Volume
LIPIcs, Volume 250
42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)
Publication Details
- published at: 2022-12-14
- Publisher: Schloss Dagstuhl – Leibniz-Zentrum für Informatik
- ISBN: 978-3-95977-261-7
- DBLP: db/conf/fsttcs/fsttcs2022
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Complete Volume
LIPIcs, Volume 250, FSTTCS 2022, Complete Volume
Abstract
LIPIcs, Volume 250, FSTTCS 2022, Complete Volume
Cite as
42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 1-792, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@Proceedings{dawar_et_al:LIPIcs.FSTTCS.2022,
title = {{LIPIcs, Volume 250, FSTTCS 2022, Complete Volume}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {1--792},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022},
URN = {urn:nbn:de:0030-drops-173910},
doi = {10.4230/LIPIcs.FSTTCS.2022},
annote = {Keywords: LIPIcs, Volume 250, FSTTCS 2022, Complete Volume}
}
Document
Front Matter
Front Matter, Table of Contents, Preface, Conference Organization
Abstract
Front Matter, Table of Contents, Preface, Conference Organization
Cite as
42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 0:i-0:xvi, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dawar_et_al:LIPIcs.FSTTCS.2022.0,
author = {Dawar, Anuj and Guruswami, Venkatesan},
title = {{Front Matter, Table of Contents, Preface, Conference Organization}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {0:i--0:xvi},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.0},
URN = {urn:nbn:de:0030-drops-173928},
doi = {10.4230/LIPIcs.FSTTCS.2022.0},
annote = {Keywords: Front Matter, Table of Contents, Preface, Conference Organization}
}
Document
Invited Talk
Algorithms for Uncertain Environments: Going Beyond the Worst-Case (Invited Talk)
Abstract
Analyzing the performance of algorithms in both the worst case and the average case are cornerstones of computer science: these are two different ways to understand how well algorithms perform. Over the past two decades, there has been a concerted effort to understand the performance of algorithms in models that go beyond these two extremes. In this talk I will discuss some of the proposed models and approaches, particularly for problems related to online algorithms, where decisions must be made sequentially without knowing future portions of the input.
Cite as
Anupam Gupta. Algorithms for Uncertain Environments: Going Beyond the Worst-Case (Invited Talk). In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, p. 1:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gupta:LIPIcs.FSTTCS.2022.1,
author = {Gupta, Anupam},
title = {{Algorithms for Uncertain Environments: Going Beyond the Worst-Case}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {1:1--1:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.1},
URN = {urn:nbn:de:0030-drops-173933},
doi = {10.4230/LIPIcs.FSTTCS.2022.1},
annote = {Keywords: Optimization under Uncertainty, Online Algorithms, Beyond Worst Case Analysis}
}
Document
Invited Talk
Why MCSP Is a More Important Problem Than SAT (Invited Talk)
Abstract
CNF Satisfiability (SAT) and its variants are generally considered the central problems in complexity theory, due to their applications in the theory of NP-completeness, logic, verification, probabilistically checkable proofs and parameterized complexity, among other areas. We challenge this conventional wisdom and argue that analysing the Minimum Circuit Size Problem (MCSP) and its relatives is more important from the perspective of fundamental problems in complexity theory, such as complexity lower bounds, minimal assumptions for cryptography, a robust theory of average-case complexity, and optimal results in hardness of approximation.
Cite as
Rahul Santhanam. Why MCSP Is a More Important Problem Than SAT (Invited Talk). In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, p. 2:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{santhanam:LIPIcs.FSTTCS.2022.2,
author = {Santhanam, Rahul},
title = {{Why MCSP Is a More Important Problem Than SAT}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {2:1--2:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.2},
URN = {urn:nbn:de:0030-drops-173943},
doi = {10.4230/LIPIcs.FSTTCS.2022.2},
annote = {Keywords: Minimum Circuit Size Problem, Satisfiability, Cryptography, Learning, Approximation}
}
Document
Invited Talk
The True Colors of Memory: A Tour of Chromatic-Memory Strategies in Zero-Sum Games on Graphs (Invited Talk)
Abstract
Two-player turn-based zero-sum games on (finite or infinite) graphs are a central framework in theoretical computer science - notably as a tool for controller synthesis, but also due to their connection with logic and automata theory. A crucial challenge in the field is to understand how complex strategies need to be to play optimally, given a type of game and a winning objective. In this invited contribution, we give a tour of recent advances aiming to characterize games where finite-memory strategies suffice (i.e., using a limited amount of information about the past). We mostly focus on so-called chromatic memory, which is limited to using colors - the basic building blocks of objectives - seen along a play to update itself. Chromatic memory has the advantage of being usable in different game graphs, and the corresponding class of strategies turns out to be of great interest to both the practical and the theoretical sides.
Cite as
Patricia Bouyer, Mickael Randour, and Pierre Vandenhove. The True Colors of Memory: A Tour of Chromatic-Memory Strategies in Zero-Sum Games on Graphs (Invited Talk). In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 3:1-3:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bouyer_et_al:LIPIcs.FSTTCS.2022.3,
author = {Bouyer, Patricia and Randour, Mickael and Vandenhove, Pierre},
title = {{The True Colors of Memory: A Tour of Chromatic-Memory Strategies in Zero-Sum Games on Graphs}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {3:1--3:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.3},
URN = {urn:nbn:de:0030-drops-173957},
doi = {10.4230/LIPIcs.FSTTCS.2022.3},
annote = {Keywords: two-player games on graphs, finite-memory strategies, chromatic memory, parity automata, \omega-regularity}
}
Document
Invited Talk
Expanders in Higher Dimensions (Invited Talk)
Abstract
Expander graphs have been studied in many areas of mathematics and in computer science with versatile applications, including coding theory, networking, computational complexity and geometry.
High-dimensional expanders are a generalization that has been studied in recent years and their promise is beginning to bear fruit. In the talk, I will survey some powerful local to global properties of high-dimensional expanders, and describe several interesting applications, ranging from convergence of random walks to construction of locally testable codes that prove the c³ conjecture (namely, codes with constant rate, constant distance, and constant locality).
Cite as
Irit Dinur. Expanders in Higher Dimensions (Invited Talk). In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, p. 4:1, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{dinur:LIPIcs.FSTTCS.2022.4,
author = {Dinur, Irit},
title = {{Expanders in Higher Dimensions}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {4:1--4:1},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.4},
URN = {urn:nbn:de:0030-drops-173967},
doi = {10.4230/LIPIcs.FSTTCS.2022.4},
annote = {Keywords: Expanders}
}
Document
Packing Arc-Disjoint 4-Cycles in Oriented Graphs
Abstract
Given a directed graph G and a positive integer k, the Arc Disjoint r-Cycle Packing problem asks whether G has k arc-disjoint r-cycles. We show that, for each integer r ≥ 3, Arc Disjoint r-Cycle Packing is NP-complete on oriented graphs with girth r. When r is even, the same result holds even when the input class is further restricted to be bipartite. On the positive side, focusing on r = 4 in oriented graphs, we study the complexity of the problem with respect to two parameterizations: solution size and vertex cover size. For the former, we give a cubic kernel with quadratic number of vertices. This is smaller than the compression size guaranteed by a reduction to the well-known 4-Set Packing. For the latter, we show fixed-parameter tractability using an unapparent integer linear programming formulation of an equivalent problem.
Cite as
Jasine Babu, R. Krithika, and Deepak Rajendraprasad. Packing Arc-Disjoint 4-Cycles in Oriented Graphs. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 5:1-5:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{babu_et_al:LIPIcs.FSTTCS.2022.5,
author = {Babu, Jasine and Krithika, R. and Rajendraprasad, Deepak},
title = {{Packing Arc-Disjoint 4-Cycles in Oriented Graphs}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {5:1--5:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.5},
URN = {urn:nbn:de:0030-drops-173979},
doi = {10.4230/LIPIcs.FSTTCS.2022.5},
annote = {Keywords: arc-disjoint cycles, bipartite digraphs, oriented graphs, parameterized complexity}
}
Document
Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions
Abstract
Submodular functions are fundamental to combinatorial optimization. Many interesting problems can be formulated as special cases of problems involving submodular functions. In this work, we consider the problem of approximating symmetric submodular functions everywhere using hypergraph cut functions. Devanur, Dughmi, Schwartz, Sharma, and Singh [Devanur et al., 2013] showed that symmetric submodular functions over n-element ground sets cannot be approximated within (n/8)-factor using a graph cut function and raised the question of approximating them using hypergraph cut functions. Our main result is that there exist symmetric submodular functions over n-element ground sets that cannot be approximated within a o(n^{1/3}/log² n)-factor using a hypergraph cut function. On the positive side, we show that symmetrized concave linear functions and symmetrized rank functions of uniform matroids and partition matroids can be constant-approximated using hypergraph cut functions.
Cite as
Calvin Beideman, Karthekeyan Chandrasekaran, Chandra Chekuri, and Chao Xu. Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 6:1-6:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{beideman_et_al:LIPIcs.FSTTCS.2022.6,
author = {Beideman, Calvin and Chandrasekaran, Karthekeyan and Chekuri, Chandra and Xu, Chao},
title = {{Approximate Representation of Symmetric Submodular Functions via Hypergraph Cut Functions}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {6:1--6:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.6},
URN = {urn:nbn:de:0030-drops-173986},
doi = {10.4230/LIPIcs.FSTTCS.2022.6},
annote = {Keywords: Submodular Functions, Hypergraphs, Approximation, Representation}
}
Document
The DAG Visit Approach for Pebbling and I/O Lower Bounds
Abstract
We introduce the notion of an r-visit of a Directed Acyclic Graph DAG G = (V,E), a sequence of the vertices of the DAG complying with a given rule r. A rule r specifies for each vertex v ∈ V a family of r-enabling sets of (immediate) predecessors: before visiting v, at least one of its enabling sets must have been visited. Special cases are the r^(top)-rule (or, topological rule), for which the only enabling set is the set of all predecessors and the r^(sin)-rule (or, singleton rule), for which the enabling sets are the singletons containing exactly one predecessor. The r-boundary complexity of a DAG G, b_r(G), is the minimum integer b such that there is an r-visit where, at each stage, for at most b of the vertices yet to be visited an enabling set has already been visited. By a reformulation of known results, it is shown that the boundary complexity of a DAG G is a lower bound to the pebbling number of the reverse DAG, G^R. Several known pebbling lower bounds can be cast in terms of the r^{(sin)}-boundary complexity. The main contributions of this paper are as follows:
- An existentially tight 𝒪(√{d_{out} n}) upper bound to the r^(sin)-boundary complexity of any DAG of n vertices and out-degree d_{out}.
