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

VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where the size of each A_i is polynomial in the number of variables (equivalently, computable by polynomial-sized algebraic branching programs (ABP)). A major open problem in geometric complexity theory (GCT) is to determine whether VBP is closed under approximation i.e. whether VBP = VBP^ ̅. The power of approximation is well understood for some restricted models of computation, e.g. the class of depth-two circuits, read-once oblivious ABPs (ROABP), monotone ABPs, depth-three circuits of bounded top fan-in, and width-two ABPs. The former three classes are known to be closed under approximation [Markus Bläser et al., 2020], whereas the approximative closure of the last one captures the entire class of polynomial families computable by polynomial-sized formulas [Bringmann et al., 2017].
In this work, we consider the subclass of VBP computed by the determinant of a symbolic matrix of the form A_0 + ∑_{i=1}^n A_i x_i where for each 1 ≤ i ≤ n, A_i is of rank one. This class has been studied extensively [Edmonds, 1968; Jack Edmonds, 1979; Murota, 1993] and efficient identity testing algorithms are known for it [Lovász, 1989; Rohit Gurjar and Thomas Thierauf, 2020]. We show that this class is closed under approximation. In the language of algebraic geometry, we show that the set obtained by taking coordinatewise products of pairs of points from (the Plücker embedding of) a Grassmannian variety is closed.

Abhranil Chatterjee, Sumanta Ghosh, Rohit Gurjar, and Roshan Raj. Border Complexity of Symbolic Determinant Under Rank One Restriction. In 38th Computational Complexity Conference (CCC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 264, pp. 2:1-2:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)

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@InProceedings{chatterjee_et_al:LIPIcs.CCC.2023.2, author = {Chatterjee, Abhranil and Ghosh, Sumanta and Gurjar, Rohit and Raj, Roshan}, title = {{Border Complexity of Symbolic Determinant Under Rank One Restriction}}, booktitle = {38th Computational Complexity Conference (CCC 2023)}, pages = {2:1--2:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2023.2}, URN = {urn:nbn:de:0030-drops-182721}, doi = {10.4230/LIPIcs.CCC.2023.2}, annote = {Keywords: Border Complexity, Symbolic Determinant, Valuated Matroid} }

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RANDOM

**Published in:** LIPIcs, Volume 207, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)

We study the matroid intersection problem from the parallel complexity perspective. Given two matroids over the same ground set, the problem asks to decide whether they have a common base and its search version asks to find a common base, if one exists. Another widely studied variant is the weighted decision version where with the two matroids, we are given small weights on the ground set elements and a target weight W, and the question is to decide whether there is a common base of weight at least W. From the perspective of parallel complexity, the relation between the search and the decision versions is not well understood. We make a significant progress on this question by giving a pseudo-deterministic parallel (NC) algorithm for the search version that uses an oracle access to the weighted decision.
The notion of pseudo-deterministic NC was recently introduced by Goldwasser and Grossman [Shafi Goldwasser and Ofer Grossman, 2017], which is a relaxation of NC. A pseudo-deterministic NC algorithm for a search problem is a randomized NC algorithm that, for a given input, outputs a fixed solution with high probability. In case the given matroids are linearly representable, our result implies a pseudo-deterministic NC algorithm (without the weighted decision oracle). This resolves an open question posed by Anari and Vazirani [Nima Anari and Vijay V. Vazirani, 2020].

Sumanta Ghosh and Rohit Gurjar. Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 207, pp. 41:1-41:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

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@InProceedings{ghosh_et_al:LIPIcs.APPROX/RANDOM.2021.41, author = {Ghosh, Sumanta and Gurjar, Rohit}, title = {{Matroid Intersection: A Pseudo-Deterministic Parallel Reduction from Search to Weighted-Decision}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2021)}, pages = {41:1--41:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-207-5}, ISSN = {1868-8969}, year = {2021}, volume = {207}, editor = {Wootters, Mary and Sanit\`{a}, Laura}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2021.41}, URN = {urn:nbn:de:0030-drops-147342}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2021.41}, annote = {Keywords: Linear Matroid, Matroid Intersection, Parallel Complexity, Pseudo-deterministic NC} }

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RANDOM

**Published in:** LIPIcs, Volume 176, Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)

We give improved explicit constructions of hitting-sets for read-once oblivious algebraic branching programs (ROABPs) and related models. For ROABPs in an unknown variable order, our hitting-set has size polynomial in (nr)^{(log n)/(max{1, log log n-log log r})}d over a field whose characteristic is zero or large enough, where n is the number of variables, d is the individual degree, and r is the width of the ROABP. A similar improved construction works over fields of arbitrary characteristic with a weaker size bound.
Based on a result of Bisht and Saxena (2020), we also give an improved explicit construction of hitting-sets for sum of several ROABPs. In particular, when the characteristic of the field is zero or large enough, we give polynomial-size explicit hitting-sets for sum of constantly many log-variate ROABPs of width r = 2^{O(log d/log log d)}.
Finally, we give improved explicit hitting-sets for polynomials computable by width-r ROABPs in any variable order, also known as any-order ROABPs. Our hitting-set has polynomial size for width r up to 2^{O(log(nd)/log log(nd))} or 2^{O(log^{1-ε} (nd))}, depending on the characteristic of the field. Previously, explicit hitting-sets of polynomial size are unknown for r = ω(1).

