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**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

When a group acts on a set, it naturally partitions it into orbits, giving rise to orbit problems. These are natural algorithmic problems, as symmetries are central in numerous questions and structures in physics, mathematics, computer science, optimization, and more. Accordingly, it is of high interest to understand their computational complexity. Recently, Bürgisser et al. (2021) gave the first polynomial-time algorithms for orbit problems of torus actions, that is, actions of commutative continuous groups on Euclidean space. In this work, motivated by theoretical and practical applications, we study the computational complexity of robust generalizations of these orbit problems, which amount to approximating the distance of orbits in ℂⁿ up to a factor γ ≥ 1. In particular, this allows deciding whether two inputs are approximately in the same orbit or far from being so. On the one hand, we prove the NP-hardness of this problem for γ = n^Ω(1/log log n) by reducing the closest vector problem for lattices to it. On the other hand, we describe algorithms for solving this problem for an approximation factor γ = exp(poly(n)). Our algorithms combine tools from invariant theory and algorithmic lattice theory, and they also provide group elements witnessing the proximity of the given orbits (in contrast to the algebraic algorithms of prior work). We prove that they run in polynomial time if and only if a version of the famous number-theoretic abc-conjecture holds - establishing a new and surprising connection between computational complexity and number theory.

Peter Bürgisser, Mahmut Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 14:1-14:48, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{burgisser_et_al:LIPIcs.CCC.2024.14, author = {B\"{u}rgisser, Peter and Do\u{g}an, Mahmut Levent and Makam, Visu and Walter, Michael and Wigderson, Avi}, title = {{Complexity of Robust Orbit Problems for Torus Actions and the abc-Conjecture}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {14:1--14:48}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.14}, URN = {urn:nbn:de:0030-drops-204100}, doi = {10.4230/LIPIcs.CCC.2024.14}, annote = {Keywords: computational invariant theory, geometric complexity theory, orbit problems, abc-conjecture, closest vector problem} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

The Stabbing Planes proof system [Paul Beame et al., 2018] was introduced to model the reasoning carried out in practical mixed integer programming solvers. As a proof system, it is powerful enough to simulate Cutting Planes and to refute the Tseitin formulas - certain unsatisfiable systems of linear equations od 2 - which are canonical hard examples for many algebraic proof systems. In a recent (and surprising) result, Dadush and Tiwari [Daniel Dadush and Samarth Tiwari, 2020] showed that these short refutations of the Tseitin formulas could be translated into quasi-polynomial size and depth Cutting Planes proofs, refuting a long-standing conjecture. This translation raises several interesting questions. First, whether all Stabbing Planes proofs can be efficiently simulated by Cutting Planes. This would allow for the substantial analysis done on the Cutting Planes system to be lifted to practical mixed integer programming solvers. Second, whether the quasi-polynomial depth of these proofs is inherent to Cutting Planes.
In this paper we make progress towards answering both of these questions. First, we show that any Stabbing Planes proof with bounded coefficients (SP*) can be translated into Cutting Planes. As a consequence of the known lower bounds for Cutting Planes, this establishes the first exponential lower bounds on SP*. Using this translation, we extend the result of Dadush and Tiwari to show that Cutting Planes has short refutations of any unsatisfiable system of linear equations over a finite field. Like the Cutting Planes proofs of Dadush and Tiwari, our refutations also incur a quasi-polynomial blow-up in depth, and we conjecture that this is inherent. As a step towards this conjecture, we develop a new geometric technique for proving lower bounds on the depth of Cutting Planes proofs. This allows us to establish the first lower bounds on the depth of Semantic Cutting Planes proofs of the Tseitin formulas.

Noah Fleming, Mika Göös, Russell Impagliazzo, Toniann Pitassi, Robert Robere, Li-Yang Tan, and Avi Wigderson. On the Power and Limitations of Branch and Cut. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 6:1-6:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{fleming_et_al:LIPIcs.CCC.2021.6, author = {Fleming, Noah and G\"{o}\"{o}s, Mika and Impagliazzo, Russell and Pitassi, Toniann and Robere, Robert and Tan, Li-Yang and Wigderson, Avi}, title = {{On the Power and Limitations of Branch and Cut}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {6:1--6:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.6}, URN = {urn:nbn:de:0030-drops-142809}, doi = {10.4230/LIPIcs.CCC.2021.6}, annote = {Keywords: Proof Complexity, Integer Programming, Cutting Planes, Branch and Cut, Stabbing Planes} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