- An existentially tight 𝒪(d_{out}/(log₂ d_{out}) log₂ n) upper bound to the r^(top)-boundary complexity of any DAG. (There are DAGs for which r^(top) provides a tight pebbling lower bound, whereas r^(sin) does not.)
- A visit partition technique for I/O lower bounds, which generalizes the S-partition I/O technique introduced by Hong and Kung in their classic paper "I/O complexity: The Red-Blue pebble game". The visit partition approach yields tight I/O bounds for some DAGs for which the S-partition technique can only yield a trivial lower bound.
Cite as
Gianfranco Bilardi and Lorenzo De Stefani. The DAG Visit Approach for Pebbling and I/O Lower Bounds. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 7:1-7:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bilardi_et_al:LIPIcs.FSTTCS.2022.7,
author = {Bilardi, Gianfranco and De Stefani, Lorenzo},
title = {{The DAG Visit Approach for Pebbling and I/O Lower Bounds}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {7:1--7:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.7},
URN = {urn:nbn:de:0030-drops-173999},
doi = {10.4230/LIPIcs.FSTTCS.2022.7},
annote = {Keywords: Pebbling, Directed Acyclic Graph, Pebbling number, I/O complexity}
}
Document
Counting and Sampling from Substructures Using Linear Algebraic Queries
Abstract
For an unknown n × n matrix A having non-negative entries, the inner product (IP) oracle takes as inputs a specified row (or a column) of A and a vector 𝐯 ∈ ℝⁿ with non-negative entries, and returns their inner product. Given two input vectors x and y in ℝⁿ with non-negative entries, and an unknown matrix A with non-negative entries with IP oracle access, we design almost optimal sublinear time algorithms for the following two fundamental matrix problems:
- Find an estimate 𝒳 for the bilinear form x^T A y such that 𝒳 ≈ x^T A y.
- Designing a sampler 𝒵 for the entries of the matrix A such that ℙ(𝒵 = (i,j)) ≈ x_i A_{ij} y_j /(x^T A y), where x_i and y_j are i-th and j-th coordinate of 𝐱 and 𝐲 respectively. As special cases of the above results, for any submatrix of an unknown matrix with non-negative entries and IP oracle access, we can efficiently estimate the sum of the entries of any submatrix, and also sample a random entry from the submatrix with probability proportional to its weight. We will show that the above results imply that if we are given IP oracle access to the adjacency matrix of a graph, with non-negative weights on the edges, then we can design sublinear time algorithms for the following two fundamental graph problems:
- Estimating the sum of the weights of the edges of an induced subgraph, and
- Sampling edges proportional to their weights from an induced subgraph. We show that compared to the classical local queries (degree, adjacency, and neighbor queries) on graphs, we can get a quadratic speedup if we use IP oracle access for the above two problems.
Apart from the above, we study several matrix problems through the lens of IP oracle, like testing if the matrix is diagonal, symmetric, doubly stochastic, etc. Note that IP oracle is in the class of linear algebraic queries used lately in a series of works by Ben-Eliezer et al. [SODA'08], Nisan [SODA'21], Rashtchian et al. [RANDOM'20], Sun et al. [ICALP'19], and Shi and Woodruff [AAAI'19]. Recently, IP oracle was used by Bishnu et al. [RANDOM'21] to estimate dissimilarities between two matrices.
Cite as
Arijit Bishnu, Arijit Ghosh, Gopinath Mishra, and Manaswi Paraashar. Counting and Sampling from Substructures Using Linear Algebraic Queries. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 8:1-8:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bishnu_et_al:LIPIcs.FSTTCS.2022.8,
author = {Bishnu, Arijit and Ghosh, Arijit and Mishra, Gopinath and Paraashar, Manaswi},
title = {{Counting and Sampling from Substructures Using Linear Algebraic Queries}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {8:1--8:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.8},
URN = {urn:nbn:de:0030-drops-174009},
doi = {10.4230/LIPIcs.FSTTCS.2022.8},
annote = {Keywords: Query complexity, Bilinear form, Uniform sampling, Weighted graphs}
}
Document
Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials
Abstract
More than three decades ago, after a series of results, Kaltofen and Trager (J. Symb. Comput. 1990) designed a randomized polynomial time algorithm for factorization of multivariate circuits. Derandomizing this algorithm, even for restricted circuit classes, is an important open problem. In particular, the case of s-sparse polynomials, having individual degree d = O(1), is very well-studied (Shpilka, Volkovich ICALP'10; Volkovich RANDOM'17; Bhargava, Saraf and Volkovich FOCS'18, JACM'20). We give a complete derandomization for this class assuming that the input is a symmetric polynomial over rationals. Generally, we prove an s^poly(d)-sparsity bound for the factors of symmetric polynomials over any field. This characterizes the known worst-case examples of sparsity blow-up for sparse polynomial factoring.
To factor f, we use techniques from convex geometry and exploit symmetry (only) in the Newton polytope of f. We prove a crucial result about convex polytopes, by introducing the concept of "low min-entropy", which might also be of independent interest.
Cite as
Pranav Bisht and Nitin Saxena. Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 9:1-9:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bisht_et_al:LIPIcs.FSTTCS.2022.9,
author = {Bisht, Pranav and Saxena, Nitin},
title = {{Derandomization via Symmetric Polytopes: Poly-Time Factorization of Certain Sparse Polynomials}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {9:1--9:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.9},
URN = {urn:nbn:de:0030-drops-174012},
doi = {10.4230/LIPIcs.FSTTCS.2022.9},
annote = {Keywords: Multivariate polynomial factorization, derandomization, sparse polynomials, symmetric polynomials, factor-sparsity, convex polytopes}
}
Document
On Solving Sparse Polynomial Factorization Related Problems
Abstract
In a recent result of Bhargava, Saraf and Volkovich [FOCS’18; JACM’20], the first factor sparsity bound for constant individual degree polynomials was shown. In particular, it was shown that any factor of a polynomial with at most s terms and individual degree bounded by d can itself have at most s^O(d²log n) terms. It is conjectured, though, that the "true" sparsity bound should be polynomial (i.e. s^poly(d)). In this paper we provide supporting evidence for this conjecture by presenting polynomial-time algorithms for several problems that would be implied by a polynomial-size sparsity bound. In particular, we give efficient (deterministic) algorithms for identity testing of Σ^[2]ΠΣΠ^[ind-deg d] circuits and testing if a sparse polynomial is an exact power. Hence, our algorithms rely on different techniques.
Cite as
Pranav Bisht and Ilya Volkovich. On Solving Sparse Polynomial Factorization Related Problems. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 10:1-10:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bisht_et_al:LIPIcs.FSTTCS.2022.10,
author = {Bisht, Pranav and Volkovich, Ilya},
title = {{On Solving Sparse Polynomial Factorization Related Problems}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {10:1--10:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.10},
URN = {urn:nbn:de:0030-drops-174023},
doi = {10.4230/LIPIcs.FSTTCS.2022.10},
annote = {Keywords: Sparse Polynomials, Identity Testing, Derandomization, Factor-Sparsity, Multivariate Polynomial Factorization}
}
Document
Complexity of Spatial Games
Abstract
Spatial games form a widely-studied class of games from biology and physics modeling the evolution of social behavior. Formally, such a game is defined by a square (d by d) payoff matrix M and an undirected graph G. Each vertex of G represents an individual, that initially follows some strategy i ∈ {1,2,…,d}. In each round of the game, every individual plays the matrix game with each of its neighbors: An individual following strategy i meeting a neighbor following strategy j receives a payoff equal to the entry (i,j) of M. Then, each individual updates its strategy to its neighbors' strategy with the highest sum of payoffs, and the next round starts. The basic computational problems consist of reachability between configurations and the average frequency of a strategy. For general spatial games and graphs, these problems are in PSPACE. In this paper, we examine restricted setting: the game is a prisoner’s dilemma; and G is a subgraph of grid. We prove that basic computational problems for spatial games with prisoner’s dilemma on a subgraph of a grid are PSPACE-hard.
Cite as
Krishnendu Chatterjee, Rasmus Ibsen-Jensen, Ismaël Jecker, and Jakub Svoboda. Complexity of Spatial Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 11:1-11:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chatterjee_et_al:LIPIcs.FSTTCS.2022.11,
author = {Chatterjee, Krishnendu and Ibsen-Jensen, Rasmus and Jecker, Isma\"{e}l and Svoboda, Jakub},
title = {{Complexity of Spatial Games}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {11:1--11:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.11},
URN = {urn:nbn:de:0030-drops-174038},
doi = {10.4230/LIPIcs.FSTTCS.2022.11},
annote = {Keywords: spatial games, computational complexity, prisoner’s dilemma, dynamical systems}
}
Document
Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product
Abstract
We establish an ε-sensitive hierarchy separation for monotone arithmetic computations. The notion of ε-sensitive monotone lower bounds was recently introduced by Hrubeš [Pavel Hrubeš, 2020]. We show the following:
- There exists a monotone polynomial over n variables in VNP that cannot be computed by 2^o(n) size monotone circuits in an ε-sensitive way as long as ε ≥ 2^(-Ω(n)).
