detecting a k-clique in k-partite regular (hyper-)graphs as hard as in the general case? Intuition suggests yes, but proving this - especially for hypergraphs - poses notable challenges. Concretely, we consider a strong notion of regularity in h-uniform hypergraphs, where we essentially require that any subset of at most h-1 is incident to a uniform number of hyperedges. Such notions are studied intensively in the combinatorial block design literature. We show that any f(k)n^{g(k)}-time algorithm for detecting k-cliques in such graphs transfers to an f'(k)n^{g(k)}-time algorithm for the general case, establishing a fine-grained equivalence between the h-uniform hyperclique hypothesis and its natural regular analogue. Equipped with this regularization result, we then fully resolve the fine-grained complexity of optimizing Boolean constraint satisfaction problems over assignments with k non-zeros. Our characterization depends on the maximum degree d of a constraint function. Specifically, if d ≤ 1, we obtain a linear-time solvable problem, if d = 2, the time complexity is essentially equivalent to k-clique detection, and if d ≥ 3 the problem requires exhaustive-search time under the 3-uniform hyperclique hypothesis. To obtain our hardness results, the regularization result plays a crucial role, enabling a very convenient approach when applied carefully. We believe that our regularization result will find further applications in the future.
Cite as

Nick Fischer, Marvin Künnemann, Mirza Redžić, and Julian Stieß. The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs. In 52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 334, pp. 78:1-78:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{fischer_et_al:LIPIcs.ICALP.2025.78,
  author =	{Fischer, Nick and K\"{u}nnemann, Marvin and Red\v{z}i\'{c}, Mirza and Stie{\ss}, Julian},
  title =	{{The Role of Regularity in (Hyper-)Clique Detection and Implications for Optimizing Boolean CSPs}},
  booktitle =	{52nd International Colloquium on Automata, Languages, and Programming (ICALP 2025)},
  pages =	{78:1--78:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-372-0},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{334},
  editor =	{Censor-Hillel, Keren and Grandoni, Fabrizio and Ouaknine, Jo\"{e}l and Puppis, Gabriele},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2025.78},
  URN =		{urn:nbn:de:0030-drops-234559},
  doi =		{10.4230/LIPIcs.ICALP.2025.78},
  annote =	{Keywords: fine-grained complexity theory, clique detections in hypergraphs, constraint satisfaction, parameterized algorithms}
}
Document
Invited Talk
DOI: 10.4230/LIPIcs.ICDT.2025.1
The Quest for Faster Join Algorithms (Invited Talk)

Authors: Paraschos Koutris, Shaleen Deep, Austen Fan, and Hangdong Zhao

Published in: LIPIcs, Volume 328, 28th International Conference on Database Theory (ICDT 2025)

Abstract
Joins are the cornerstone of relational databases. Surprisingly, even after several decades of research in the systems and theory database community, we still lack an understanding of how to design the fastest possible join algorithm. In this talk, we will present the exciting progress the database theory community has achieved in join algorithms over the last two decades. The talk will revolve around five key ideas fundamentally shaping this research area: tree decompositions, data partitioning, leveraging statistical information, enumeration, and algebraic techniques.
Cite as

Paraschos Koutris, Shaleen Deep, Austen Fan, and Hangdong Zhao. The Quest for Faster Join Algorithms (Invited Talk). In 28th International Conference on Database Theory (ICDT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 328, pp. 1:1-1:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{koutris_et_al:LIPIcs.ICDT.2025.1,
  author =	{Koutris, Paraschos and Deep, Shaleen and Fan, Austen and Zhao, Hangdong},
  title =	{{The Quest for Faster Join Algorithms}},
  booktitle =	{28th International Conference on Database Theory (ICDT 2025)},
  pages =	{1:1--1:12},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-364-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{328},
  editor =	{Roy, Sudeepa and Kara, Ahmet},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2025.1},
  URN =		{urn:nbn:de:0030-drops-229428},
  doi =		{10.4230/LIPIcs.ICDT.2025.1},
  annote =	{Keywords: Conjunctive Queries, Joins, Tree Decompositions, Enumeration, Semirings}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2025.20
A Formal Language Perspective on Factorized Representations