Zeyu Guo and Rohit Gurjar. Improved Explicit Hitting-Sets for ROABPs. In Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 176, pp. 4:1-4:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{guo_et_al:LIPIcs.APPROX/RANDOM.2020.4, author = {Guo, Zeyu and Gurjar, Rohit}, title = {{Improved Explicit Hitting-Sets for ROABPs}}, booktitle = {Approximation, Randomization, and Combinatorial Optimization. Algorithms and Techniques (APPROX/RANDOM 2020)}, pages = {4:1--4:16}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-164-1}, ISSN = {1868-8969}, year = {2020}, volume = {176}, editor = {Byrka, Jaros{\l}aw and Meka, Raghu}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.APPROX/RANDOM.2020.4}, URN = {urn:nbn:de:0030-drops-126076}, doi = {10.4230/LIPIcs.APPROX/RANDOM.2020.4}, annote = {Keywords: polynomial identity testing, hitting-set, ROABP, arithmetic branching programs, derandomization} }

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Track A: Algorithms, Complexity and Games

**Published in:** LIPIcs, Volume 168, 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)

A set function f : 2^E → ℝ on the subsets of a set E is called submodular if it satisfies a natural diminishing returns property: for any S ⊆ E and x,y ∉ S, we have f(S ∪ {x,y}) - f(S ∪ {y}) ≤ f(S ∪ {x}) - f(S). Submodular minimization problem asks for finding the minimum value a given submodular function takes. We give an algebraic algorithm for this problem for a special class of submodular functions that are "linearly representable". It is known that every submodular function f can be decomposed into a sum of two monotone submodular functions, i.e., there exist two non-decreasing submodular functions f₁,f₂ such that f(S) = f₁(S) + f₂(E ⧵ S) for each S ⊆ E. Our class consists of those submodular functions f, for which each of f₁ and f₂ is a sum of k rank functions on families of subspaces of 𝔽ⁿ, for some field 𝔽.
Our algebraic algorithm for this class of functions can be parallelized, and thus, puts the problem of finding the minimizing set in the complexity class randomized NC. Further, we derandomize our algorithm so that it needs only O(log²(kn|E|)) many random bits.
We also give reductions from two combinatorial optimization problems to linearly representable submodular minimization, and thus, get such parallel algorithms for these problems. These problems are (i) covering a directed graph by k a-arborescences and (ii) packing k branchings with given root sets in a directed graph.

Rohit Gurjar and Rajat Rathi. Linearly Representable Submodular Functions: An Algebraic Algorithm for Minimization. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 61:1-61:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{gurjar_et_al:LIPIcs.ICALP.2020.61, author = {Gurjar, Rohit and Rathi, Rajat}, title = {{Linearly Representable Submodular Functions: An Algebraic Algorithm for Minimization}}, booktitle = {47th International Colloquium on Automata, Languages, and Programming (ICALP 2020)}, pages = {61:1--61:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-138-2}, ISSN = {1868-8969}, year = {2020}, volume = {168}, editor = {Czumaj, Artur and Dawar, Anuj and Merelli, Emanuela}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2020.61}, URN = {urn:nbn:de:0030-drops-124687}, doi = {10.4230/LIPIcs.ICALP.2020.61}, annote = {Keywords: Submodular Minimization, Parallel Algorithms, Derandomization, Algebraic Algorithms} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

We present a geometric approach towards derandomizing the {Isolation Lemma} by Mulmuley, Vazirani, and Vazirani. In particular, our approach produces a quasi-polynomial family of weights, where each weight is an integer and quasi-polynomially bounded, that can isolate a vertex in any 0/1 polytope for which each face lies in an affine space defined by a totally unimodular matrix. This includes the polytopes given by totally unimodular constraints and generalizes the recent derandomization of the Isolation Lemma for {bipartite perfect matching} and {matroid intersection}. We prove our result by associating a {lattice} to each face of the polytope and showing that if there is a totally unimodular kernel matrix for this lattice, then the number of vectors of length within 3/2 of the shortest vector in it is polynomially bounded. The proof of this latter geometric fact is combinatorial and follows from a polynomial bound on the number of circuits of size within 3/2 of the shortest circuit in a regular matroid. This is the technical core of the paper and relies on a variant of Seymour's decomposition theorem for regular matroids. It generalizes an influential result by Karger on the number of minimum cuts in a graph to regular matroids.