- There exists a polynomial over n variables that can be computed by polynomial size monotone circuits but cannot be computed by any monotone arithmetic branching program (ABP) of n^o(log n) size, even in an ε-sensitive fashion as long as ε ≥ n^(-Ω(log n)).
- There exists a polynomial over n variables that can be computed by polynomial size monotone ABPs but cannot be computed in n^o(log n) size by monotone formulas even in an ε-sensitive way, when ε ≥ n^(-Ω(log n)).
- There exists a polynomial over n variables that can be computed by width-4 polynomial size monotone arithmetic branching programs (ABPs) but cannot be computed in 2^o(n^{1/d}) size by monotone, unbounded fan-in formulas of product depth d even in an ε-sensitive way, when ε ≥ 2^(-Ω(n^{1/d})). This yields an ε-sensitive separation of constant-depth monotone formulas and constant-width monotone ABPs. The novel feature of our separations is that in each case the polynomial exhibited is obtained from a graph inner-product polynomial by choosing an appropriate graph topology. The closely related graph inner-product Boolean function for expander graphs was invented by Hayes [Thomas P. Hayes, 2011], also independently by Pitassi [Toniann Pitassi, 2009], in the context of best-partition multiparty communication complexity.
Cite as
Arkadev Chattopadhyay, Utsab Ghosal, and Partha Mukhopadhyay. Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 12:1-12:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chattopadhyay_et_al:LIPIcs.FSTTCS.2022.12,
author = {Chattopadhyay, Arkadev and Ghosal, Utsab and Mukhopadhyay, Partha},
title = {{Robustly Separating the Arithmetic Monotone Hierarchy via Graph Inner-Product}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {12:1--12:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.12},
URN = {urn:nbn:de:0030-drops-174045},
doi = {10.4230/LIPIcs.FSTTCS.2022.12},
annote = {Keywords: Algebraic Complexity, Discrepancy, Lower Bounds, Monotone Computations}
}
Document
Inscribing or Circumscribing a Histogon to a Convex Polygon
Abstract
We consider two optimization problems of approximating a convex polygon, one by a largest inscribed histogon and the other by a smallest circumscribed histogon. An axis-aligned histogon is an axis-aligned rectilinear polygon such that every horizontal edge has an integer length. A histogon of orientation θ is a copy of an axis-aligned histogon rotated by θ in counterclockwise direction. The goal is to find a largest inscribed histogon and a smallest circumscribed histogon over all orientations in [0,π). Depending on whether the horizontal width of a histogon is predetermined or not, we consider several different versions of the problem and present exact algorithms. These optimization problems belong to shape analysis, classification, and simplification, and they have applications in various cost-optimization problems.
Cite as
Jaehoon Chung, Sang Won Bae, Chan-Su Shin, Sang Duk Yoon, and Hee-Kap Ahn. Inscribing or Circumscribing a Histogon to a Convex Polygon. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 13:1-13:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{chung_et_al:LIPIcs.FSTTCS.2022.13,
author = {Chung, Jaehoon and Bae, Sang Won and Shin, Chan-Su and Yoon, Sang Duk and Ahn, Hee-Kap},
title = {{Inscribing or Circumscribing a Histogon to a Convex Polygon}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {13:1--13:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.13},
URN = {urn:nbn:de:0030-drops-174054},
doi = {10.4230/LIPIcs.FSTTCS.2022.13},
annote = {Keywords: Shape simplification, Shape analysis, Histogon, Convex polygon}
}
Document
More Verifier Efficient Interactive Protocols for Bounded Space
Abstract
Let TISP[T, S], BPTISP[T, S], NTISP[T, S] and CoNTISP[T, S] be the set of languages recognized by deterministic, randomized, nondeterministic, and co-nondeterministic algorithms, respectively, running in time T and space S. Let ITIME[T_V, T_P] be the set of languages recognized by an interactive protocol where the verifier runs in time T_V and the prover runs in time T_P.
For S = Ω(log(n)) and T constructible in time log(T) S + n, we prove:
TISP[T, S] ⊆ ITIME[Õ(log(T) S + n), 2^O(S)]
BPTISP[T, S] ⊆ ITIME[Õ(log(T) S + n), 2^O(S)]
NTISP[T, S] ⊆ ITIME[Õ(log(T)^2 S + n), 2^O(S)]
CoNTISP[T, S] ⊆ ITIME[Õ(log(T)^2 S + n), 2^O(S)] .
The best prior verifier time is from Shamir [Shamir, 1992; Lund et al., 1990]: TISP[T, S] ⊆ ITIME[Õ(log(T)(S + n)), 2^O(log(T)(S + n))].
Our prover is faster, and our verifier is faster when S = o(n).
The best prior prover time uses ideas from Goldwasser, Kalai, and Rothblum [Goldwasser et al., 2015]:
NTISP[T, S] ⊆ ITIME[Õ(log(T) S² + n), 2^O(S)].
Our verifier is faster when log(T) = o(S), and for deterministic algorithms.
To our knowledge, no previous interactive protocol for TISP simultaneously has the same verifier time and prover time as ours. In our opinion, our protocol is also simpler than previous protocols.
Cite as
Joshua Cook. More Verifier Efficient Interactive Protocols for Bounded Space. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 14:1-14:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{cook:LIPIcs.FSTTCS.2022.14,
author = {Cook, Joshua},
title = {{More Verifier Efficient Interactive Protocols for Bounded Space}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {14:1--14:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.14},
URN = {urn:nbn:de:0030-drops-174067},
doi = {10.4230/LIPIcs.FSTTCS.2022.14},
annote = {Keywords: Interactive Proofs, Verifier Time, Randomized Space, Nondeterministic Space, Fine Grain Complexity}
}
Document
Improved Quantum Query Upper Bounds Based on Classical Decision Trees
Abstract
We consider the following question in query complexity: Given a classical query algorithm in the form of a decision tree, when does there exist a quantum query algorithm with a speed-up (i.e., that makes fewer queries) over the classical one? We provide a general construction based on the structure of the underlying decision tree, and prove that this can give us an up-to-quadratic quantum speed-up in the number of queries. In particular, our results give a bounded-error quantum query algorithm of cost O(√s) to compute a Boolean function (more generally, a relation) that can be computed by a classical (even randomized) decision tree of size s. This recovers an O(√n) algorithm for the Search problem, for example.
Lin and Lin [Theory of Computing'16] and Beigi and Taghavi [Quantum'20] showed results of a similar flavor. Their upper bounds are in terms of a quantity which we call the "guessing complexity" of a decision tree. We identify that the guessing complexity of a decision tree equals its rank, a notion introduced by Ehrenfeucht and Haussler [Information and Computation'89] in the context of learning theory. This answers a question posed by Lin and Lin, who asked whether the guessing complexity of a decision tree is related to any measure studied in classical complexity theory. We also show a polynomial separation between rank and its natural randomized analog for the complete binary AND-OR tree.
Beigi and Taghavi constructed span programs and dual adversary solutions for Boolean functions given classical decision trees computing them and an assignment of non-negative weights to edges of the tree. We explore the effect of changing these weights on the resulting span program complexity and objective value of the dual adversary bound, and capture the best possible weighting scheme by an optimization program. We exhibit a solution to this program and argue its optimality from first principles. We also exhibit decision trees for which our bounds are strictly stronger than those of Lin and Lin, and Beigi and Taghavi. This answers a question of Beigi and Taghavi, who asked whether different weighting schemes in their construction could yield better upper bounds.
Cite as
Arjan Cornelissen, Nikhil S. Mande, and Subhasree Patro. Improved Quantum Query Upper Bounds Based on Classical Decision Trees. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 15:1-15:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{cornelissen_et_al:LIPIcs.FSTTCS.2022.15,
author = {Cornelissen, Arjan and Mande, Nikhil S. and Patro, Subhasree},
title = {{Improved Quantum Query Upper Bounds Based on Classical Decision Trees}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {15:1--15:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.15},
URN = {urn:nbn:de:0030-drops-174071},
doi = {10.4230/LIPIcs.FSTTCS.2022.15},
annote = {Keywords: Quantum Query Complexity, Decision Trees, Decision Tree Rank}
}
Document
On the VNP-Hardness of Some Monomial Symmetric Polynomials
Abstract
A polynomial P ∈ 𝔽[x_1,…,x_n] is said to be symmetric if it is invariant under any permutation of its input variables. The study of symmetric polynomials is a classical topic in mathematics, specifically in algebraic combinatorics and representation theory. More recently, they have been studied in several works in computer science, especially in algebraic complexity theory.
In this paper, we prove the computational hardness of one of the most basic kinds of symmetric polynomials: the monomial symmetric polynomials, which are obtained by summing all distinct permutations of a single monomial. This family of symmetric functions is a natural basis for the space of symmetric polynomials (over any field), and generalizes many well-studied families such as the elementary symmetric polynomials and the power-sum symmetric polynomials.
We show that certain families of monomial symmetric polynomials are VNP-complete with respect to oracle reductions. This stands in stark contrast to the case of elementary and power symmetric polynomials, both of which have constant-depth circuits of polynomial size.
