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**Published in:** LIPIcs, Volume 264, 38th Computational Complexity Conference (CCC 2023)

We prove lower bounds for the Minimum Circuit Size Problem (MCSP) in the Sum-of-Squares (SoS) proof system. Our main result is that for every Boolean function f: {0,1}ⁿ → {0,1}, SoS requires degree Ω(s^{1-ε}) to prove that f does not have circuits of size s (for any s > poly(n)). As a corollary we obtain that there are no low degree SoS proofs of the statement NP ⊈ P/poly.
We also show that for any 0 < α < 1 there are Boolean functions with circuit complexity larger than 2^{n^α} but SoS requires size 2^{2^Ω(n^α)} to prove this. In addition we prove analogous results on the minimum monotone circuit size for monotone Boolean slice functions.
Our approach is quite general. Namely, we show that if a proof system Q has strong enough constraint satisfaction problem lower bounds that only depend on good expansion of the constraint-variable incidence graph and, furthermore, Q is expressive enough that variables can be substituted by local Boolean functions, then the MCSP problem is hard for Q.

Per Austrin and Kilian Risse. Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 31:1-31:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{austrin_et_al:LIPIcs.CCC.2023.31, author = {Austrin, Per and Risse, Kilian}, title = {{Sum-Of-Squares Lower Bounds for the Minimum Circuit Size Problem}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {31:1--31:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-282-2}, ISSN = {1868-8969}, year = {2023}, volume = {264}, editor = {Ta-Shma, Amnon}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.31}, URN = {urn:nbn:de:0030-drops-183011}, doi = {10.4230/LIPIcs.CCC.2023.31}, annote = {Keywords: Proof Complexity, Sum of Squares, Minimum Circuit Size Problem} }

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APPROX

**Published in:** LIPIcs, Volume 145, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)

Assuming the Unique Games Conjecture, we show that existing approximation algorithms for some Boolean Max-2-CSPs with cardinality constraints are optimal. In particular, we prove that Max-Cut with cardinality constraints is UG-hard to approximate within ~~0.858, and that Max-2-Sat with cardinality constraints is UG-hard to approximate within ~~0.929. In both cases, the previous best hardness results were the same as the hardness of the corresponding unconstrained Max-2-CSP (~~0.878 for Max-Cut, and ~~0.940 for Max-2-Sat).
The hardness for Max-2-Sat applies to monotone Max-2-Sat instances, meaning that we also obtain tight inapproximability for the Max-k-Vertex-Cover problem.

Per Austrin and Aleksa Stanković. Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 145, pp. 24:1-24:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{austrin_et_al:LIPIcs.APPROX-RANDOM.2019.24, author = {Austrin, Per and Stankovi\'{c}, Aleksa}, title = {{Global Cardinality Constraints Make Approximating Some Max-2-CSPs Harder}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2019)}, pages = {24:1--24:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-125-2}, ISSN = {1868-8969}, year = {2019}, volume = {145}, editor = {Achlioptas, Dimitris and V\'{e}gh, L\'{a}szl\'{o} A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX-RANDOM.2019.24}, URN = {urn:nbn:de:0030-drops-112394}, doi = {10.4230/LIPIcs.APPROX-RANDOM.2019.24}, annote = {Keywords: Constraint satisfaction problems, global cardinality constraints, semidefinite programming, inapproximability, Unique Games Conjecture, Max-Cut, Max-2-Sat} }

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

We study tensor networks as a model of arithmetic computation for evaluating multilinear maps. These capture any algorithm based on low border rank tensor decompositions, such as O(n^{omega+epsilon}) time matrix multiplication, and in addition many other algorithms such as O(n log n) time discrete Fourier transform and O^*(2^n) time for computing the permanent of a matrix. However tensor networks sometimes yield faster algorithms than those that follow from low-rank decompositions. For instance the fastest known O(n^{(omega +epsilon)t}) time algorithms for counting 3t-cliques can be implemented with tensor networks, even though the underlying tensor has border rank n^{3t} for all t >= 2. For counting homomorphisms of a general pattern graph P into a host graph on n vertices we obtain an upper bound of O(n^{(omega+epsilon)bw(P)/2}) where bw(P) is the branchwidth of P. This essentially matches the bound for counting cliques, and yields small improvements over previous algorithms for many choices of P.
While powerful, the model still has limitations, and we are able to show a number of unconditional lower bounds for various multilinear maps, including:
b) an Omega(n^{bw(P)}) time lower bound for counting homomorphisms from P to an n-vertex graph, matching the upper bound if omega = 2. In particular for P a v-clique this yields an Omega(n^{ceil[2v/3]}) time lower bound for counting v-cliques, and for P a k-uniform v-hyperclique we obtain an Omega(n^v) time lower bound for k >= 3, ruling out tensor networks as an approach to obtaining non-trivial algorithms for hyperclique counting and the Max-3-CSP problem.
c) an Omega(2^{0.918n}) time lower bound for the permanent of an n x n matrix.

