arXiv subjects

Raghunath Tewari

Publications and source records attributed to Raghunath Tewari.

16 recordsLinked to original sources

arXiv2025

Efficient Isolation of Perfect Matching in O(log n) Genus Bipartite Graphs

We show that given an embedding of an $O(\log n)$ genus bipartite graph, one can construct an edge weight function in logarithmic space, with respect to which the minimum weight perfect matching in the graph is unique, if one exists. As a consequence, we obtain that deciding whether such a graph has a perfect matching or not is in SPL. In 1999, Reinhardt, Allender and Zhou proved that if one can construct a polynomially bounded weight function for a graph in logspace such that it isolates a minimum weight perfect matching in the graph, then the perfect matching problem can be solved in SPL. In this paper, we give a deterministic logspace construction of such a weight function for $O(\log n)$ genus bipartite graphs.

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arXiv2025

Trading Determinism for Time: The k-Reach Problem

Kallampally and Tewari showed in 2016 that there can be a trade-off between determinism and time in space-bounded computations. This they did by describing an unambiguous non-deterministic algorithm to solve Directed Graph Reachability that requires O(log^2 n) space and simultaneously runs in polynomial time. Savitch's 1970 algorithm that solves the same problem deterministically also requires O(log^2 n) space but doesn't guarantee polynomial running time and hence the trade off. We describe a new problem for which we can show a similar trade off between determinism and time. We consider a collection P of f directed paths. We show that the problem of finding reachability from one vertex to another in the union G of these path graphs via a path that switches amongst the paths in P at most k times can be solved in O(klog f+log n) space but the algorithm doesn't guarantee polynomial runtime. On the other hand, we also show that the same problem can be solved by an unambiguous non-deterministic algorithm that simultaneously runs in O(klog f+log n) space and polynomial time. Since these two algorithms are not dependent on Savitch, therefore this example sheds new light on how such a trade off between determinism and time happens in space-bounded computations and makes the phenomenon less elusive.

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arXiv2024

On Solving Reachability in Grid Digraphs using a Psuedoseparator

The reachability problem asks to decide if there exists a path from one vertex to another in a digraph. In a grid digraph, the vertices are the points of a two-dimensional square grid, and an edge can occur between a vertex and its immediate horizontal and vertical neighbors only. Asano and Doerr (CCCG'11) presented the first simultaneous time-space bound for reachability in grid digraphs by solving the problem in polynomial time and $O(n^{1/2 + ε})$ space. In 2018, the space complexity was improved to $\tilde{O}(n^{1/3})$ by Ashida and Nakagawa (SoCG'18). In this paper, we show that there exists a polynomial-time algorithm that uses $O(n^{1/4 + ε})$ space to solve the reachability problem in a grid digraph containing $n$ vertices. We define and construct a new separator-like device called pseudoseparator to develop a divide-and-conquer algorithm. This algorithm works in a space-efficient manner to solve reachability.

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arXiv2024

Lossy Catalytic Computation

A catalytic Turing machine is a variant of a Turing machine in which there exists an auxiliary tape in addition to the input tape and the work tape. This auxiliary tape is initially filled with arbitrary content. The machine can read and write on the auxiliary tape, but it is constrained to restore its initial content when it halts. Studying such a model and finding its powers and limitations has practical applications. In this paper, we study catalytic Turing machines with O(log n)-sized work tape and polynomial-sized auxiliary tape that are allowed to lose at most constant many bits of the auxiliary tape when they halt. We show that such catalytic Turing machines can only decide the same set of languages as standard catalytic Turing machines with the same size work and auxiliary tape.

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arXiv2021

Dynamic Meta-theorems for Distance and Matching

Reachability, distance, and matching are some of the most fundamental graph problems that have been of particular interest in dynamic complexity theory in recent years [DKMSZ18, DMVZ18, DKMTVZ20]. Reachability can be maintained with first-order update formulas, or equivalently in DynFO in general graphs with n nodes [DKMSZ18], even under O(log n/loglog n) changes per step [DMVZ18]. In the context of how large the number of changes can be handled, it has recently been shown [DKMTVZ20] that under a polylogarithmic number of changes, reachability is in DynFOpar in planar, bounded treewidth, and related graph classes -- in fact in any graph where small non-zero circulation weights can be computed in NC. We continue this line of investigation and extend the meta-theorem for reachability to distance and bipartite maximum matching with the same bounds. These are amongst the most general classes of graphs known where we can maintain these problems deterministically without using a majority quantifier and even maintain witnesses. For the bipartite matching result, modifying the approach from [FGT], we convert the static non-zero circulation weights to dynamic matching-isolating weights. While reachability is in DynFOar under O(log n/loglog n) changes, no such bound is known for either distance or matching in any non-trivial class of graphs under non-constant changes. We show that, in the same classes of graphs as before, bipartite maximum matching is in DynFOar under O(log n/loglog n) changes per step. En route to showing this we prove that the rank of a matrix can be maintained in DynFOar, also under O(log n/loglog n) entry changes, improving upon the previous O(1) bound [DKMSZ18]. This implies similar extension for the non-uniform DynFO bound for maximum matching in general graphs and an alternate algorithm for maintaining reachability under O(log n/loglog n) changes [DMVZ18].