Authors: Benny Kimelfeld, Wim Martens, and Matthias Niewerth

Published in: LIPIcs, Volume 328, 28th International Conference on Database Theory (ICDT 2025)

Abstract
Factorized representations (FRs) are a well-known tool to succinctly represent results of join queries and have been originally defined using the named database perspective. We define FRs in the unnamed database perspective and use them to establish several new connections. First, unnamed FRs can be exponentially more succinct than named FRs, but this difference can be alleviated by imposing a disjointness condition on columns. Conversely, named FRs can also be exponentially more succinct than unnamed FRs. Second, unnamed FRs are the same as (i.e., isomorphic to) context-free grammars for languages in which each word has the same length. This tight connection allows us to transfer a wide range of results on context-free grammars to database factorization; of which we offer a selection in the paper. Third, when we generalize unnamed FRs to arbitrary sets of tuples, they become a generalization of path multiset representations, a formalism that was recently introduced to succinctly represent sets of paths in the context of graph database query evaluation.
Cite as

Benny Kimelfeld, Wim Martens, and Matthias Niewerth. A Formal Language Perspective on Factorized Representations. In 28th International Conference on Database Theory (ICDT 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 328, pp. 20:1-20:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{kimelfeld_et_al:LIPIcs.ICDT.2025.20,
  author =	{Kimelfeld, Benny and Martens, Wim and Niewerth, Matthias},
  title =	{{A Formal Language Perspective on Factorized Representations}},
  booktitle =	{28th International Conference on Database Theory (ICDT 2025)},
  pages =	{20:1--20:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-364-5},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{328},
  editor =	{Roy, Sudeepa and Kara, Ahmet},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2025.20},
  URN =		{urn:nbn:de:0030-drops-229614},
  doi =		{10.4230/LIPIcs.ICDT.2025.20},
  annote =	{Keywords: Databases, relational databases, graph databases, factorized databases, regular path queries, compact representations}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.55
Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Linear Integer Arithmetic

Authors: Geri Gokaj and Marvin Künnemann

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
In the last three decades, the k-SUM hypothesis has emerged as a satisfying explanation of long-standing time barriers for a variety of algorithmic problems. Yet to this day, the literature knows of only few proven consequences of a refutation of this hypothesis. Taking a descriptive complexity viewpoint, we ask: What is the largest logically defined class of problems captured by the k-SUM problem? To this end, we introduce a class FOP_ℤ of problems corresponding to deciding sentences in Presburger arithmetic/linear integer arithmetic over finite subsets of integers. We establish two large fragments for which the k-SUM problem is complete under fine-grained reductions: 1) The k-SUM problem is complete for deciding the sentences with k existential quantifiers. 2) The 3-SUM problem is complete for all 3-quantifier sentences of FOP_ℤ expressible using at most 3 linear inequalities. Specifically, a faster-than-n^{⌈k/2⌉ ± o(1)} algorithm for k-SUM (or faster-than-n^{2 ± o(1)} algorithm for 3-SUM, respectively) directly translate to polynomial speedups of a general algorithm for all sentences in the respective fragment. Observing a barrier for proving completeness of 3-SUM for the entire class FOP_ℤ, we turn to the question which other - seemingly more general - problems are complete for FOP_ℤ. In this direction, we establish FOP_ℤ-completeness of the problem pair of Pareto Sum Verification and Hausdorff Distance under n Translations under the L_∞/L₁ norm in ℤ^d. In particular, our results invite to investigate Pareto Sum Verification as a high-dimensional generalization of 3-SUM.
Cite as