Cite as
Radu Curticapean, Nutan Limaye, and Srikanth Srinivasan. On the VNP-Hardness of Some Monomial Symmetric Polynomials. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 16:1-16:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{curticapean_et_al:LIPIcs.FSTTCS.2022.16,
author = {Curticapean, Radu and Limaye, Nutan and Srinivasan, Srikanth},
title = {{On the VNP-Hardness of Some Monomial Symmetric Polynomials}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {16:1--16:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.16},
URN = {urn:nbn:de:0030-drops-174081},
doi = {10.4230/LIPIcs.FSTTCS.2022.16},
annote = {Keywords: algebraic complexity, symmetric polynomial, permanent, Sidon set}
}
Document
Online Piercing of Geometric Objects
Abstract
We consider the online version of the piercing set problem where geometric objects arrive one by one. The online algorithm must maintain a piercing set for the arrived objects by making irrevocable decisions. First, we show that any deterministic online algorithm that solves this problem has a competitive ratio of at least Ω(n), which even holds when the objects are one-dimensional intervals. On the other hand, piercing unit objects is equivalent to the unit covering problem which is well-studied in the online model. Due to this, all the results related to the online unit covering problem are preserved for the online unit piercing problem when the objects are translated from each other. Surprisingly, no upper bound was known for the unit covering problem when unit objects are anything other than balls and hypercubes. In this paper, we introduce the notion of α-aspect and α-aspect_∞ objects. We give an upper bound of competitive ratio for α-aspect and α-aspect_∞ objects in ℝ³ and ℝ^d, respectively, with a scaling factor in the range [1,k]. We also propose a lower bound of the competitive ratio for bounded scaled objects like α-aspect objects in ℝ², axis-aligned hypercubes in ℝ^d, and balls in ℝ² and ℝ³. For piercing α-aspect_∞ objects in ℝ^d, we show that a simple deterministic algorithm achieves a competitive ratio of at most (2/α)^d((1+α)^d-1) (⌈log_(1+α)(2k/α)⌉)+1. This result is very general in nature. One can obtain upper bounds for specific objects by specifying the value of α. By putting the value of k = 1 to the above result, we get an upper bound of the competitive ratio for the unit covering problem for various types of objects.
Cite as
Minati De, Saksham Jain, Sarat Varma Kallepalli, and Satyam Singh. Online Piercing of Geometric Objects. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 17:1-17:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{de_et_al:LIPIcs.FSTTCS.2022.17,
author = {De, Minati and Jain, Saksham and Kallepalli, Sarat Varma and Singh, Satyam},
title = {{Online Piercing of Geometric Objects}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {17:1--17:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.17},
URN = {urn:nbn:de:0030-drops-174090},
doi = {10.4230/LIPIcs.FSTTCS.2022.17},
annote = {Keywords: piercing set problem, online algorithm, competitive ratio, unit covering problem, geometric objects}
}
Document
Half-Guarding Weakly-Visible Polygons and Terrains
Abstract
We consider a variant of the art gallery problem where all guards are limited to seeing 180degree. Guards that can only see in one direction are called half-guards. We give a polynomial time approximation scheme for vertex guarding the vertices of a weakly-visible polygon with half-guards. We extend this to vertex guarding the boundary of a weakly-visible polygon with half-guards. We also show NP-hardness for vertex guarding a weakly-visible polygon with half-guards. Lastly, we show that the orientation of half-guards is critical in terrain guarding. Depending on the orientation of the half-guards, the problem is either very easy (polynomial time solvable) or very hard (NP-hard).
Cite as
Nandhana Duraisamy, Hannah Miller Hillberg, Ramesh K. Jallu, Erik Krohn, Anil Maheshwari, Subhas C. Nandy, and Alex Pahlow. Half-Guarding Weakly-Visible Polygons and Terrains. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 18:1-18:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{duraisamy_et_al:LIPIcs.FSTTCS.2022.18,
author = {Duraisamy, Nandhana and Hillberg, Hannah Miller and Jallu, Ramesh K. and Krohn, Erik and Maheshwari, Anil and Nandy, Subhas C. and Pahlow, Alex},
title = {{Half-Guarding Weakly-Visible Polygons and Terrains}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {18:1--18:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.18},
URN = {urn:nbn:de:0030-drops-174103},
doi = {10.4230/LIPIcs.FSTTCS.2022.18},
annote = {Keywords: Art Gallery Problem, Approximation Algorithm, NP-Hardness, Monotone Polygons, Half-Guards}
}
Document
A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications
Abstract
Recently [Mai and Vazirani, 2018] identified and initiated work on a new problem, namely understanding structural relationships between the lattices of solutions of two "nearby" instances of stable matching. They also gave an application of their work to finding a robust stable matching. However, the types of changes they allowed in going from instance A to B were very restricted, namely any one agent executes an upward shift.
In this paper, we allow any one agent to permute its preference list arbitrarily. Let M_A and M_B be the sets of stable matchings of the resulting pair of instances A and B, and let ℒ_A and ℒ_B be the corresponding lattices of stable matchings. We prove that the matchings in M_A ∩ M_B form a sublattice of both ℒ_A and ℒ_B and those in M_A ⧵ M_B form a join semi-sublattice. These properties enable us to obtain a polynomial time algorithm for not only finding a stable matching in M_A ∩ M_B, but also for obtaining the partial order, as promised by Birkhoff’s Representation Theorem [Birkhoff, 1937]. As a result, we can generate all matchings in this sublattice.
Our algorithm also helps solve a version of the robust stable matching problem. We discuss another potential application, namely obtaining new insights into the incentive compatibility properties of the Gale-Shapley Deferred Acceptance Algorithm.
Cite as
Rohith Reddy Gangam, Tung Mai, Nitya Raju, and Vijay V. Vazirani. A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 19:1-19:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gangam_et_al:LIPIcs.FSTTCS.2022.19,
author = {Gangam, Rohith Reddy and Mai, Tung and Raju, Nitya and Vazirani, Vijay V.},
title = {{A Structural and Algorithmic Study of Stable Matching Lattices of "Nearby" Instances, with Applications}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {19:1--19:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.19},
URN = {urn:nbn:de:0030-drops-174114},
doi = {10.4230/LIPIcs.FSTTCS.2022.19},
annote = {Keywords: stable matching, robust solutions, finite distributive lattice, Birkhoff’s Representation Theorem}
}
Document
Degree-Restricted Strength Decompositions and Algebraic Branching Programs
Abstract
We analyze Kumar’s recent quadratic algebraic branching program size lower bound proof method (CCC 2017) for the power sum polynomial. We present a refinement of this method that gives better bounds in some cases.
The lower bound relies on Noether-Lefschetz type conditions on the hypersurface defined by the homogeneous polynomial. In the explicit example that we provide, the lower bound is proved resorting to classical intersection theory.
Furthermore, we use similar methods to improve the known lower bound methods for slice rank of polynomials. We consider a sequence of polynomials that have been studied before by Shioda and show that for these polynomials the improved lower bound matches the known upper bound.
Cite as
Fulvio Gesmundo, Purnata Ghosal, Christian Ikenmeyer, and Vladimir Lysikov. Degree-Restricted Strength Decompositions and Algebraic Branching Programs. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 20:1-20:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{gesmundo_et_al:LIPIcs.FSTTCS.2022.20,
author = {Gesmundo, Fulvio and Ghosal, Purnata and Ikenmeyer, Christian and Lysikov, Vladimir},
title = {{Degree-Restricted Strength Decompositions and Algebraic Branching Programs}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {20:1--20:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.20},
URN = {urn:nbn:de:0030-drops-174127},
doi = {10.4230/LIPIcs.FSTTCS.2022.20},
annote = {Keywords: Lower bounds, Slice rank, Strength of polynomials, Algebraic branching programs}
}
Document
A Simple Polynomial Time Algorithm for Max Cut on Laminar Geometric Intersection Graphs
Abstract
In a geometric intersection graph, given a collection of n geometric objects as input, each object corresponds to a vertex and there is an edge between two vertices if and only if the corresponding objects intersect. In this work, we present a somewhat surprising result: a polynomial time algorithm for max cut on laminar geometric intersection graphs. In a laminar geometric intersection graph, if two objects intersect, then one of them will completely lie inside the other. To the best of our knowledge, for max cut this is the first class of (non-trivial) geometric intersection graphs with an exact solution in polynomial time. Our algorithm uses a simple greedy strategy. However, proving its correctness requires non-trivial ideas.
Next, we design almost-linear time algorithms (in terms of n) for laminar axis-aligned boxes by combining the properties of laminar objects with vertical ray shooting data structures. Note that the edge-set of the graph is not explicitly given as input; only the n geometric objects are given as input.
Cite as
Utkarsh Joshi, Saladi Rahul, and Josson Joe Thoppil. A Simple Polynomial Time Algorithm for Max Cut on Laminar Geometric Intersection Graphs. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 21:1-21:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{joshi_et_al:LIPIcs.FSTTCS.2022.21,
author = {Joshi, Utkarsh and Rahul, Saladi and Thoppil, Josson Joe},
title = {{A Simple Polynomial Time Algorithm for Max Cut on Laminar Geometric Intersection Graphs}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {21:1--21:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.21},
URN = {urn:nbn:de:0030-drops-174139},
doi = {10.4230/LIPIcs.FSTTCS.2022.21},
annote = {Keywords: Geometric intersection graphs, Max cut, Vertical ray shooting}
}
Document
Stable Matchings with One-Sided Ties and Approximate Popularity
Abstract
We consider a matching problem in a bipartite graph G = (A ∪ B, E) where vertices in A rank their neighbors in a strict order of preference while vertices in B are allowed to have weak rankings, i.e., ties are allowed in their preferences. Stable matchings always exist in G and are easy to find, however popular matchings need not exist and it is NP-complete to decide if one exists. This motivates the "approximately popular" matching problem.
A well-known measure of approximate popularity is low unpopularity factor. We show that when each tie in G has length at most k, there always exists a stable matching whose unpopularity factor is at most k. Our proof is algorithmic and we compute such a stable matching in polynomial time. Our result can be considered to be a generalization of Gärdenfors' result (1975) which showed that when rankings are strict, every stable matching is popular.
