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Samir Datta

Publications and source records attributed to Samir Datta.

35 records · Page 2Linked to original sources
arXiv2020

Dynamic complexity of Reachability: How many changes can we handle?

In 2015, it was shown that reachability for arbitrary directed graphs can be updated by first-order formulas after inserting or deleting single edges. Later, in 2018, this was extended for changes of size $\frac{\log n}{\log \log n}$, where $n$ is the size of the graph. Changes of polylogarithmic size can be handled when also majority quantifiers may be used. In this paper we extend these results by showing that, for changes of polylogarithmic size, first-order update formulas suffice for maintaining (1) undirected reachability, and (2) directed reachability under insertions. For classes of directed graphs for which efficient parallel algorithms can compute non-zero circulation weights, reachability can be maintained with update formulas that may use "modulo 2" quantifiers under changes of polylogarithmic size. Examples for these classes include the class of planar graphs and graphs with bounded treewidth. The latter is shown here. As the logics we consider cannot maintain reachability under changes of larger sizes, our results are optimal with respect to the size of the changes.

cs.LO↗
arXiv2019

A Strategy for Dynamic Programs: Start over and Muddle through

In the setting of DynFO, dynamic programs update the stored result of a query whenever the underlying data changes. This update is expressed in terms of first-order logic. We introduce a strategy for constructing dynamic programs that utilises periodic computation of auxiliary data from scratch and the ability to maintain a query for a limited number of change steps. We show that if some program can maintain a query for log n change steps after an AC$^1$-computable initialisation, it can be maintained by a first-order dynamic program as well, i.e., in DynFO. As an application, it is shown that decision and optimisation problems defined by monadic second-order (MSO) formulas are in DynFO, if only change sequences that produce graphs of bounded treewidth are allowed. To establish this result, a Feferman-Vaught-type composition theorem for MSO is established that might be useful in its own right.

cs.LO↗
arXiv2018

Reachability and Distances under Multiple Changes

Recently it was shown that the transitive closure of a directed graph can be updated using first-order formulas after insertions and deletions of single edges in the dynamic descriptive complexity framework by Dong, Su, and Topor, and Patnaik and Immerman. In other words, Reachability is in DynFO. In this article we extend the framework to changes of multiple edges at a time, and study the Reachability and Distance queries under these changes. We show that the former problem can be maintained in DynFO$(+, \times)$ under changes affecting O($\frac{\log n}{\log \log n}$) nodes, for graphs with $n$ nodes. If the update formulas may use a majority quantifier then both Reachability and Distance can be maintained under changes that affect O($\log^c n$) nodes, for fixed $c \in \mathbb{N}$. Some preliminary results towards showing that distances are in DynFO are discussed.

cs.LO↗
arXiv2018

Shortest $k$-Disjoint Paths via Determinants

The well-known $k$-disjoint path problem ($k$-DPP) asks for pairwise vertex-disjoint paths between $k$ specified pairs of vertices $(s_i, t_i)$ in a given graph, if they exist. The decision version of the shortest $k$-DPP asks for the length of the shortest (in terms of total length) such paths. Similarly the search and counting versions ask for one such and the number of such shortest set of paths, respectively. We restrict attention to the shortest $k$-DPP instances on undirected planar graphs where all sources and sinks lie on a single face or on a pair of faces. We provide efficient sequential and parallel algorithms for the search versions of the problem answering one of the main open questions raised by Colin de Verdiere and Schrijver for the general one-face problem. We do so by providing a randomised $NC^2$ algorithm along with an $O(n^ω)$ time randomised sequential algorithm. We also obtain deterministic algorithms with similar resource bounds for the counting and search versions. In contrast, previously, only the sequential complexity of decision and search versions of the "well-ordered" case has been studied. For the one-face case, sequential versions of our routines have better running times for constantly many terminals. In addition, the earlier best known sequential algorithms (e.g. Borradaile et al.) were randomised while ours are also deterministic. The algorithms are based on a bijection between a shortest $k$-tuple of disjoint paths in the given graph and cycle covers in a related digraph. This allows us to non-trivially modify established techniques relating counting cycle covers to the determinant. We further need to do a controlled inclusion-exclusion to produce a polynomial sum of determinants such that all "bad" cycle covers cancel out in the sum allowing us to count "good" cycle covers.

cs.DS↗
arXiv2017

Reachability is in DynFO

Patnaik and Immerman introduced the dynamic complexity class DynFO of database queries that can be maintained by first-order dynamic programs with the help of auxiliary relations under insertions and deletions of edges (Patnaik and Immerman 1997). This article confirms their conjecture that the Reachability query is in DynFO. As a byproduct it is shown that the rank of a matrix with small values can be maintained in DynFO(+,x). It is further shown that the (size of the) maximum matching of a graph can be maintained in non-uniform DynFO, another extension of DynFO, with non-uniform initialisation of the auxiliary relations.