Geri Gokaj and Marvin Künnemann. Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Linear Integer Arithmetic. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 55:1-55:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gokaj_et_al:LIPIcs.ITCS.2025.55,
  author =	{Gokaj, Geri and K\"{u}nnemann, Marvin},
  title =	{{Completeness Theorems for k-SUM and Geometric Friends: Deciding Fragments of Linear Integer Arithmetic}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{55:1--55:25},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.55},
  URN =		{urn:nbn:de:0030-drops-226835},
  doi =		{10.4230/LIPIcs.ITCS.2025.55},
  annote =	{Keywords: fine-grained complexity theory, descriptive complexity, presburger arithmetic, completeness results, k-SUM}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2025.58
The Computational Complexity of Factored Graphs

Authors: Shreya Gupta, Boyang Huang, Russell Impagliazzo, Stanley Woo, and Christopher Ye

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
While graphs and abstract data structures can be large and complex, practical instances are often regular or highly structured. If the instance has sufficient structure, we might hope to compress the object into a more succinct representation. An efficient algorithm (with respect to the compressed input size) could then lead to more efficient computations than algorithms taking the explicit, uncompressed object as input. This leads to a natural question: when does knowing the input instance has a more succinct representation make computation easier? We initiate the study of the computational complexity of problems on factored graphs: graphs that are given as a formula of products and unions on smaller graphs. For any graph problem, we define a parameterized version that takes factored graphs as input, parameterized by the number of (smaller) ordinary graphs used to construct the factored graph. In this setting, we characterize the parameterized complexity of several natural graph problems, exhibiting a variety of complexities. We show that a decision version of lexicographically first maximal independent set is XP-complete, and therefore unconditionally not fixed-parameter tractable (FPT). On the other hand, we show that clique counting is FPT. Finally, we show that reachability is XNL-complete. Moreover, XNL is contained in FPT if and only if NL is contained in some fixed polynomial time.
Cite as

Shreya Gupta, Boyang Huang, Russell Impagliazzo, Stanley Woo, and Christopher Ye. The Computational Complexity of Factored Graphs. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 58:1-58:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{gupta_et_al:LIPIcs.ITCS.2025.58,
  author =	{Gupta, Shreya and Huang, Boyang and Impagliazzo, Russell and Woo, Stanley and Ye, Christopher},
  title =	{{The Computational Complexity of Factored Graphs}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{58:1--58:19},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.58},
  URN =		{urn:nbn:de:0030-drops-226865},
  doi =		{10.4230/LIPIcs.ITCS.2025.58},
  annote =	{Keywords: Parameterized Complexity, Fine-grained complexity, Fixed-parameter tractability, Graph algorithms}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2023.17
Conjunctive Queries with Free Access Patterns Under Updates

Authors: Ahmet Kara, Milos Nikolic, Dan Olteanu, and Haozhe Zhang

Published in: LIPIcs, Volume 255, 26th International Conference on Database Theory (ICDT 2023)

Abstract
We study the problem of answering conjunctive queries with free access patterns under updates. A free access pattern is a partition of the free variables of the query into input and output. The query returns tuples over the output variables given a tuple of values over the input variables. We introduce a fully dynamic evaluation approach for such queries. We also give a syntactic characterisation of those queries that admit constant time per single-tuple update and whose output tuples can be enumerated with constant delay given an input tuple. Finally, we chart the complexity trade-off between the preprocessing time, update time and enumeration delay for such queries. For a class of queries, our approach achieves optimal, albeit non-constant, update time and delay. Their optimality is predicated on the Online Matrix-Vector Multiplication conjecture. Our results recover prior work on the dynamic evaluation of conjunctive queries without access patterns.
Cite as

Ahmet Kara, Milos Nikolic, Dan Olteanu, and Haozhe Zhang. Conjunctive Queries with Free Access Patterns Under Updates. In 26th International Conference on Database Theory (ICDT 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 255, pp. 17:1-17:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kara_et_al:LIPIcs.ICDT.2023.17,
  author =	{Kara, Ahmet and Nikolic, Milos and Olteanu, Dan and Zhang, Haozhe},
  title =	{{Conjunctive Queries with Free Access Patterns Under Updates}},
  booktitle =	{26th International Conference on Database Theory (ICDT 2023)},
  pages =	{17:1--17:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-270-9},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{255},
  editor =	{Geerts, Floris and Vandevoort, Brecht},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2023.17},
  URN =		{urn:nbn:de:0030-drops-177599},
  doi =		{10.4230/LIPIcs.ICDT.2023.17},
  annote =	{Keywords: fully dynamic algorithm, enumeration delay, complexity trade-off, dichotomy}
}
Document
DOI: 10.4230/LIPIcs.CSL.2023.29
Evaluation Trade-Offs for Acyclic Conjunctive Queries