A graph G is called self-ordered (a.k.a asymmetric) if the identity permutation is its only automorphism. Equivalently, there is a unique isomorphism from G to any graph that is isomorphic to G. We say that G = (V,E) is robustly self-ordered if the size of the symmetric difference between E and the edge-set of the graph obtained by permuting V using any permutation π:V → V is proportional to the number of non-fixed-points of π. In this work, we initiate the study of the structure, construction and utility of robustly self-ordered graphs.
We show that robustly self-ordered bounded-degree graphs exist (in abundance), and that they can be constructed efficiently, in a strong sense. Specifically, given the index of a vertex in such a graph, it is possible to find all its neighbors in polynomial-time (i.e., in time that is poly-logarithmic in the size of the graph).
We provide two very different constructions, in tools and structure. The first, a direct construction, is based on proving a sufficient condition for robust self-ordering, which requires that an auxiliary graph is expanding. The second construction is iterative, boosting the property of robust self-ordering from smaller to larger graphs. Structuraly, the first construction always yields expanding graphs, while the second construction may produce graphs that have many tiny (sub-logarithmic) connected components.
We also consider graphs of unbounded degree, seeking correspondingly unbounded robustness parameters. We again demonstrate that such graphs (of linear degree) exist (in abundance), and that they can be constructed efficiently, in a strong sense. This turns out to require very different tools. Specifically, we show that the construction of such graphs reduces to the construction of non-malleable two-source extractors (with very weak parameters but with some additional natural features).
We demonstrate that robustly self-ordered bounded-degree graphs are useful towards obtaining lower bounds on the query complexity of testing graph properties both in the bounded-degree and the dense graph models. Indeed, their robustness offers efficient, local and distance preserving reductions from testing problems on ordered structures (like sequences) to the unordered (effectively unlabeled) graphs. One of the results that we obtain, via such a reduction, is a subexponential separation between the query complexities of testing and tolerant testing of graph properties in the bounded-degree graph model.

Oded Goldreich and Avi Wigderson. Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 12:1-12:74, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{goldreich_et_al:LIPIcs.CCC.2021.12, author = {Goldreich, Oded and Wigderson, Avi}, title = {{Robustly Self-Ordered Graphs: Constructions and Applications to Property Testing}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {12:1--12:74}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.12}, URN = {urn:nbn:de:0030-drops-142867}, doi = {10.4230/LIPIcs.CCC.2021.12}, annote = {Keywords: Asymmetric graphs, expanders, testing graph properties, two-source extractors, non-malleable extractors, coding theory, tolerant testing, random graphs} }

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**Published in:** LIPIcs, Volume 200, 36th Computational Complexity Conference (CCC 2021)

An action of a group on a vector space partitions the latter into a set of orbits. We consider three natural and useful algorithmic "isomorphism" or "classification" problems, namely, orbit equality, orbit closure intersection, and orbit closure containment. These capture and relate to a variety of problems within mathematics, physics and computer science, optimization and statistics. These orbit problems extend the more basic null cone problem, whose algorithmic complexity has seen significant progress in recent years.
In this paper, we initiate a study of these problems by focusing on the actions of commutative groups (namely, tori). We explain how this setting is motivated from questions in algebraic complexity, and is still rich enough to capture interesting combinatorial algorithmic problems. While the structural theory of commutative actions is well understood, no general efficient algorithms were known for the aforementioned problems. Our main results are polynomial time algorithms for all three problems. We also show how to efficiently find separating invariants for orbits, and how to compute systems of generating rational invariants for these actions (in contrast, for polynomial invariants the latter is known to be hard). Our techniques are based on a combination of fundamental results in invariant theory, linear programming, and algorithmic lattice theory.

