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Rohit Gurjar

Publications and source records attributed to Rohit Gurjar.

18 recordsLinked to original sources
arXiv2026

Fair Allocation under Conflict Constraints

We study the fair allocation of indivisible items subject to conflict constraints. In this framework, the items are represented as the vertices of a graph, with edges corresponding to conflicts between pairs of items. Each agent is assigned an independent set of items from the graph. Our goal is to achieve a fair and efficient allocation of these items. Fairness pertains to satisfying envy-freeness up to one item (EF1), while efficiency is defined by maximality, meaning that no unallocated item can be feasibly assigned to any agent. First, we explore the case of two agents. For monotone valuations, we show that a maximal EF1 allocation always exists on any graph. Our existence proof relies on a color-switching technique, which locally modifies a maximal allocation while preserving feasibility and restoring EF1. We further show that such allocations can be computed in pseudopolynomial time in general, and in polynomial time for additive valuations on arbitrary graphs, as well as for monotone valuations on interval and bipartite graphs. By contrast, once monotonicity is dropped, maximal EF1 allocations need not exist even for identical additive valuations, and deciding existence becomes NP-hard. Next, we consider the case with a general number of agents. Again, we arrive at a negative result: An EF1 and maximal allocation fails to exist even for three agents under identical monotone valuations, and determining the existence of such an allocation is NP-hard. On the positive side, we show that under identical non-monotone additive valuations on a path graph, an EF[1,1] and maximal allocation always exists. This result involves a novel application of the "cycle plus triangles" theorem.

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arXiv2026

Planarizing Gadgets for (k, l)-tight Graphs Do Not Exist

The problem of recognizing (k, l)-tight graphs is a fundamental problem that has close connections to well studied problems like graph rigidity. The problem is better understood for planar graphs as compared to general graphs. For example, deterministic NC-algorithms for the problem are known for planar graphs, but no such algorithm is known for general graphs. A common approach to reduce a graph problem to the planar case is to use planarizing gadgets. Our main contribution is to show that, unconditionally, planarizing gadgets for the problem of recognizing (k, l)-tight graphs do not exist.

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arXiv2026

Learning Read-Once Determinants and the Principal Minor Assignment Problem

A symbolic determinant under rank-one restriction computes a polynomial of the form $\det(A_0+A_1y_1+\ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are square matrices over a field $\mathbb{F}$ and $rank(A_i)=1$ for each $i\in[n]$. This class of polynomials has been studied extensively, since the work of Edmonds (1967), in the context of linear matroids, matching, matrix completion and polynomial identity testing. We study the following learning problem for this class: Given black-box access to an $n$-variate polynomial $f=\det(A_0+A_1y_1+ \ldots+A_ny_n)$, where $A_0,A_1,\ldots,A_n$ are unknown square matrices over $\mathbb{F}$ and rank$(A_i)=1$ for each $i\in[n]$, find a square matrix $B_0$ and rank-one square matrices $B_1,\ldots,B_n$ over $\mathbb{F}$ such that $f=\det(B_0+B_1y_1+\ldots+B_ny_n)$. In this work, we give a randomized poly(n) time algorithm to solve this problem. As the above-mentioned class is known to be equivalent to the class of read-once determinants (RODs), we will refer to the problem as learning RODs. The algorithm for learning RODs is obtained by connecting with a well-known open problem in linear algebra, namely the Principal Minor Assignment Problem (PMAP), which asks to find (if possible) a matrix having prescribed principal minors. PMAP has also been studied in machine learning to learn the kernel matrix of a determinantal point process. Here, we study a natural black-box version of PMAP: Given black-box access to an $n$-variate polynomial $f = \det(A + Y)$, where $A \in \mathbb{F}^{n \times n}$ is unknown and $Y = diag(y_1,\ldots,y_n)$, find a $B\in\mathbb{F}^{n\times n}$ such that $f=det(B+Y)$. We show that black-box PMAP can be solved in randomized poly(n) time, and further, it is randomized polynomial-time equivalent to learning RODs. We resolve black-box PMAP by investigating a property of dense matrices that we call the rank-one extension property.

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arXiv2024

Characterizing and Testing Principal Minor Equivalence of Matrices

Two matrices are said to be principal minor equivalent if they have equal corresponding principal minors of all orders. We give a characterization of principal minor equivalence and a deterministic polynomial time algorithm to check if two given matrices are principal minor equivalent. Earlier such results were known for certain special cases like symmetric matrices, skew-symmetric matrices with {0, 1, -1}-entries, and matrices with no cuts (i.e., for any non-trivial partition of the indices, the top right block or the bottom left block must have rank more than 1). As an immediate application, we get an algorithm to check if the determinantal point processes corresponding to two given kernel matrices (not necessarily symmetric) are the same. As another application, we give a deterministic polynomial-time test to check equality of two multivariate polynomials, each computed by a symbolic determinant with a rank 1 constraint on coefficient matrices.