There are several applications where the size of the matching is its most important attribute. What one seeks here is a maximum matching M such that there is no maximum matching more popular than M. When rankings are weak, it is NP-hard to decide if G admits such a matching. When ties are one-sided and of length at most k, we show a polynomial time algorithm to find a maximum matching whose unpopularity factor within the set of maximum matchings is at most 2k.
Cite as
Telikepalli Kavitha. Stable Matchings with One-Sided Ties and Approximate Popularity. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 22:1-22:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kavitha:LIPIcs.FSTTCS.2022.22,
author = {Kavitha, Telikepalli},
title = {{Stable Matchings with One-Sided Ties and Approximate Popularity}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {22:1--22:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.22},
URN = {urn:nbn:de:0030-drops-174144},
doi = {10.4230/LIPIcs.FSTTCS.2022.22},
annote = {Keywords: Bipartite graphs, Maximum matchings, Unpopularity factor}
}
Document
Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing
Abstract
We study the generalized multidimensional bin packing problem (GVBP) that generalizes both geometric packing and vector packing. Here, we are given n rectangular items where the i-th item has width w(i), height h(i), and d nonnegative weights v₁(i), v₂(i), …, v_d(i). Our goal is to get an axis-parallel non-overlapping packing of the items into square bins so that for all j ∈ [d], the sum of the j-th weight of items in each bin is at most 1. This is a natural problem arising in logistics, resource allocation, and scheduling. Despite being well-studied in practice, approximation algorithms for this problem have rarely been explored.
We first obtain two simple algorithms for GVBP having asymptotic approximation ratios 6(d+1) and 3(1 + ln(d+1) + ε). We then extend the Round-and-Approx (R&A) framework [Bansal et al., 2009; Bansal and Khan, 2014] to wider classes of algorithms, and show how it can be adapted to GVBP. Using more sophisticated techniques, we obtain better approximation algorithms for GVBP, and we get further improvement by combining them with the R&A framework. This gives us an asymptotic approximation ratio of 2(1 + ln((d+4)/2)) + ε for GVBP, which improves to 2.919+ε for the special case of d = 1. We obtain further improvement when the items are allowed to be rotated. We also present algorithms for a generalization of GVBP where the items are high dimensional cuboids.
Cite as
Arindam Khan, Eklavya Sharma, and K. V. N. Sreenivas. Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 23:1-23:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{khan_et_al:LIPIcs.FSTTCS.2022.23,
author = {Khan, Arindam and Sharma, Eklavya and Sreenivas, K. V. N.},
title = {{Geometry Meets Vectors: Approximation Algorithms for Multidimensional Packing}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {23:1--23:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.23},
URN = {urn:nbn:de:0030-drops-174151},
doi = {10.4230/LIPIcs.FSTTCS.2022.23},
annote = {Keywords: Bin packing, rectangle packing, multidimensional packing, approximation algorithms}
}
Document
Black Box Absolute Reconstruction for Sums of Powers of Linear Forms
Abstract
We study the decomposition of multivariate polynomials as sums of powers of linear forms. We give a randomized algorithm for the following problem: If a homogeneous polynomial f ∈ K[x_1 , . . . , x_n] (where K ⊆ ℂ) of degree d is given as a blackbox, decide whether it can be written as a linear combination of d-th powers of linearly independent complex linear forms. The main novel features of the algorithm are:
- For d = 3, we improve by a factor of n on the running time from the algorithm in [Pascal Koiran and Mateusz Skomra, 2021]. The price to be paid for this improvement is that the algorithm now has two-sided error.
- For d > 3, we provide the first randomized blackbox algorithm for this problem that runs in time poly(n,d) (in an algebraic model where only arithmetic operations and equality tests are allowed). Previous algorithms for this problem [Kayal, 2011] as well as most of the existing reconstruction algorithms for other classes appeal to a polynomial factorization subroutine. This requires extraction of complex polynomial roots at unit cost and in standard models such as the unit-cost RAM or the Turing machine this approach does not yield polynomial time algorithms.
- For d > 3, when f has rational coefficients (i.e. K = ℚ), the running time of the blackbox algorithm is polynomial in n,d and the maximal bit size of any coefficient of f. This yields the first algorithm for this problem over ℂ with polynomial running time in the bit model of computation. These results are true even when we replace ℂ by ℝ. We view the problem as a tensor decomposition problem and use linear algebraic methods such as checking the simultaneous diagonalisability of the slices of a tensor. The number of such slices is exponential in d. But surprisingly, we show that after a random change of variables, computing just 3 special slices is enough. We also show that our approach can be extended to the computation of the actual decomposition. In forthcoming work we plan to extend these results to overcomplete decompositions, i.e., decompositions in more than n powers of linear forms.
Cite as
Pascal Koiran and Subhayan Saha. Black Box Absolute Reconstruction for Sums of Powers of Linear Forms. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{koiran_et_al:LIPIcs.FSTTCS.2022.24,
author = {Koiran, Pascal and Saha, Subhayan},
title = {{Black Box Absolute Reconstruction for Sums of Powers of Linear Forms}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {24:1--24:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.24},
URN = {urn:nbn:de:0030-drops-174163},
doi = {10.4230/LIPIcs.FSTTCS.2022.24},
annote = {Keywords: reconstruction algorithms, tensor decomposition, sums of powers of linear forms, simultaneous diagonalisation, algebraic algorithm, black box}
}
Document
When You Come at the King You Best Not Miss
Abstract
A tournament is an orientation of a complete graph. We say that a vertex x in a tournament T controls another vertex y if there exists a directed path of length at most two from x to y. A vertex is called a king if it controls every vertex of the tournament. It is well known that every tournament has a king. We follow Shen, Sheng, and Wu [Jian Shen et al., 2003] in investigating the query complexity of finding a king, that is, the number of arcs in T one has to know in order to surely identify at least one vertex as a king.
The aforementioned authors showed that one always has to query at least Ω(n^{4/3}) arcs and provided a strategy that queries at most O(n^{3/2}). While this upper bound has not yet been improved for the original problem, [Biswas et al., 2017] proved that with O(n^{4/3}) queries one can identify a semi-king, meaning a vertex which controls at least half of all vertices.
Our contribution is a novel strategy which improves upon the number of controlled vertices: using O(n^{4/3} polylog n) queries, we can identify a (1/2+2/17)-king. To achieve this goal we use a novel structural result for tournaments.
Cite as
Oded Lachish, Felix Reidl, and Chhaya Trehan. When You Come at the King You Best Not Miss. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 25:1-25:12, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{lachish_et_al:LIPIcs.FSTTCS.2022.25,
author = {Lachish, Oded and Reidl, Felix and Trehan, Chhaya},
title = {{When You Come at the King You Best Not Miss}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {25:1--25:12},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.25},
URN = {urn:nbn:de:0030-drops-174177},
doi = {10.4230/LIPIcs.FSTTCS.2022.25},
annote = {Keywords: Digraphs, tournaments, kings, query complexity}
}
Document
Complexity of Fault Tolerant Query Complexity
Abstract
In the model of fault tolerant decision trees introduced by Kenyon and Yao, there is a known upper bound E on the total number of queries that may be faulty (i.e., get the wrong bit). We consider this computational problem: Given as input the truth table of a function f: {0,1}ⁿ → {0,1} and a value of E, find the minimum possible height (worst-case number of queries) of any decision tree that computes f while tolerating up to E many faults. We design an algorithm for this problem that runs in time Õ(binom(n+E,E)⋅(2E+3)ⁿ), which is polynomial in the size of the truth table when E is a constant. This generalizes a standard algorithm for the non-fault tolerant setting.
Cite as
Ramita Maharjan and Thomas Watson. Complexity of Fault Tolerant Query Complexity. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 26:1-26:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{maharjan_et_al:LIPIcs.FSTTCS.2022.26,
author = {Maharjan, Ramita and Watson, Thomas},
title = {{Complexity of Fault Tolerant Query Complexity}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {26:1--26:11},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.26},
URN = {urn:nbn:de:0030-drops-174185},
doi = {10.4230/LIPIcs.FSTTCS.2022.26},
annote = {Keywords: Fault, Tolerant, Query, Complexity}
}
Document
Romeo and Juliet Meeting in Forest like Regions
Abstract
The game of rendezvous with adversaries is a game on a graph played by two players: Facilitator and Divider. Facilitator has two agents and Divider has a team of k ≥ 1 agents. While the initial positions of Facilitator’s agents are fixed, Divider gets to select the initial positions of his agents. Then, they take turns to move their agents to adjacent vertices (or stay put) with Facilitator’s goal to bring both her agents at same vertex and Divider’s goal to prevent it. The computational question of interest is to determine if Facilitator has a winning strategy against Divider with k agents. Fomin, Golovach, and Thilikos [WG, 2021] introduced this game and proved that it is PSPACE-hard and co-W[2]-hard parameterized by the number of agents.
This hardness naturally motivates the structural parameterization of the problem. The authors proved that it admits an FPT algorithm when parameterized by the modular width and the number of allowed rounds. However, they left open the complexity of the problem from the perspective of other structural parameters. In particular, they explicitly asked whether the problem admits an FPT or XP-algorithm with respect to the treewidth of the input graph. We answer this question in the negative and show that Rendezvous is co-NP-hard even for graphs of constant treewidth. Further, we show that the problem is co-W[1]-hard when parameterized by the feedback vertex set number and the number of agents, and is unlikely to admit a polynomial kernel when parameterized by the vertex cover number and the number of agents. Complementing these hardness results, we show that the Rendezvous is FPT when parameterized by both the vertex cover number and the solution size. Finally, for graphs of treewidth at most two and girds, we show that the problem can be solved in polynomial time.