Per Austrin, Petteri Kaski, and Kaie Kubjas. Tensor Network Complexity of Multilinear Maps. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 7:1-7:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{austrin_et_al:LIPIcs.ITCS.2019.7, author = {Austrin, Per and Kaski, Petteri and Kubjas, Kaie}, title = {{Tensor Network Complexity of Multilinear Maps}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {7:1--7:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.7}, URN = {urn:nbn:de:0030-drops-101006}, doi = {10.4230/LIPIcs.ITCS.2019.7}, annote = {Keywords: arithmetic complexity, lower bound, multilinear map, tensor network} }

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**Published in:** LIPIcs, Volume 47, 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)

The SUBSET SUM problem asks whether a given set of n positive integers contains a subset of elements that sum up to a given target t. It is an outstanding open question whether the O^*(2^{n/2})-time algorithm for SUBSET SUM by Horowitz and Sahni [J. ACM 1974] can be beaten in the worst-case setting by a "truly faster", O^*(2^{(0.5-delta)*n})-time algorithm, with some constant delta > 0. Continuing an earlier work [STACS 2015], we study SUBSET SUM parameterized by the maximum bin size beta, defined as the largest number of subsets of the n input integers that yield the same sum. For every epsilon > 0 we give a truly faster algorithm for instances with beta <= 2^{(0.5-epsilon)*n}, as well as instances with beta >= 2^{0.661n}. Consequently, we also obtain a characterization in terms of the popular density parameter n/log_2(t): if all instances of density at least 1.003 admit a truly faster algorithm, then so does every instance. This goes against the current intuition that instances of density 1 are the hardest, and therefore is a step toward answering the open question in the affirmative. Our results stem from a novel combinatorial analysis of mixings of earlier algorithms for SUBSET SUM and a study of an extremal question in additive combinatorics connected to the problem of Uniquely Decodable Code Pairs in information theory.

Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Dense Subset Sum May Be the Hardest. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 13:1-13:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{austrin_et_al:LIPIcs.STACS.2016.13, author = {Austrin, Per and Kaski, Petteri and Koivisto, Mikko and Nederlof, Jesper}, title = {{Dense Subset Sum May Be the Hardest}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {13:1--13:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-001-9}, ISSN = {1868-8969}, year = {2016}, volume = {47}, editor = {Ollinger, Nicolas and Vollmer, Heribert}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.13}, URN = {urn:nbn:de:0030-drops-57143}, doi = {10.4230/LIPIcs.STACS.2016.13}, annote = {Keywords: subset sum, additive combinatorics, exponential-time algorithm, homo-morphic hashing, littlewood–offord problem} }

Document

**Published in:** LIPIcs, Volume 30, 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)

We study the exact time complexity of the Subset Sum problem. Our focus is on instances that lack additive structure in the sense that the sums one can form from the subsets of the given integers are not strongly concentrated on any particular integer value. We present a randomized algorithm that runs in O(2^0.3399nB^4) time on instances with the property that no value can arise as a sum of more than B different subsets of the n given integers.

Per Austrin, Petteri Kaski, Mikko Koivisto, and Jesper Nederlof. Subset Sum in the Absence of Concentration. In 32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 30, pp. 48-61, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{austrin_et_al:LIPIcs.STACS.2015.48, author = {Austrin, Per and Kaski, Petteri and Koivisto, Mikko and Nederlof, Jesper}, title = {{Subset Sum in the Absence of Concentration}}, booktitle = {32nd International Symposium on Theoretical Aspects of Computer Science (STACS 2015)}, pages = {48--61}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-78-1}, ISSN = {1868-8969}, year = {2015}, volume = {30}, editor = {Mayr, Ernst W. and Ollinger, Nicolas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2015.48}, URN = {urn:nbn:de:0030-drops-49034}, doi = {10.4230/LIPIcs.STACS.2015.48}, annote = {Keywords: subset sum, additive combinatorics, exponential-time algorithm, homomorphic hashing, Littlewood--Offord problem} }

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