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arXiv2024

Geometric Bipartite Matching is in NC

In this work, we study the parallel complexity of the Euclidean minimum-weight perfect matching (EWPM) problem. Here our graph is the complete bipartite graph $G$ on two sets of points $A$ and $B$ in $\mathbb{R}^2$ and the weight of each edge is the Euclidean distance between the corresponding points. The weighted perfect matching problem on general bipartite graphs is known to be in RNC [Mulmuley, Vazirani, and Vazirani, 1987], and Quasi-NC [Fenner, Gurjar, and Thierauf, 2016]. Both of these results work only when the weights are of $O(\log n)$ bits. It is a long-standing open question to show the problem to be in NC. First, we show that for EWPM, a linear number of bits of approximation is required to distinguish between the minimum-weight perfect matching and other perfect matchings. Next, we show that the EWPM problem that allows up to $\frac{1}{poly(n)}$ additive error, is in NC.

cs.CG↗
arXiv2024

Fractional Linear Matroid Matching is in quasi-NC

The matching and linear matroid intersection problems are solvable in quasi-NC, meaning that there exist deterministic algorithms that run in polylogarithmic time and use quasi-polynomially many parallel processors. However, such a parallel algorithm is unknown for linear matroid matching, which generalizes both of these problems. In this work, we propose a quasi-NC algorithm for fractional linear matroid matching, which is a relaxation of linear matroid matching and commonly generalizes fractional matching and linear matroid intersection. Our algorithm builds upon the connection of fractional matroid matching to non-commutative Edmonds' problem recently revealed by Oki and Soma~(2023). As a corollary, we also solve black-box non-commutative Edmonds' problem with rank-two skew-symmetric coefficients.

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arXiv2024

Fair Interval Scheduling of Indivisible Chores

We study the problem of fairly assigning a set of discrete tasks (or chores) among a set of agents with additive valuations. Each chore is associated with a start and finish time, and each agent can perform at most one chore at any given time. The goal is to find a fair and efficient schedule of the chores, where fairness pertains to satisfying envy-freeness up to one chore (EF1) and efficiency pertains to maximality (i.e., no unallocated chore can be feasibly assigned to any agent). Our main result is a polynomial-time algorithm for computing an EF1 and maximal schedule for two agents under monotone valuations when the conflict constraints constitute an arbitrary interval graph. The algorithm uses a coloring technique in interval graphs that may be of independent interest. For an arbitrary number of agents with identical additive valuations, we show the existence of an EF1 and maximal schedule when the constraints constitute a path graph. This result uses a reduction to the ``cycle-plus-triangles'' theorem. Using different techniques, we provide an efficient algorithm for finding such a schedule when there are four or more agents and the valuations are further assumed to be dichotomous. We also show that stronger fairness and efficiency properties, including envy-freeness up to any chore (EFX) along with maximality and EF1 along with Pareto optimality, cannot be achieved.

cs.GT↗
arXiv2023

Border Complexity of Symbolic Determinant under Rank One Restriction

VBP is the class of polynomial families that can be computed by the determinant of a symbolic matrix of the form $A_0 + \sum_{i=1}^n A_ix_i$ 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. 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 [Bl"{a}ser, Ikenmeyer, Mahajan, Pandey, and Saurabh (2020)], whereas the approximative closure of the last one captures the whole class of polynomial families computable by polynomial-sized formulas [Bringmann, Ikenmeyer, and Zuiddam (2017)]. In this work, we consider the subclass of VBP computed by the determinant of a symbolic matrix of the form $A_0 + \sum_{i=1}^n A_ix_i$ where for each $1\leq i \leq n$, $A_i$ is of rank one. It has been studied extensively [Edmonds(1968), Edmonds(1979)] and efficient identity testing algorithms are known [Lov"{a}sz (1989), Gurjar and 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.

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arXiv2018

On the Number of Circuits in Regular Matroids (with Connections to Lattices and Codes)

We show that for any regular matroid on $m$ elements and any $α\geq 1$, the number of $α$-minimum circuits, or circuits whose size is at most an $α$-multiple of the minimum size of a circuit in the matroid is bounded by $m^{O(α^2)}$. This generalizes a result of Karger for the number of $α$-minimum cuts in a graph. As a consequence, we obtain similar bounds on the number of $α$-shortest vectors in "totally unimodular" lattices and on the number of $α$-minimum weight codewords in "regular" codes.

cs.DS↗
arXiv2018

Identity Testing for Constant-Width, and Any-Order, Read-Once Oblivious Arithmetic Branching Programs

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 order of the variables. The best previously known hitting set for this case had size $(nw)^{O(\log n)}$ 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 size for the known-order case to $n^{O(\log w)}$. In particular, this gives the first polynomial-size hitting set for constant-width ROABP (known-order). However, our hitting set only works when the characteristic of the field is zero or large enough. To construct the hitting set, we use the concept of the rank of the partial derivative matrix. Unlike previous approaches which build up from mapping variables to monomials, we map variables to polynomials directly. The second case we consider is that of polynomials computable by width-$w$ ROABPs in any order of the variables. The best previously known hitting set for this case had size $d^{O(\log w)}(nw)^{O(\log \log w)}$, where $d$ is the individual degree. We improve the hitting-set size to $(ndw)^{O(\log \log w)}$.