Peter Bürgisser, M. Levent Doğan, Visu Makam, Michael Walter, and Avi Wigderson. Polynomial Time Algorithms in Invariant Theory for Torus Actions. In 36th Computational Complexity Conference (CCC 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 200, pp. 32:1-32:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{burgisser_et_al:LIPIcs.CCC.2021.32, author = {B\"{u}rgisser, Peter and Do\u{g}an, M. Levent and Makam, Visu and Walter, Michael and Wigderson, Avi}, title = {{Polynomial Time Algorithms in Invariant Theory for Torus Actions}}, booktitle = {36th Computational Complexity Conference (CCC 2021)}, pages = {32:1--32:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-193-1}, ISSN = {1868-8969}, year = {2021}, volume = {200}, editor = {Kabanets, Valentine}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2021.32}, URN = {urn:nbn:de:0030-drops-143062}, doi = {10.4230/LIPIcs.CCC.2021.32}, annote = {Keywords: computational invariant theory, geometric complexity theory, orbit closure intersection problem} }

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**Published in:** LIPIcs, Volume 169, 35th Computational Complexity Conference (CCC 2020)

We consider the problem of computing succinct encodings of lists of generators for invariant rings for group actions. Mulmuley conjectured that there are always polynomial sized such encodings for invariant rings of SL_n(ℂ)-representations. We provide simple examples that disprove this conjecture (under standard complexity assumptions).
We develop a general framework, denoted algebraic circuit search problems, that captures many important problems in algebraic complexity and computational invariant theory. This framework encompasses various proof systems in proof complexity and some of the central problems in invariant theory as exposed by the Geometric Complexity Theory (GCT) program, including the aforementioned problem of computing succinct encodings for generators for invariant rings.

Ankit Garg, Christian Ikenmeyer, Visu Makam, Rafael Oliveira, Michael Walter, and Avi Wigderson. Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings. In 35th Computational Complexity Conference (CCC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 169, pp. 12:1-12:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{garg_et_al:LIPIcs.CCC.2020.12, author = {Garg, Ankit and Ikenmeyer, Christian and Makam, Visu and Oliveira, Rafael and Walter, Michael and Wigderson, Avi}, title = {{Search Problems in Algebraic Complexity, GCT, and Hardness of Generators for Invariant Rings}}, booktitle = {35th Computational Complexity Conference (CCC 2020)}, pages = {12:1--12:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-156-6}, ISSN = {1868-8969}, year = {2020}, volume = {169}, editor = {Saraf, Shubhangi}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2020.12}, URN = {urn:nbn:de:0030-drops-125645}, doi = {10.4230/LIPIcs.CCC.2020.12}, annote = {Keywords: generators for invariant rings, succinct encodings} }

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

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

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

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

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Arithmetic complexity, the study of the cost of computing polynomials via additions and multiplications, is considered (for many good reasons) simpler to understand than Boolean complexity, namely computing Boolean functions via logical gates. And indeed, we seem to have significantly more lower bound techniques and results in arithmetic complexity than in Boolean complexity. Despite many successes and rapid progress, however, foundational challenges, like proving super-polynomial lower bounds on circuit or formula size for explicit polynomials, or super-linear lower bounds on explicit 3-dimensional tensors, remain elusive.
At the same time (and possibly for similar reasons), we have plenty more excuses, in the form of "barrier results" for failing to prove basic lower bounds in Boolean complexity than in arithmetic complexity. Efforts to find barriers to arithmetic lower bound techniques seem harder, and despite some attempts we have no excuses of similar quality for these failures in arithmetic complexity. This paper aims to add to this study.
In this paper we address rank methods, which were long recognized as encompassing and abstracting almost all known arithmetic lower bounds to-date, including the most recent impressive successes. Rank methods (under the name of flattenings) are also in wide use in algebraic geometry for proving tensor rank and symmetric tensor rank lower bounds. Our main results are barriers to these methods. In particular,
1. Rank methods cannot prove better than (2^d)*n^(d/2) lower bound on the tensor rank of any d-dimensional tensor of side n. (In particular, they cannot prove super-linear, indeed even >8n tensor rank lower bounds for any 3-dimensional tensors.)
2. Rank methods cannot prove (d+1)n^(d/2) on the Waring rank of any n-variate polynomial of degree d. (In particular, they cannot prove such lower bounds on stronger models, including depth-3 circuits.)
The proofs of these bounds use simple linear-algebraic arguments, leveraging connections between the symbolic rank of matrix polynomials and the usual rank of their evaluations. These techniques can perhaps be extended to barriers for other arithmetic models on which progress has halted.
To see how these barrier results directly inform the state-of-art in arithmetic complexity we note the following.
First, the bounds above nearly match the best explicit bounds we know for these models, hence offer an explanations why the rank methods got stuck there. Second, the bounds above are a far cry (quadratically away) from the true complexity (e.g. of random polynomials) in these models, which if achieved (by any methods), are known to imply super-polynomial formula lower bounds.
We also explain the relation of our barrier results to other attempts, and in particular how they significantly differ from the recent attempts to find analogues of "natural proofs" for arithmetic complexity. Finally, we discuss the few arithmetic lower bound approaches which fall outside rank methods, and some natural directions our barriers suggest.