Authors: Ahmet Kara, Milos Nikolic, Dan Olteanu, and Haozhe Zhang

Published in: LIPIcs, Volume 252, 31st EACSL Annual Conference on Computer Science Logic (CSL 2023)

Abstract
We consider the evaluation of acyclic conjunctive queries, where the evaluation time is decomposed into preprocessing time and enumeration delay. In a seminal paper at CSL'07, Bagan, Durand, and Grandjean showed that acyclic queries can be evaluated with linear preprocessing time and linear enumeration delay. If the query is free-connex, the enumeration delay becomes constant. Further prior work showed that constant enumeration delay can be achieved for arbitrary acyclic conjunctive queries at the expense of a preprocessing time that is characterised by the fractional hypertree width. We introduce an approach that exposes a trade-off between preprocessing time and enumeration delay for acyclic conjunctive queries. The aforementioned prior works represent extremes in this trade-off space. Yet our approach also allows for the enumeration delay and the preprocessing time between these extremes, in particular the delay may lie between constant and linear time. Our approach decomposes the given query into subqueries and achieves for each subquery a trade-off that depends on a parameter controlling the times for preprocessing and enumeration. The complexity of the query is given by the Pareto optimal points of a bi-objective optimisation program whose inputs are possible query decompositions and parameter values.
Cite as

Ahmet Kara, Milos Nikolic, Dan Olteanu, and Haozhe Zhang. Evaluation Trade-Offs for Acyclic Conjunctive Queries. In 31st EACSL Annual Conference on Computer Science Logic (CSL 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 252, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{kara_et_al:LIPIcs.CSL.2023.29,
  author =	{Kara, Ahmet and Nikolic, Milos and Olteanu, Dan and Zhang, Haozhe},
  title =	{{Evaluation Trade-Offs for Acyclic Conjunctive Queries}},
  booktitle =	{31st EACSL Annual Conference on Computer Science Logic (CSL 2023)},
  pages =	{29:1--29:20},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-264-8},
  ISSN =	{1868-8969},
  year =	{2023},
  volume =	{252},
  editor =	{Klin, Bartek and Pimentel, Elaine},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2023.29},
  URN =		{urn:nbn:de:0030-drops-174907},
  doi =		{10.4230/LIPIcs.CSL.2023.29},
  annote =	{Keywords: acyclic queries, query evaluation, enumeration delay}
}
Document
DOI: 10.4230/LIPIcs.IPEC.2021.3
The Fine-Grained Complexity of Multi-Dimensional Ordering Properties

Authors: Haozhe An, Mohit Gurumukhani, Russell Impagliazzo, Michael Jaber, Marvin Künnemann, and Maria Paula Parga Nina

Published in: LIPIcs, Volume 214, 16th International Symposium on Parameterized and Exact Computation (IPEC 2021)

Abstract
We define a class of problems whose input is an n-sized set of d-dimensional vectors, and where the problem is first-order definable using comparisons between coordinates. This class captures a wide variety of tasks, such as complex types of orthogonal range search, model-checking first-order properties on geometric intersection graphs, and elementary questions on multidimensional data like verifying Pareto optimality of a choice of data points. Focusing on constant dimension d, we show that any k-quantifier, d-dimensional such problem is solvable in O(n^{k-1} log^{d-1} n) time. Furthermore, this algorithm is conditionally tight up to subpolynomial factors: we show that assuming the 3-uniform hyperclique hypothesis, there is a k-quantifier, (3k-3)-dimensional problem in this class that requires time Ω(n^{k-1-o(1)}). Towards identifying a single representative problem for this class, we study the existence of complete problems for the 3-quantifier setting (since 2-quantifier problems can already be solved in near-linear time O(nlog^{d-1} n), and k-quantifier problems with k > 3 reduce to the 3-quantifier case). We define a problem Vector Concatenated Non-Domination VCND_d (Given three sets of vectors X,Y and Z of dimension d,d and 2d, respectively, is there an x ∈ X and a y ∈ Y so that their concatenation x∘y is not dominated by any z ∈ Z, where vector u is dominated by vector v if u_i ≤ v_i for each coordinate 1 ≤ i ≤ d), and determine it as the "unique" candidate to be complete for this class (under fine-grained assumptions).
Cite as