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arXiv2018

Isolating a Vertex via Lattices: Polytopes with Totally Unimodular Faces

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.

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arXiv2017

Pseudorandom Bits for Oblivious Branching Programs

We construct a pseudorandom generator which fools read-$k$ oblivious branching programs and, more generally, any linear length oblivious branching program, assuming that the sequence according to which the bits are read is known in advance. For polynomial width branching programs, the seed lengths in our constructions are $\tilde{O}(n^{1-1/2^{k-1}})$ (for the read-$k$ case) and $O(n/ \log \log n)$ (for the linear length case). Previously, the best construction for these models required seed length $(1-Ω(1))n$.

cs.CC↗
arXiv2016

Bipartite Perfect Matching is in quasi-NC

We show that the bipartite perfect matching problem is in quasi-NC$^2$. That is, it has uniform circuits of quasi-polynomial size $n^{O(\log n)}$, and $O(log^2 n)$ depth. Previously, only an exponential upper bound was known on the size of such circuits with poly-logarithmic depth. We obtain our result by an almost complete derandomization of the famous Isolation Lemma when applied to yield an efficient randomized parallel algorithm for the bipartite perfect matching problem.

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arXiv2015

Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs

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).

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arXiv2014

Derandomizing Isolation Lemma for $K_{3,3}$-free and $K_5$-free Bipartite Graphs

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 $K_{3,3}$-free and $K_5$-free bipartite graphs, i.e. 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 $K_{3,3}$-free and $K_5$-free bipartite graphs is in SPL. It also gives an alternate proof for an already known result -- reachability for $K_{3,3}$-free and $K_5$-free graphs is in UL.

cs.CC↗
arXiv2014

Hitting-sets for ROABP and Sum of Set-Multilinear circuits

We give a $n^{O(\log n)}$-time ($n$ is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious algebraic branching programs (ROABP). The best result known for this class was $n^{O(\log^2 n)}$ due to Forbes-Saptharishi-Shpilka (STOC 2014), and that too only for multilinear ROABP. We get rid of their exponential dependence on the individual degree. With this, we match the time-complexity for the unknown order ROABP with the known order ROABP (due to Forbes-Shpilka (FOCS 2013)) and also with the depth-$3$ set-multilinear circuits (due to Agrawal-Saha-Saxena (STOC 2013)). Our proof is simpler and involves a new technique called basis isolation. The depth-$3$ model has recently gained much importance, as it has become a stepping-stone to understanding general arithmetic circuits. Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper, we take a step towards designing such hitting-sets. We give the first subexponential whitebox PIT for the sum of constantly many set-multilinear depth-$3$ circuits. To achieve this, we define notions of distance and base sets. Distance, for a multilinear depth-$3$ circuit, measures how far are the partitions from a mere refinement. We design a hitting-set in time $n^{O(d \log n)}$ for $d$-distance. Further, we give an extension of our result to models where the distance is large but it is small when restricted to certain base sets (of variables). We also explore a new model of ROABP where the factor-matrices are invertible (called invertible-factor ROABP). We design a hitting-set in time poly($n^{w^2}$) for width-$w$ invertible-factor ROABP. Further, we could do without the invertibility restriction when $w=2$. Previously, the best result for width-$2$ ROABP was quasi-polynomial time (Forbes-Saptharishi-Shpilka, STOC 2014).

cs.CC↗
arXiv2013

Hitting-sets for low-distance multilinear depth-3

The depth-$3$ model has recently gained much importance, as it has become a stepping-stone to understanding general arithmetic circuits. Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper we take a step towards designing such hitting-sets. We define a notion of distance for multilinear depth-$3$ circuits (say, in $n$ variables and $k$ product gates) that measures how far are the partitions from a mere refinement. The $1$-distance strictly subsumes the set-multilinear model, while $n$-distance captures general multilinear depth-$3$. We design a hitting-set in time poly($n^{δ\log k}$) for $δ$-distance. Further, we give an extension of our result to models where the distance is large (close to $n$) but it is small when restricted to certain variables. This implies the first subexponential whitebox PIT for the sum of constantly many set-multilinear depth-$3$ circuits. We also explore a new model of read-once algebraic branching programs (ROABP) where the factor-matrices are invertible (called invertible-factor ROABP). We design a hitting-set in time poly($\text{size}^{w^2}$) for width-$w$ invertible-factor ROABP. Further, we could do without the invertibility restriction when $w=2$. Previously, the best result for width-$2$ ROABP was quasi-polynomial time (Forbes-Saptharishi-Shpilka, arXiv 2013). The common thread in all these results is the phenomenon of low-support `rank concentration'. We exploit the structure of these models to prove rank-concentration after a `small shift' in the variables. Our proof techniques are stronger than the results of Agrawal-Saha-Saxena (STOC 2013) and Forbes-Saptharishi-Shpilka (arXiv 2013); giving us quasi-polynomial-time hitting-sets for models where no subexponential whitebox algorithms were known before.

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