Cite as
Neeldhara Misra, Manas Mulpuri, Prafullkumar Tale, and Gaurav Viramgami. Romeo and Juliet Meeting in Forest like Regions. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 27:1-27:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{misra_et_al:LIPIcs.FSTTCS.2022.27,
author = {Misra, Neeldhara and Mulpuri, Manas and Tale, Prafullkumar and Viramgami, Gaurav},
title = {{Romeo and Juliet Meeting in Forest like Regions}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {27:1--27:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.27},
URN = {urn:nbn:de:0030-drops-174194},
doi = {10.4230/LIPIcs.FSTTCS.2022.27},
annote = {Keywords: Games on Graphs, Dynamic Separators, W\lbrack1\rbrack-hardness, Structural Parametersization, Treewidth}
}
Document
New Characterizations of Core Imputations of Matching and b-Matching Games
Abstract
We give new characterizations of core imputations for the following games:
1) The assignment game.
2) Concurrent games, i.e., general graph matching games having non-empty core.
3) The unconstrained bipartite b-matching game (edges can be matched multiple times).
4) The constrained bipartite b-matching game (edges can be matched at most once).
The classic paper of Shapley and Shubik [Shapley and Shubik, 1971] showed that core imputations of the assignment game are precisely optimal solutions to the dual of the LP-relaxation of the game. Building on this, Deng et al. [Deng et al., 1999] gave a general framework which yields analogous characterizations for several fundamental combinatorial games. Interestingly enough, their framework does not apply to the last two games stated above. In turn, we show that some of the core imputations of these games correspond to optimal dual solutions and others do not. This leads to the tantalizing question of understanding the origins of the latter.
We also present new characterizations of the profits accrued by agents and teams in core imputations of the first two games. Our characterization for the first game is stronger than that for the second; the underlying reason is that the characterization of vertices of the Birkhoff polytope is stronger than that of the Balinski polytope.
Cite as
Vijay V. Vazirani. New Characterizations of Core Imputations of Matching and b-Matching Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 28:1-28:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{vazirani:LIPIcs.FSTTCS.2022.28,
author = {Vazirani, Vijay V.},
title = {{New Characterizations of Core Imputations of Matching and b-Matching Games}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {28:1--28:13},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.28},
URN = {urn:nbn:de:0030-drops-174207},
doi = {10.4230/LIPIcs.FSTTCS.2022.28},
annote = {Keywords: LP-duality theory, cooperative game theory, core of a game, assignment game, general graph matching game, bipartite b-matching game}
}
Document
Algorithms and Hardness Results for Computing Cores of Markov Chains
Abstract
Given a Markov chain M = (V, v_0, δ), with state space V and a starting state v_0, and a probability threshold ε, an ε-core is a subset C of states that is left with probability at most ε. More formally, C ⊆ V is an ε-core, iff ℙ[reach (V\C)] ≤ ε. Cores have been applied in a wide variety of verification problems over Markov chains, Markov decision processes, and probabilistic programs, as a means of discarding uninteresting and low-probability parts of a probabilistic system and instead being able to focus on the states that are likely to be encountered in a real-world run. In this work, we focus on the problem of computing a minimal ε-core in a Markov chain. Our contributions include both negative and positive results: (i) We show that the decision problem on the existence of an ε-core of a given size is NP-complete. This solves an open problem posed in [Jan Kretínský and Tobias Meggendorfer, 2020]. We additionally show that the problem remains NP-complete even when limited to acyclic Markov chains with bounded maximal vertex degree; (ii) We provide a polynomial time algorithm for computing a minimal ε-core on Markov chains over control-flow graphs of structured programs. A straightforward combination of our algorithm with standard branch prediction techniques allows one to apply the idea of cores to find a subset of program lines that are left with low probability and then focus any desired static analysis on this core subset.
Cite as
Ali Ahmadi, Krishnendu Chatterjee, Amir Kafshdar Goharshady, Tobias Meggendorfer, Roodabeh Safavi, and Ðorđe Žikelić. Algorithms and Hardness Results for Computing Cores of Markov Chains. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 29:1-29:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ahmadi_et_al:LIPIcs.FSTTCS.2022.29,
author = {Ahmadi, Ali and Chatterjee, Krishnendu and Goharshady, Amir Kafshdar and Meggendorfer, Tobias and Safavi, Roodabeh and \v{Z}ikeli\'{c}, Ðor{\d}e},
title = {{Algorithms and Hardness Results for Computing Cores of Markov Chains}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {29:1--29:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.29},
URN = {urn:nbn:de:0030-drops-174216},
doi = {10.4230/LIPIcs.FSTTCS.2022.29},
annote = {Keywords: Markov Chains, Cores, Complexity}
}
Document
Computing Threshold Budgets in Discrete-Bidding Games
Abstract
In a two-player zero-sum graph game, the players move a token throughout the graph to produce an infinite play, which determines the winner of the game. Bidding games are graph games in which in each turn, an auction (bidding) determines which player moves the token: the players have budgets, and in each turn, both players simultaneously submit bids that do not exceed their available budgets, the higher bidder moves the token, and pays the bid to the lower bidder. We distinguish between continuous- and discrete-bidding games. In the latter, the granularity of the players' bids is restricted, e.g., bids must be given in cents. Continuous-bidding games are well understood, however, from a practical standpoint, discrete-bidding games are more appealing.
In this paper we focus on discrete-bidding games. We study the problem of finding threshold budgets; namely, a necessary and sufficient initial budget for winning the game. Previously, the properties of threshold budgets were only studied for reachability games. For parity discrete-bidding games, thresholds were known to exist, but their structure was not understood. We describe two algorithms for finding threshold budgets in parity discrete-bidding games. The first algorithm is a fixed-point algorithm, and it reveals the structure of the threshold budgets in these games. Second, we show that the problem of finding threshold budgets is in NP and coNP for parity discrete-bidding games. Previously, only exponential-time algorithms where known for reachability and parity objectives. A corollary of this proof is a construction of strategies that use polynomial-size memory.
Cite as
Guy Avni and Suman Sadhukhan. Computing Threshold Budgets in Discrete-Bidding Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 30:1-30:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{avni_et_al:LIPIcs.FSTTCS.2022.30,
author = {Avni, Guy and Sadhukhan, Suman},
title = {{Computing Threshold Budgets in Discrete-Bidding Games}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {30:1--30:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.30},
URN = {urn:nbn:de:0030-drops-174222},
doi = {10.4230/LIPIcs.FSTTCS.2022.30},
annote = {Keywords: Discrete bidding games, Richman games, parity games, reachability games}
}
Document
Dependency Matrices for Multiplayer Strategic Dependencies
Abstract
In multi-player games, players take their decisions on the basis of their knowledge about what other players have done, or currently do, or even, in some cases, will do. An ability to reason in games with temporal dependencies between players' decisions is a challenging topic, in particular because it involves imperfect information. In this work, we propose a theoretical framework based on dependency matrices that includes many instances of strategic dependencies in multi-player imperfect information games. For our framework to be well-defined, we get inspiration from quantified linear-time logic where each player has to label the timeline with truth values of the propositional variable she owns. We study the problem of the existence of a winning strategy for a coalition of players, show it is undecidable in general, and exhibit an interesting subclass of dependency matrices that makes the problem decidable: the class of perfect-information dependency matrices.
Cite as
Dylan Bellier, Sophie Pinchinat, and François Schwarzentruber. Dependency Matrices for Multiplayer Strategic Dependencies. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bellier_et_al:LIPIcs.FSTTCS.2022.31,
author = {Bellier, Dylan and Pinchinat, Sophie and Schwarzentruber, Fran\c{c}ois},
title = {{Dependency Matrices for Multiplayer Strategic Dependencies}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {31:1--31:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.31},
URN = {urn:nbn:de:0030-drops-174230},
doi = {10.4230/LIPIcs.FSTTCS.2022.31},
annote = {Keywords: Temporal dependency, Delay games, Strategic reasoning, Temporal logic}
}
Document
Semilinear Representations for Series-Parallel Atomic Congestion Games
Abstract
We consider atomic congestion games on series-parallel networks, and study the structure of the sets of Nash equilibria and social local optima on a given network when the number of players varies. We establish that these sets are definable in Presburger arithmetic and that they admit semilinear representations whose all period vectors have a common direction. As an application, we prove that the prices of anarchy and stability converge to 1 as the number of players goes to infinity, and show how to exploit these semilinear representations to compute these ratios precisely for a given network and number of players.
Cite as
Nathalie Bertrand, Nicolas Markey, Suman Sadhukhan, and Ocan Sankur. Semilinear Representations for Series-Parallel Atomic Congestion Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bertrand_et_al:LIPIcs.FSTTCS.2022.32,
author = {Bertrand, Nathalie and Markey, Nicolas and Sadhukhan, Suman and Sankur, Ocan},
title = {{Semilinear Representations for Series-Parallel Atomic Congestion Games}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {32:1--32:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.32},
URN = {urn:nbn:de:0030-drops-174243},
doi = {10.4230/LIPIcs.FSTTCS.2022.32},
annote = {Keywords: congestion games, Nash equilibria, Presburger arithmetic, semilinear sets, price of anarchy}
}
Document
Playing (Almost-)Optimally in Concurrent Büchi and Co-Büchi Games
Abstract
We study two-player concurrent stochastic games on finite graphs, with Büchi and co-Büchi objectives. The goal of the first player is to maximize the probability of satisfying the given objective. Following Martin’s determinacy theorem for Blackwell games, we know that such games have a value. Natural questions are then: does there exist an optimal strategy, that is, a strategy achieving the value of the game? what is the memory required for playing (almost-)optimally?
The situation is rather simple to describe for turn-based games, where positional pure strategies suffice to play optimally in games with parity objectives. Concurrency makes the situation intricate and heterogeneous. For most ω-regular objectives, there do indeed not exist optimal strategies in general. For some objectives (that we will mention), infinite memory might also be required for playing optimally or almost-optimally.