Haozhe An, Mohit Gurumukhani, Russell Impagliazzo, Michael Jaber, Marvin Künnemann, and Maria Paula Parga Nina. The Fine-Grained Complexity of Multi-Dimensional Ordering Properties. In 16th International Symposium on Parameterized and Exact Computation (IPEC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 214, pp. 3:1-3:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{an_et_al:LIPIcs.IPEC.2021.3,
  author =	{An, Haozhe and Gurumukhani, Mohit and Impagliazzo, Russell and Jaber, Michael and K\"{u}nnemann, Marvin and Nina, Maria Paula Parga},
  title =	{{The Fine-Grained Complexity of Multi-Dimensional Ordering Properties}},
  booktitle =	{16th International Symposium on Parameterized and Exact Computation (IPEC 2021)},
  pages =	{3:1--3:15},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-216-7},
  ISSN =	{1868-8969},
  year =	{2021},
  volume =	{214},
  editor =	{Golovach, Petr A. and Zehavi, Meirav},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2021.3},
  URN =		{urn:nbn:de:0030-drops-153869},
  doi =		{10.4230/LIPIcs.IPEC.2021.3},
  annote =	{Keywords: Fine-grained complexity, First-order logic, Orthogonal vectors}
}
Document
DOI: 10.4230/LIPIcs.ICDT.2019.4
Counting Triangles under Updates in Worst-Case Optimal Time

Authors: Ahmet Kara, Hung Q. Ngo, Milos Nikolic, Dan Olteanu, and Haozhe Zhang

Published in: LIPIcs, Volume 127, 22nd International Conference on Database Theory (ICDT 2019)

Abstract
We consider the problem of incrementally maintaining the triangle count query under single-tuple updates to the input relations. We introduce an approach that exhibits a space-time tradeoff such that the space-time product is quadratic in the size of the input database and the update time can be as low as the square root of this size. This lowest update time is worst-case optimal conditioned on the Online Matrix-Vector Multiplication conjecture. The classical and factorized incremental view maintenance approaches are recovered as special cases of our approach within the space-time tradeoff. In particular, they require linear-time maintenance under updates, which is suboptimal. Our approach can also count all triangles in a static database in the worst-case optimal time needed for enumerating them.
Cite as

Ahmet Kara, Hung Q. Ngo, Milos Nikolic, Dan Olteanu, and Haozhe Zhang. Counting Triangles under Updates in Worst-Case Optimal Time. In 22nd International Conference on Database Theory (ICDT 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 127, pp. 4:1-4:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{kara_et_al:LIPIcs.ICDT.2019.4,
  author =	{Kara, Ahmet and Ngo, Hung Q. and Nikolic, Milos and Olteanu, Dan and Zhang, Haozhe},
  title =	{{Counting Triangles under Updates in Worst-Case Optimal Time}},
  booktitle =	{22nd International Conference on Database Theory (ICDT 2019)},
  pages =	{4:1--4:18},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-101-6},
  ISSN =	{1868-8969},
  year =	{2019},
  volume =	{127},
  editor =	{Barcelo, Pablo and Calautti, Marco},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICDT.2019.4},
  URN =		{urn:nbn:de:0030-drops-103068},
  doi =		{10.4230/LIPIcs.ICDT.2019.4},
  annote =	{Keywords: incremental view maintenance, amortized analysis, data skew}
}
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