Klim Efremenko, Ankit Garg, Rafael Oliveira, and Avi Wigderson. Barriers for Rank Methods in Arithmetic Complexity. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 1:1-1:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{efremenko_et_al:LIPIcs.ITCS.2018.1, author = {Efremenko, Klim and Garg, Ankit and Oliveira, Rafael and Wigderson, Avi}, title = {{Barriers for Rank Methods in Arithmetic Complexity}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {1:1--1:19}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.1}, URN = {urn:nbn:de:0030-drops-83506}, doi = {10.4230/LIPIcs.ITCS.2018.1}, annote = {Keywords: Lower Bounds, Barriers, Partial Derivatives, Flattenings, Algebraic Complexity} }

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**Published in:** LIPIcs, Volume 94, 9th Innovations in Theoretical Computer Science Conference (ITCS 2018)

Alternating minimization heuristics seek to solve a (difficult) global optimization task through iteratively solving a sequence of (much easier) local optimization tasks on different parts (or blocks) of the input parameters. While popular and widely applicable, very few examples of this heuristic are rigorously shown to converge to optimality, and even fewer to do so efficiently.
In this paper we present a general framework which is amenable to rigorous analysis, and expose its applicability. Its main feature is that the local optimization domains are each a group of invertible matrices, together naturally acting on tensors, and the optimization problem is minimizing the norm of an input tensor under this joint action. The solution of this optimization problem captures a basic problem in Invariant Theory, called the null-cone problem.
This algebraic framework turns out to encompass natural computational problems in combinatorial optimization, algebra, analysis, quantum information theory, and geometric complexity theory. It includes and extends to high dimensions the recent advances on (2-dimensional) operator scaling.
Our main result is a fully polynomial time approximation scheme for this general problem, which may be viewed as a multi-dimensional scaling algorithm. This directly leads to progress on some of the problems in the areas above, and a unified view of others. We explain how faster convergence of an algorithm for the same problem will allow resolving central open problems.
Our main techniques come from Invariant Theory, and include its rich non-commutative duality theory, and new bounds on the bitsizes of coefficients of invariant polynomials. They enrich the algorithmic toolbox of this very computational field of mathematics, and are directly related to some challenges in geometric complexity theory (GCT).

Peter Bürgisser, Ankit Garg, Rafael Oliveira, Michael Walter, and Avi Wigderson. Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory. In 9th Innovations in Theoretical Computer Science Conference (ITCS 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 94, pp. 24:1-24:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{burgisser_et_al:LIPIcs.ITCS.2018.24, author = {B\"{u}rgisser, Peter and Garg, Ankit and Oliveira, Rafael and Walter, Michael and Wigderson, Avi}, title = {{Alternating Minimization, Scaling Algorithms, and the Null-Cone Problem from Invariant Theory}}, booktitle = {9th Innovations in Theoretical Computer Science Conference (ITCS 2018)}, pages = {24:1--24:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-060-6}, ISSN = {1868-8969}, year = {2018}, volume = {94}, editor = {Karlin, Anna R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2018.24}, URN = {urn:nbn:de:0030-drops-83510}, doi = {10.4230/LIPIcs.ITCS.2018.24}, annote = {Keywords: alternating minimization, tensors, scaling, quantum marginal problem, null cone, invariant theory, geometric complexity theory} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