We also provide characterizations of local interactions of the players to ensure positionality of (almost-)optimal strategies for Büchi and co-Büchi objectives. This characterization relies on properties of game forms underpinning the formalism for defining local interactions of the two players. These well-behaved game forms are like elementary bricks which, when they behave well in isolation, can be assembled in graph games and ensure the good property for the whole game.
Cite as
Benjamin Bordais, Patricia Bouyer, and Stéphane Le Roux. Playing (Almost-)Optimally in Concurrent Büchi and Co-Büchi Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 33:1-33:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{bordais_et_al:LIPIcs.FSTTCS.2022.33,
author = {Bordais, Benjamin and Bouyer, Patricia and Le Roux, St\'{e}phane},
title = {{Playing (Almost-)Optimally in Concurrent B\"{u}chi and Co-B\"{u}chi Games}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {33:1--33:18},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.33},
URN = {urn:nbn:de:0030-drops-174258},
doi = {10.4230/LIPIcs.FSTTCS.2022.33},
annote = {Keywords: Concurrent Games, Optimal Strategies, B\"{u}chi Objective, co-B\"{u}chi Objective}
}
Document
Ambiguity Through the Lens of Measure Theory
Abstract
In this paper, we establish a strong link between the ambiguity for finite words of a Büchi automaton and the ambiguity for infinite words of the same automaton. This link is based on measure theory. More precisely, we show that such an automaton is unambiguous, in the sense that no finite word labels two runs with the same starting state and the same ending state if and only if for each state, the set of infinite sequences labelling two runs starting from that state has measure zero. The measure used to define these negligible sets, that is sets of measure zero, can be any measure computed by a weighted automaton which is compatible with the Büchi automaton. This latter condition is very natural: the measure must only put weight on sets wA^ℕ where w is the label of some run in the Büchi automaton.
Cite as
Olivier Carton. Ambiguity Through the Lens of Measure Theory. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 34:1-34:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{carton:LIPIcs.FSTTCS.2022.34,
author = {Carton, Olivier},
title = {{Ambiguity Through the Lens of Measure Theory}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {34:1--34:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.34},
URN = {urn:nbn:de:0030-drops-174269},
doi = {10.4230/LIPIcs.FSTTCS.2022.34},
annote = {Keywords: ambiguity, B\"{u}chi automata, measure theory}
}
Document
Phase Semantics for Linear Logic with Least and Greatest Fixed Points
Abstract
The truth semantics of linear logic (i.e. phase semantics) is often overlooked despite having a wide range of applications and deep connections with several denotational semantics. In phase semantics, one is concerned about the provability of formulas rather than the contents of their proofs (or refutations). Linear logic equipped with the least and greatest fixpoint operators (μMALL) has been an active field of research for the past one and a half decades. Various proof systems are known viz. finitary and non-wellfounded, based on explicit and implicit (co)induction respectively.
In this paper, we extend the phase semantics of multiplicative additive linear logic (a.k.a. MALL) to μMALL with explicit (co)induction (i.e. μMALL^{ind}). We introduce a Tait-style system for μMALL called μMALL_ω where proofs are wellfounded but potentially infinitely branching. We study its phase semantics and prove that it does not have the finite model property.
Cite as
Abhishek De, Farzad Jafarrahmani, and Alexis Saurin. Phase Semantics for Linear Logic with Least and Greatest Fixed Points. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 35:1-35:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{de_et_al:LIPIcs.FSTTCS.2022.35,
author = {De, Abhishek and Jafarrahmani, Farzad and Saurin, Alexis},
title = {{Phase Semantics for Linear Logic with Least and Greatest Fixed Points}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {35:1--35:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.35},
URN = {urn:nbn:de:0030-drops-174272},
doi = {10.4230/LIPIcs.FSTTCS.2022.35},
annote = {Keywords: Linear logic, fixed points, phase semantics, closure ordinals, cut elimination}
}
Document
Natural Colors of Infinite Words
Abstract
While finite automata have minimal DFAs as a simple and natural normal form, deterministic omega-automata do not currently have anything similar. One reason for this is that a normal form for omega-regular languages has to speak about more than acceptance - for example, to have a normal form for a parity language, it should relate every infinite word to some natural color for this language. This raises the question of whether or not a concept such as a natural color of an infinite word (for a given language) exists, and, if it does, how it relates back to automata.
We define the natural color of a word purely based on an omega-regular language, and show how this natural color can be traced back from any deterministic parity automaton after two cheap and simple automaton transformations. The resulting streamlined automaton does not necessarily accept every word with its natural color, but it has a "co-run", which is like a run, but can once move to a language equivalent state, whose color is the natural color, and no co-run with a higher color exists.
The streamlined automaton defines, for every color c, a good-for-games co-Büchi automaton that recognizes the words whose natural colors with respect to the represented language are at least c. This provides a canonical representation for every ω-regular language, because good-for-games co-Büchi automata have a canonical minimal - and cheap to obtain - representation for every co-Büchi language.
Cite as
Rüdiger Ehlers and Sven Schewe. Natural Colors of Infinite Words. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 36:1-36:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ehlers_et_al:LIPIcs.FSTTCS.2022.36,
author = {Ehlers, R\"{u}diger and Schewe, Sven},
title = {{Natural Colors of Infinite Words}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {36:1--36:17},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.36},
URN = {urn:nbn:de:0030-drops-174280},
doi = {10.4230/LIPIcs.FSTTCS.2022.36},
annote = {Keywords: parity automata, automata over infinite words, \omega-regular languages}
}
Document
Synthesizing Dominant Strategies for Liveness
Abstract
Reactive synthesis automatically derives a strategy that satisfies a given specification. However, requiring a strategy to meet the specification in every situation is, in many cases, too hard of a requirement. Particularly in compositional synthesis of distributed systems, individual winning strategies for the processes often do not exist. Remorsefree dominance, a weaker notion than winning, accounts for such situations: dominant strategies are only required to be as good as any alternative strategy, i.e. , they are allowed to violate the specification if no other strategy would have satisfied it in the same situation. The composition of dominant strategies is only guaranteed to be dominant for safety properties, though; preventing the use of dominance in compositional synthesis for liveness specifications. Yet, safety properties are often not expressive enough. In this paper, we thus introduce a new winning condition for strategies, called delay-dominance, that overcomes this weakness of remorsefree dominance: we show that it is compositional for many safety and liveness specifications, enabling a compositional synthesis algorithm based on delay-dominance for general specifications. Furthermore, we introduce an automaton construction for recognizing delay-dominant strategies and prove its soundness and completeness. The resulting automaton is of single-exponential size in the squared length of the specification and can immediately be used for safraless synthesis procedures. Thus, synthesis of delay-dominant strategies is, as synthesis of winning strategies, in 2EXPTIME.
Cite as
Bernd Finkbeiner and Noemi Passing. Synthesizing Dominant Strategies for Liveness. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 37:1-37:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{finkbeiner_et_al:LIPIcs.FSTTCS.2022.37,
author = {Finkbeiner, Bernd and Passing, Noemi},
title = {{Synthesizing Dominant Strategies for Liveness}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {37:1--37:19},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.37},
URN = {urn:nbn:de:0030-drops-174298},
doi = {10.4230/LIPIcs.FSTTCS.2022.37},
annote = {Keywords: Dominant Strategies, Compositional Synthesis, Reactive Synthesis}
}
Document
Low-Latency Sliding Window Algorithms for Formal Languages
Abstract
Low-latency sliding window algorithms for regular and context-free languages are studied, where latency refers to the worst-case time spent for a single window update or query. For every regular language L it is shown that there exists a constant-latency solution that supports adding and removing symbols independently on both ends of the window (the so-called two-way variable-size model). We prove that this result extends to all visibly pushdown languages. For deterministic 1-counter languages we present a 𝒪(log n) latency sliding window algorithm for the two-way variable-size model where n refers to the window size. We complement these results with a conditional lower bound: there exists a fixed real-time deterministic context-free language L such that, assuming the OMV (online matrix vector multiplication) conjecture, there is no sliding window algorithm for L with latency n^(1/2-ε) for any ε > 0, even in the most restricted sliding window model (one-way fixed-size model). The above mentioned results all refer to the unit-cost RAM model with logarithmic word size. For regular languages we also present a refined picture using word sizes 𝒪(1), 𝒪(log log n), and 𝒪(log n).
Cite as
Moses Ganardi, Louis Jachiet, Markus Lohrey, and Thomas Schwentick. Low-Latency Sliding Window Algorithms for Formal Languages. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 38:1-38:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{ganardi_et_al:LIPIcs.FSTTCS.2022.38,
author = {Ganardi, Moses and Jachiet, Louis and Lohrey, Markus and Schwentick, Thomas},
title = {{Low-Latency Sliding Window Algorithms for Formal Languages}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {38:1--38:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.38},
URN = {urn:nbn:de:0030-drops-174301},
doi = {10.4230/LIPIcs.FSTTCS.2022.38},
annote = {Keywords: Streaming algorithms, regular languages, context-free languages}
}
Document
The Design and Regulation of Exchanges: A Formal Approach
Abstract
We use formal methods to specify, design, and monitor continuous double auctions, which are widely used to match buyers and sellers at exchanges of foreign currencies, stocks, and commodities. We identify three natural properties of such auctions and formally prove that these properties completely determine the input-output relationship. We then formally verify that a natural algorithm satisfies these properties. All definitions, theorems, and proofs are formalized in an interactive theorem prover. We extract a verified program of our algorithm to build an automated checker that is guaranteed to detect errors in the trade logs of exchanges if they generate transactions that violate any of the natural properties.