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arXiv2021

Reachability and Matching in Single Crossing Minor Free Graphs

We show that for each single crossing graph $H$, a polynomially bounded weight function for all $H$-minor free graphs $G$ can be constructed in Logspace such that it gives nonzero weights to all the cycles in $G$. This class of graphs subsumes almost all classes of graphs for which such a weight function is known to be constructed in Logspace. As a consequence, we obtain that for the class of $H$-minor free graphs where $H$ is a single crossing graph, reachability can be solved in UL, and bipartite maximum matching can be solved in SPL, which are small subclasses of the parallel complexity class NC. In the restrictive case of bipartite graphs, our maximum matching result improves upon the recent result of Eppstein and Vazirani, where they show an NC bound for constructing perfect matching in general single crossing minor free graphs.

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arXiv2020

Time Space Optimal Algorithm for Computing Separators in Bounded Genus Graphs

A graph separator is a subset of vertices of a graph whose removal divides the graph into small components. Computing small graph separators for various classes of graphs is an important computational task. In this paper, we present a polynomial time algorithm that uses $O(g^{1/2}n^{1/2}\log n)$-space to find an $O(g^{1/2}n^{1/2})$-sized separator of a graph having $n$ vertices and embedded on a surface of genus $g$.

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arXiv2020

Two Player Hidden Pointer Chasing and Multi-Pass Lower Bounds in Turnstile Streams

The authors have withdrawn this paper due to an error in the proof of Lemma 3.4. -------------------------------------------------------------------------------------------- The authors have withdrawn this paper due to an error in the proof of Lemma 3.4z(Assadi, Chen, and Khanna, 2019) define a 4-player hidden-pointer-chasing ($\mathsf{HPC}^4$), and using it, give strong multi-pass lower bounds for graph problems in the streaming model of computation and a lower bound on the query complexity of sub-modular minimization. We present a two-player version ($\mathsf{HPC}^2$) of $\mathsf{HPC}^4$ that has matching communication complexity to $\mathsf{HPC}^4$. Our formulation allows us to lower bound its communication complexity with a simple direct-sum argument. Using this lower bound on the communication complexity of $\mathsf{HPC}^2$, we retain the streaming and query complexity lower bounds by (Assadi, Chen, and Khanna, 2019). Further, by giving reductions from $\mathsf{HPC}^2$, we prove new multi-pass space lower bounds for graph problems in turnstile streams. In particular, we show that any algorithm which computes the exact weight of the maximum weighted matching in an $n$-vertex graph requires $\tilde{O}(n^{2})$ space unless it makes $ω(\log n)$ passes over the turnstile stream, and that any algorithm which computes the minimum $s\text{-}t$ distance in an $n$-vertex graph requires $n^{2-o(1)}$ space unless it makes $n^{Ω(1)}$ passes over the turnstile stream. Our reductions can be modified to use $\mathsf{HPC}^4$ as well.

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arXiv2019

Reachability in High Treewidth Graphs

Reachability is the problem of deciding whether there is a path from one vertex to the other in the graph. Standard graph traversal algorithms such as DFS and BFS take linear time to decide reachability however their space complexity is also linear. On the other hand, Savitch's algorithm takes quasipolynomial time although the space bound is $O(\log^2 n)$. Here, we study space efficient algorithms for deciding reachability that runs simultaneously in polynomial time. In this paper, we show that given an $n$ vertex directed graph of treewidth $w$ along with its tree decomposition, there exists an algorithm running in polynomial time and $O(w\log n)$ space, that solves reachability in the graph.

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arXiv2018

Circuit Complexity of Bounded Planar Cutwidth Graph Matching

Recently, perfect matching in bounded planar cutwidth bipartite graphs (\BGGM) was shown to be in ACC$^0$ by Hansen et al.. They also conjectured that the problem is in AC$^0$. In this paper, we disprove their conjecture by showing that the problem is not in AC$^0[p^α]$ for every prime $p$. Our results show that the previous upper bound is almost tight. Our techniques involve giving a reduction from Parity to BGGM. A further improvement in lower bounds is difficult since we do not have an algebraic characterization for AC$^0[m]$ where $m$ is not a prime power. Moreover, this will also imply a separation of AC$^0[m]$ from P. Our results also imply a better lower bound for perfect matching in general bounded planar cutwidth graphs.