The sensitivity of a Boolean function f is the maximum, over all inputs x, of the number of sensitive coordinates of x (namely the number of Hamming neighbors of x with different f-value). The well-known sensitivity conjecture of Nisan (see also Nisan and Szegedy) states that every sensitivity-s Boolean function can be computed by a polynomial over the reals of degree s^{O(1)}. The best known upper bounds on degree, however, are exponential rather than polynomial in s.
Our main result is an approximate version of the conjecture: every Boolean function with sensitivity s can be eps-approximated (in l_2) by a polynomial whose degree is s * polylog(1/eps). This is the first improvement on the folklore bound of s/eps. We prove this via a new "switching lemma for low-sensitivity functions" which establishes that a random restriction of a low-sensitivity function is very likely to have low decision tree depth. This is analogous to the well-known switching lemma for AC^0 circuits.
Our proof analyzes the combinatorial structure of the graph G_f of sensitive edges of a Boolean function f. Understanding the structure of this graph is of independent interest as a means of understanding Boolean functions. We propose several new complexity measures for Boolean functions based on this graph, including tree sensitivity and component dimension, which may be viewed as relaxations of worst-case sensitivity, and we introduce some new techniques, such as proper walks and shifting, to analyze these measures. We use these notions to show that the graph of a function of full degree must be sufficiently complex, and that random restrictions of low-sensitivity functions are unlikely to lead to such complex graphs.
We postulate a robust analogue of the sensitivity conjecture: if most inputs to a Boolean function f have low sensitivity, then most of the Fourier mass of f is concentrated on small subsets. We prove a lower bound on tree sensitivity in terms of decision tree depth, and show that a polynomial strengthening of this lower bound implies the robust conjecture. We feel that studying the graph G_f is interesting in its own right, and we hope that some of the notions and techniques we introduce in this work will be of use in its further study.

Parikshit Gopalan, Rocco A. Servedio, and Avi Wigderson. Degree and Sensitivity: Tails of Two Distributions. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 13:1-13:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gopalan_et_al:LIPIcs.CCC.2016.13, author = {Gopalan, Parikshit and Servedio, Rocco A. and Wigderson, Avi}, title = {{Degree and Sensitivity: Tails of Two Distributions}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {13:1--13:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.13}, URN = {urn:nbn:de:0030-drops-58488}, doi = {10.4230/LIPIcs.CCC.2016.13}, annote = {Keywords: Boolean functions, random restrictions, Fourier analysis} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We give upper and lower bounds on the power of subsystems of the Ideal Proof System (IPS), the algebraic proof system recently proposed by Grochow and Pitassi, where the circuits comprising the proof come from various restricted algebraic circuit classes. This mimics an established research direction in the boolean setting for subsystems of Extended Frege proofs whose lines are circuits from restricted boolean circuit classes. Essentially all of the subsystems considered in this paper can simulate the well-studied Nullstellensatz proof system, and prior to this work there were no known lower bounds when measuring proof size by the algebraic complexity of the polynomials (except with respect to degree, or to sparsity).
Our main contributions are two general methods of converting certain algebraic lower bounds into proof complexity ones. Both require stronger arithmetic lower bounds than common, which should hold not for a specific polynomial but for a whole family defined by it. These may be likened to some of the methods by which Boolean circuit lower bounds are turned into related proof-complexity ones, especially the "feasible interpolation" technique. We establish algebraic lower bounds of these forms for several explicit polynomials, against a variety of classes, and infer the relevant proof complexity bounds. These yield separations between IPS subsystems, which we complement by simulations to create a partial structure theory for IPS systems.
Our first method is a functional lower bound, a notion of Grigoriev and Razborov, which is a function f' from n-bit strings to a field, such that any polynomial f agreeing with f' on the boolean cube requires large algebraic circuit complexity. We develop functional lower bounds for a variety of circuit classes (sparse polynomials, depth-3 powering formulas, read-once algebraic branching programs and multilinear formulas) where f'(x) equals 1/p(x) for a constant-degree polynomial p depending on the relevant circuit class. We believe these lower bounds are of independent interest in algebraic complexity, and show that they also imply lower bounds for the size of the corresponding IPS refutations for proving that the relevant polynomial p is non-zero over the boolean cube. In particular, we show super-polynomial lower bounds for refuting variants of the subset-sum axioms in these IPS subsystems.
Our second method is to give lower bounds for multiples, that is, to give explicit polynomials whose all (non-zero) multiples require large algebraic circuit complexity. By extending known techniques, we give lower bounds for multiples for various restricted circuit classes such sparse polynomials, sums of powers of low-degree polynomials, and roABPs. These results are of independent interest, as we argue that lower bounds for multiples is the correct notion for instantiating the algebraic hardness versus randomness paradigm of Kabanets and Impagliazzo. Further, we show how such lower bounds for multiples extend to lower bounds for refutations in the corresponding IPS subsystem.