Cite as
Mohit Garg and Suneel Sarswat. The Design and Regulation of Exchanges: A Formal Approach. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 39:1-39:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{garg_et_al:LIPIcs.FSTTCS.2022.39,
author = {Garg, Mohit and Sarswat, Suneel},
title = {{The Design and Regulation of Exchanges: A Formal Approach}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {39:1--39:21},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.39},
URN = {urn:nbn:de:0030-drops-174318},
doi = {10.4230/LIPIcs.FSTTCS.2022.39},
annote = {Keywords: Double Auctions, Formal Specification and Verification, Financial Markets}
}
Document
Parikh Automata over Infinite Words
Abstract
Parikh automata extend finite automata by counters that can be tested for membership in a semilinear set, but only at the end of a run, thereby preserving many of the desirable algorithmic properties of finite automata. Here, we study the extension of the classical framework onto infinite inputs: We introduce reachability, safety, Büchi, and co-Büchi Parikh automata on infinite words and study expressiveness, closure properties, and the complexity of verification problems.
We show that almost all classes of automata have pairwise incomparable expressiveness, both in the deterministic and the nondeterministic case; a result that sharply contrasts with the well-known hierarchy in the ω-regular setting. Furthermore, emptiness is shown decidable for Parikh automata with reachability or Büchi acceptance, but undecidable for safety and co-Büchi acceptance. Most importantly, we show decidability of model checking with specifications given by deterministic Parikh automata with safety or co-Büchi acceptance, but also undecidability for all other types of automata. Finally, solving games is undecidable for all types.
Cite as
Shibashis Guha, Ismaël Jecker, Karoliina Lehtinen, and Martin Zimmermann. Parikh Automata over Infinite Words. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 40:1-40:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{guha_et_al:LIPIcs.FSTTCS.2022.40,
author = {Guha, Shibashis and Jecker, Isma\"{e}l and Lehtinen, Karoliina and Zimmermann, Martin},
title = {{Parikh Automata over Infinite Words}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {40:1--40:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.40},
URN = {urn:nbn:de:0030-drops-174327},
doi = {10.4230/LIPIcs.FSTTCS.2022.40},
annote = {Keywords: Parikh automata, \omega-automata, Infinite Games}
}
Document
New Analytic Techniques for Proving the Inherent Ambiguity of Context-Free Languages
Abstract
This article extends the work of Flajolet [Philippe Flajolet, 1987] on the relation between generating series and inherent ambiguity. We first propose an analytic criterion to prove the infinite inherent ambiguity of some context-free languages, and apply it to give a purely combinatorial proof of the infinite ambiguity of Shamir’s language. Then we show how Ginsburg and Ullian’s criterion on unambiguous bounded languages translates into a useful criterion on generating series, which generalises and simplifies the proof of the recent criterion of Makarov [Vladislav Makarov, 2021]. We then propose a new criterion based on generating series to prove the inherent ambiguity of languages with interlacing patterns, like {a^nb^ma^pb^q | n≠p or m≠q, with n,m,p,q ∈ ℕ^*}. We illustrate the applicability of these two criteria on many examples.
Cite as
Florent Koechlin. New Analytic Techniques for Proving the Inherent Ambiguity of Context-Free Languages. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 41:1-41:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{koechlin:LIPIcs.FSTTCS.2022.41,
author = {Koechlin, Florent},
title = {{New Analytic Techniques for Proving the Inherent Ambiguity of Context-Free Languages}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {41:1--41:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.41},
URN = {urn:nbn:de:0030-drops-174331},
doi = {10.4230/LIPIcs.FSTTCS.2022.41},
annote = {Keywords: Inherent ambiguity, Infinite ambiguity, Ambiguity, Generating series, Context-free languages, Bounded languages}
}
Document
Synthesis of Privacy-Preserving Systems
Abstract
Synthesis is the automated construction of a system from its specification. In many cases, we want to maintain the privacy of the system and the environment, thus limit the information that they share with each other or with an observer of the interaction. We introduce a framework for synthesis that addresses privacy in a simple yet powerful way. Our method is based on specification formalisms that include an unknown truth value. When the system and the environment interact, they may keep the truth values of some input and output signals private, which may cause the satisfaction value of specifications to become unknown. The input to the synthesis problem contains, in addition to the specification φ, also secrets ψ_1,…,ψ_k. During the interaction, the system directs the environment which input signals should stay private. The system then realizes the specification if in all interactions, the satisfaction value of the specification φ is true, whereas the satisfaction value of the secrets ψ_1,…,ψ_k is unknown. Thus, the specification is satisfied without the secrets being revealed. We describe our framework for specifications and secrets in LTL, and extend the framework also to the multi-valued specification formalism LTL[F], which enables the specification of the quality of computations. When both the specification and secrets are in LTL[F], one can trade-off the satisfaction value of the specification with the extent to which the satisfaction values of the secrets are revealed. We show that the complexity of the problem in all settings is 2EXPTIME-complete, thus it is not harder than synthesis with no privacy requirements.
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Orna Kupferman and Ofer Leshkowitz. Synthesis of Privacy-Preserving Systems. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 42:1-42:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{kupferman_et_al:LIPIcs.FSTTCS.2022.42,
author = {Kupferman, Orna and Leshkowitz, Ofer},
title = {{Synthesis of Privacy-Preserving Systems}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {42:1--42:23},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.42},
URN = {urn:nbn:de:0030-drops-174342},
doi = {10.4230/LIPIcs.FSTTCS.2022.42},
annote = {Keywords: Synthesis, Privacy, LTL, Games}
}
Document
A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation
Abstract
We look at classes of languages associated to fragments of first-order logic BΣ₁, in which quantifier alternations are disallowed. Each such fragment is fully determined by choosing the set of predicates on positions that may be used. Equipping first-order logic with the linear ordering and possibly the successor relation as predicates yields two natural fragments, which were investigated by Simon and Knast, who proved that these two variants have decidable membership: "does an input regular language belong to the class ?". We extend their results in two orthogonal directions.
- First, instead of membership, we explore the more general separation problem: decide if two regular languages can be separated by a language from the class under study.
- Second, we use more general inputs: classes 𝒢 of group languages (i.e., recognized by a DFA in which each letter induces a permutation of the states) and extensions thereof, written 𝒢^+. We rely on a characterization of BΣ₁ by the operator BPol: given an input class 𝒞, it outputs a class BPol(𝒞) that corresponds to a variant of BΣ₁ equipped with special predicates associated to 𝒞. The classes BPol(𝒢) and BPol(𝒢^+) capture many natural variants of BΣ₁ which use predicates such as the linear ordering, the successor, the modular predicates or the alphabetic modular predicates.
We show that separation is decidable for BPol(𝒢) and BPol(𝒢^+) when this is the case for 𝒢. This was already known for BPol(𝒢) and for two particular classes of the form BPol(𝒢^+). Yet, the algorithms were indirect and relied on involved frameworks, yielding poor upper complexity bounds. In contrast, our approach is direct. We work only with elementary concepts (mainly, finite automata). Our main contribution consists in polynomial time Turing reductions from both BPol(𝒢)- and BPol(𝒢^+)-separation to 𝒢-separation. This yields polynomial time algorithms for several key variants of BΣ₁, including those equipped with the linear ordering and possibly the successor and/or the modular predicates.
Cite as
Thomas Place and Marc Zeitoun. A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 43:1-43:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{place_et_al:LIPIcs.FSTTCS.2022.43,
author = {Place, Thomas and Zeitoun, Marc},
title = {{A Generic Polynomial Time Approach to Separation by First-Order Logic Without Quantifier Alternation}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {43:1--43:22},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.43},
URN = {urn:nbn:de:0030-drops-174356},
doi = {10.4230/LIPIcs.FSTTCS.2022.43},
annote = {Keywords: Automata, Separation, Covering, Concatenation hierarchies, Group languages}
}
Document
A Technique to Speed up Symmetric Attractor-Based Algorithms for Parity Games
Abstract
The classic McNaughton-Zielonka algorithm for solving parity games has excellent performance in practice, but its worst-case asymptotic complexity is worse than that of the state-of-the-art algorithms. This work pinpoints the mechanism that is responsible for this relative underperformance and proposes a new technique that eliminates it. The culprit is the wasteful manner in which the results obtained from recursive calls are indiscriminately discarded by the algorithm whenever subgames on which the algorithm is run change. Our new technique is based on firstly enhancing the algorithm to compute attractor decompositions of subgames instead of just winning strategies on them, and then on making it carefully use attractor decompositions computed in prior recursive calls to reduce the size of subgames on which further recursive calls are made. We illustrate the new technique on the classic example of the recursive McNaughton-Zielonka algorithm, but it can be applied to other symmetric attractor-based algorithms that were inspired by it, such as the quasi-polynomial versions of the McNaughton-Zielonka algorithm based on universal trees.
Cite as
K. S. Thejaswini, Pierre Ohlmann, and Marcin Jurdziński. A Technique to Speed up Symmetric Attractor-Based Algorithms for Parity Games. In 42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 250, pp. 44:1-44:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
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@InProceedings{thejaswini_et_al:LIPIcs.FSTTCS.2022.44,
author = {Thejaswini, K. S. and Ohlmann, Pierre and Jurdzi\'{n}ski, Marcin},
title = {{A Technique to Speed up Symmetric Attractor-Based Algorithms for Parity Games}},
booktitle = {42nd IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (FSTTCS 2022)},
pages = {44:1--44:20},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-261-7},
ISSN = {1868-8969},
year = {2022},
volume = {250},
editor = {Dawar, Anuj and Guruswami, Venkatesan},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2022.44},
URN = {urn:nbn:de:0030-drops-174365},
doi = {10.4230/LIPIcs.FSTTCS.2022.44},
annote = {Keywords: Parity games, Attractor decomposition, Quasipolynomial Algorithms, Universal trees}
}