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arXiv2016

Trading Determinism for Time in Space Bounded Computations

Savitch showed in $1970$ that nondeterministic logspace (NL) is contained in deterministic $\mathcal{O}(\log^2 n)$ space but his algorithm requires quasipolynomial time. The question whether we can have a deterministic algorithm for every problem in NL that requires polylogarithmic space and simultaneously runs in polynomial time was left open. In this paper we give a partial solution to this problem and show that for every language in NL there exists an unambiguous nondeterministic algorithm that requires $\mathcal{O}(\log^2 n)$ space and simultaneously runs in polynomial time.

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arXiv2015

Simultaneous Time-Space Upper Bounds for Certain Problems in Planar Graphs

In this paper, we show that given a weighted, directed planar graph $G$, and any $ε>0$, there exists a polynomial time and $O(n^{\frac{1}{2}+ε})$ space algorithm that computes the shortest path between two fixed vertices in $G$. We also consider the {\RB} problem, which states that given a graph $G$ whose edges are colored either red or blue and two fixed vertices $s$ and $t$ in $G$, is there a path from $s$ to $t$ in $G$ that alternates between red and blue edges. The {\RB} problem in planar DAGs is {\NL}-complete. We exhibit a polynomial time and $O(n^{\frac{1}{2}+ε})$ space algorithm (for any $ε>0$) for the {\RB} problem in planar DAG. In the last part of this paper, we consider the problem of deciding and constructing the perfect matching present in a planar bipartite graph and also a similar problem which is to find a Hall-obstacle in a planar bipartite graph. We show the time-space bound of these two problems are same as the bound of shortest path problem in a directed planar graph.

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arXiv2015

An $O(n^ε)$ Space and Polynomial Time Algorithm for Reachability in Directed Layered Planar Graphs

Given a graph $G$ and two vertices $s$ and $t$ in it, {\em graph reachability} is the problem of checking whether there exists a path from $s$ to $t$ in $G$. We show that reachability in directed layered planar graphs can be decided in polynomial time and $O(n^ε)$ space, for any $ε> 0$. The previous best known space bound for this problem with polynomial time was approximately $O(\sqrt{n})$ space \cite{INPVW13}. Deciding graph reachability in {\SC} is an important open question in complexity theory and in this paper we make progress towards resolving this question.

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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.

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arXiv2010

Space Complexity of Perfect Matching in Bounded Genus Bipartite Graphs

We investigate the space complexity of certain perfect matching problems over bipartite graphs embedded on surfaces of constant genus (orientable or non-orientable). We show that the problems of deciding whether such graphs have (1) a perfect matching or not and (2) a unique perfect matching or not, are in the logspace complexity class \SPL. Since \SPL\ is contained in the logspace counting classes $\oplusŁ$ (in fact in \modk\ for all $k\geq 2$), \CeqL, and \PL, our upper bound places the above-mentioned matching problems in these counting classes as well. We also show that the search version, computing a perfect matching, for this class of graphs is in $\FL^{\SPL}$. Our results extend the same upper bounds for these problems over bipartite planar graphs known earlier. As our main technical result, we design a logspace computable and polynomially bounded weight function which isolates a minimum weight perfect matching in bipartite graphs embedded on surfaces of constant genus. We use results from algebraic topology for proving the correctness of the weight function.

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arXiv2010

On the Power of Unambiguity in Logspace

We report progress on the \NL vs \UL problem. [-] We show unconditionally that the complexity class $\ReachFewL\subseteq\UL$. This improves on the earlier known upper bound $\ReachFewL \subseteq \FewL$. [-] We investigate the complexity of min-uniqueness - a central notion in studying the \NL vs \UL problem. We show that min-uniqueness is necessary and sufficient for showing $\NL =\UL$. We revisit the class $\OptL[\log n]$ and show that {\sc ShortestPathLength} - computing the length of the shortest path in a DAG, is complete for $\OptL[\log n]$. We introduce $\UOptL[\log n]$, an unambiguous version of $\OptL[\log n]$, and show that (a) $\NL =\UL$ if and only if $\OptL[\log n] = \UOptL[\log n]$, (b) $\LogFew \leq \UOptL[\log n] \leq \SPL$. [-] We show that the reachability problem over graphs embedded on 3 pages is complete for \NL. This contrasts with the reachability problem over graphs embedded on 2 pages which is logspace equivalent to the reachability problem in planar graphs and hence is in \UL.

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