Rohit Gurjar, Thomas Thierauf, and Nisheeth K. Vishnoi. Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 74:1-74:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{gurjar_et_al:LIPIcs.ICALP.2018.74, author = {Gurjar, Rohit and Thierauf, Thomas and Vishnoi, Nisheeth K.}, title = {{Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {74:1--74:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.74}, URN = {urn:nbn:de:0030-drops-90782}, doi = {10.4230/LIPIcs.ICALP.2018.74}, annote = {Keywords: Derandomization, Isolation Lemma, Total unimodularity, Near-shortest vectors in Lattices, Regular matroids} }

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

We give improved hitting-sets for two special cases of Read-once Oblivious Arithmetic Branching Programs (ROABP). First is the case of an ROABP with known variable order. The best hitting-set known for this case had cost (nw)^{O(log(n))}, where n is the number of variables and w is the width of the ROABP. Even for a constant-width ROABP, nothing better than a quasi-polynomial bound was known. We improve the hitting-set complexity for the known-order case to n^{O(log(w))}. In particular, this gives the first polynomial time hitting-set for constant-width ROABP (known-order). However, our hitting-set works only over those fields whose characteristic is zero or large enough. To construct the hitting-set, we use the concept of the rank of partial derivative matrix. Unlike previous approaches whose starting point is a monomial map, we use a polynomial map directly.
The second case we consider is that of commutative ROABP. The best known hitting-set for this case had cost d^{O(log(w))}(nw)^{O(log(log(w)))}, where d is the individual degree. We improve this hitting-set complexity to (ndw)^{O(log(log(w)))}. We get this by achieving rank concentration more efficiently.

Rohit Gurjar, Arpita Korwar, and Nitin Saxena. Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 29:1-29:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{gurjar_et_al:LIPIcs.CCC.2016.29, author = {Gurjar, Rohit and Korwar, Arpita and Saxena, Nitin}, title = {{Identity Testing for Constant-Width, and Commutative, Read-Once Oblivious ABPs}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {29:1--29:16}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.29}, URN = {urn:nbn:de:0030-drops-58438}, doi = {10.4230/LIPIcs.CCC.2016.29}, annote = {Keywords: PIT, hitting-set, constant-width ROABPs, commutative ROABPs} }

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

The perfect matching problem has a randomized NC algorithm, using the celebrated Isolation Lemma of Mulmuley, Vazirani and Vazirani. The Isolation Lemma states that giving a random weight assignment to the edges of a graph ensures that it has a unique minimum weight perfect matching, with a good probability. We derandomize this lemma for K3,3-free and K5-free bipartite graphs. That is, we give a deterministic log-space construction of such a weight assignment for these graphs. Such a construction was known previously for planar bipartite graphs. Our result implies that the perfect matching problem for K3,3-free and K5-free bipartite graphs is in SPL. It also gives an alternate proof for an already known result – reachability for K3,3-free and K5-free graphs is in UL.

Rahul Arora, Ashu Gupta, Rohit Gurjar, and Raghunath Tewari. Derandomizing Isolation Lemma for K3,3-free and K5-free Bipartite Graphs. In 33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 47, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arora_et_al:LIPIcs.STACS.2016.10, author = {Arora, Rahul and Gupta, Ashu and Gurjar, Rohit and Tewari, Raghunath}, title = {{Derandomizing Isolation Lemma for K3,3-free and K5-free Bipartite Graphs}}, booktitle = {33rd Symposium on Theoretical Aspects of Computer Science (STACS 2016)}, pages = {10:1--10:15}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.STACS.2016.10}, URN = {urn:nbn:de:0030-drops-57116}, doi = {10.4230/LIPIcs.STACS.2016.10}, annote = {Keywords: bipartite matching, derandomization, isolation lemma, SPL, minor-free graph} }

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

A read-once oblivious arithmetic branching program (ROABP) is an arithmetic branching program (ABP) where each variable occurs in at most one layer. We give the first polynomial time whitebox identity test for a polynomial computed by a sum of constantly many ROABPs. We also give a corresponding blackbox algorithm with quasi-polynomial time complexity n^(O(log(n))). In both the cases, our time complexity is double exponential in the number of ROABPs.
ROABPs are a generalization of set-multilinear depth-3 circuits. The prior results for the sum of constantly many set-multilinear depth-3 circuits were only slightly better than brute-force, i.e. exponential-time.
Our techniques are a new interplay of three concepts for ROABP: low evaluation dimension, basis isolating weight assignment and low-support rank concentration. We relate basis isolation to rank concentration and extend it to a sum of two ROABPs using evaluation dimension (or partial derivatives).

Rohit Gurjar, Arpita Korwar, Nitin Saxena, and Thomas Thierauf. Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs. In 30th Conference on Computational Complexity (CCC 2015). Leibniz International Proceedings in Informatics (LIPIcs), Volume 33, pp. 323-346, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2015)

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@InProceedings{gurjar_et_al:LIPIcs.CCC.2015.323, author = {Gurjar, Rohit and Korwar, Arpita and Saxena, Nitin and Thierauf, Thomas}, title = {{Deterministic Identity Testing for Sum of Read-once Oblivious Arithmetic Branching Programs}}, booktitle = {30th Conference on Computational Complexity (CCC 2015)}, pages = {323--346}, 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-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2015.323}, URN = {urn:nbn:de:0030-drops-50647}, doi = {10.4230/LIPIcs.CCC.2015.323}, annote = {Keywords: PIT, Hitting-set, Sum of ROABPs, Evaluation Dimension, Rank Concentration} }

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