Michael A. Forbes, Amir Shpilka, Iddo Tzameret, and Avi Wigderson. Proof Complexity Lower Bounds from Algebraic Circuit Complexity. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 32:1-32:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{forbes_et_al:LIPIcs.CCC.2016.32, author = {Forbes, Michael A. and Shpilka, Amir and Tzameret, Iddo and Wigderson, Avi}, title = {{Proof Complexity Lower Bounds from Algebraic Circuit Complexity}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {32:1--32:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.32}, URN = {urn:nbn:de:0030-drops-58321}, doi = {10.4230/LIPIcs.CCC.2016.32}, annote = {Keywords: Proof Complexity, Algebraic Complexity, Nullstellensatz, Subset-Sum} }

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**Published in:** LIPIcs, Volume 33, 30th Conference on Computational Complexity (CCC 2015)

We consider randomness extraction by AC0 circuits. The main parameter, n, is the length of the source, and all other parameters are functions of it. The additional extraction parameters are the min-entropy bound k=k(n), the seed length r=r(n), the output length m=m(n), and the (output) deviation bound epsilon=epsilon(n).
For k <=e n/\log^(omega(1))(n), we show that AC0-extraction is possible if and only if m/r <= 1+ poly(log(n)) * k/n; that is, the extraction rate m/r exceeds the trivial rate (of one) by an additive amount that is proportional to the min-entropy rate k/n. In particular, non-trivial AC0-extraction (i.e., m >= r+1) is possible if and only if k * r > n/poly(log(n)). For k >= n/log^(O(1))(n),
we show that AC0-extraction of r+Omega(r) bits is possible when r=O(log(n)), but leave open the question of whether more bits can be extracted in this case.
The impossibility result is for constant epsilon, and the possibility result supports epsilon=1/poly(n). The impossibility result is for (possibly) non-uniform AC0, whereas the possibility result hold for uniform AC0. All our impossibility results hold even for the model of bit-fixing sources, where k coincides with the number of non-fixed (i.e., random) bits.
We also consider deterministic AC0 extraction from various classes of restricted sources. In particular, for any constant $\delta>0$, we give explicit AC0 extractors for poly(1/delta) independent sources that are each of min-entropy rate delta; and four sources suffice for delta=0.99. Also, we give non-explicit AC0 extractors for bit-fixing sources of entropy rate 1/poly(log(n)) (i.e., having n/poly(log(n)) unfixed bits). This shows that the known analysis of the "restriction method" (for making a circuit constant by fixing as few variables as possible) is tight for AC0 even if the restriction is picked deterministically depending on the circuit.

Oded Goldreich, Emanuele Viola, and Avi Wigderson. On Randomness Extraction in AC0. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 601-668, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{goldreich_et_al:LIPIcs.CCC.2015.601, author = {Goldreich, Oded and Viola, Emanuele and Wigderson, Avi}, title = {{On Randomness Extraction in AC0}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {601--668}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-81-1}, ISSN = {1868-8969}, year = {2015}, volume = {33}, editor = {Zuckerman, David}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.601}, URN = {urn:nbn:de:0030-drops-50515}, doi = {10.4230/LIPIcs.CCC.2015.601}, annote = {Keywords: AC0, randomness extractors, general min-entropy sources, block sources, bit-fixing sources, multiple-source extraction} }

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**Published in:** LIPIcs, Volume 4, IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science (2009)

Randomness extractors are efficient algorithms which convert weak random sources into nearly perfect ones. While such purification of randomness was the original motivation for constructing extractors, these constructions turn out to have strong pseudorandom properties which found applications in diverse areas of computer science and combinatorics. We will highlight some of the
applications, as well as recent constructions achieving near-optimal
extraction.

Avi Wigderson. Randomness extractors -- applications and constructions. In IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science. Leibniz International Proceedings in Informatics (LIPIcs), Volume 4, pp. 471-473, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2009)

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@InProceedings{wigderson:LIPIcs.FSTTCS.2009.2340, author = {Wigderson, Avi}, title = {{Randomness extractors -- applications and constructions}}, booktitle = {IARCS Annual Conference on Foundations of Software Technology and Theoretical Computer Science}, pages = {471--473}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-939897-13-2}, ISSN = {1868-8969}, year = {2009}, volume = {4}, editor = {Kannan, Ravi and Narayan Kumar, K.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.FSTTCS.2009.2340}, URN = {urn:nbn:de:0030-drops-23407}, doi = {10.4230/LIPIcs.FSTTCS.2009.2340}, annote = {Keywords: Randomness extractors, weak random sources, purification of randomness} }

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