cs.LO↗
arXiv2015

Counting Euler Tours in Undirected Bounded Treewidth Graphs

We show that counting Euler tours in undirected bounded tree-width graphs is tractable even in parallel - by proving a $\#SAC^1$ upper bound. This is in stark contrast to #P-completeness of the same problem in general graphs. Our main technical contribution is to show how (an instance of) dynamic programming on bounded \emph{clique-width} graphs can be performed efficiently in parallel. Thus we show that the sequential result of Espelage, Gurski and Wanke for efficiently computing Hamiltonian paths in bounded clique-width graphs can be adapted in the parallel setting to count the number of Hamiltonian paths which in turn is a tool for counting the number of Euler tours in bounded tree-width graphs. Our technique also yields parallel algorithms for counting longest paths and bipartite perfect matchings in bounded-clique width graphs. While establishing that counting Euler tours in bounded tree-width graphs can be computed by non-uniform monotone arithmetic circuits of polynomial degree (which characterize $\#SAC^1$) is relatively easy, establishing a uniform $\#SAC^1$ bound needs a careful use of polynomial interpolation.

cs.CC↗
arXiv2015

Graph properties in node-query setting: effect of breaking symmetry

The query complexity of graph properties is well-studied when queries are on edges. We investigate the same when queries are on nodes. In this setting a graph $G = (V, E)$ on $n$ vertices and a property $\mathcal{P}$ are given. A black-box access to an unknown subset $S \subseteq V$ is provided via queries of the form `Does $i \in S$?'. We are interested in the minimum number of queries needed in worst case in order to determine whether $G[S]$, the subgraph of $G$ induced on $S$, satisfies $\mathcal{P}$. Apart from being combinatorially rich, this setting allows us to initiate a systematic study of breaking symmetry in the context of query complexity of graph properties. In particular, we focus on hereditary graph properties. The monotone functions in the node-query setting translate precisely to the hereditary graph properties. The famous Evasiveness Conjecture asserts that even with a minimal symmetry assumption on $G$, namely that of vertex-transitivity, the query complexity for any hereditary graph property in our setting is the worst possible, i.e., $n$. We show that in the absence of any symmetry on $G$ it can fall as low as $O(n^{1/(d + 1) })$ where $d$ denotes the minimum possible degree of a minimal forbidden sub-graph for $\mathcal{P}$. In particular, every hereditary property benefits at least quadratically. The main question left open is: can it go exponentially low for some hereditary property? We show that the answer is no for any hereditary property with {finitely many} forbidden subgraphs by exhibiting a bound of $Ω(n^{1/k})$ for some constant $k$ depending only on the property. For general ones we rule out the possibility of the query complexity falling down to constant by showing $Ω(\log n/ \log \log n)$ bound. Interestingly, our lower bound proofs rely on the famous Sunflower Lemma due to Erdös and Rado.

cs.CC↗
arXiv2014

Bounded Treewidth and Space-Efficient Linear Algebra

Motivated by a recent result of Elberfeld, Jakoby and Tantau showing that $\mathsf{MSO}$ properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and as our main result prove that it is in Logspace. It is important to notice that the determinant is neither an $\mathsf{MSO}$-property nor counts the number of solutions of an $\mathsf{MSO}$-predicate. This technique yields Logspace algorithms for counting the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs. We demonstrate some linear algebraic applications of the determinant algorithm by describing Logspace procedures for the characteristic polynomial, the powers of a weighted bounded tree-width graph and feasibility of a system of linear equations where the underlying bipartite graph has bounded tree-width. Finally, we complement our upper bounds by proving $\mathsf{L}$-hardness of the problems of computing the determinant, and of powering a bounded tree-width matrix. We also show the $\mathsf{GapL}$-hardness of Iterated Matrix Multiplication where each matrix has bounded tree-width.

cs.CC↗
arXiv2013

Tree-width and Logspace: Determinants and Counting Euler Tours

Motivated by the recent result of [EJT10] showing that MSO properties are Logspace computable on graphs of bounded tree-width, we consider the complexity of computing the determinant of the adjacency matrix of a bounded tree-width graph and prove that it is L-complete. It is important to notice that the determinant is neither an MSO-property nor counts the number of solutions of an MSO-predicate. We extend this technique to count the number of spanning arborescences and directed Euler tours in bounded tree-width digraphs, and further to counting the number of spanning trees and the number of Euler tours in undirected graphs, all in L. Notice that undirected Euler tours are not known to be MSO-expressible and the corresponding counting problem is in fact #P-hard for general graphs. Counting undirected Euler tours in bounded tree-width graphs was not known to be polynomial time computable till very recently Chebolu et al [CCM13] gave a polynomial time algorithm for this problem (concurrently and independently of this work). Finally, we also show some linear algebraic extensions of the determinant algorithm to show how to compute the charcteristic polynomial and trace of the powers of a bounded tree-width graph in L.

cs.CC↗
arXiv2011

Computing Bits of Algebraic Numbers

We initiate the complexity theoretic study of the problem of computing the bits of (real) algebraic numbers. This extends the work of Yap on computing the bits of transcendental numbers like π, in Logspace. Our main result is that computing a bit of a fixed real algebraic number is in C=NC1\subseteq Logspace when the bit position has a verbose (unary) representation and in the counting hierarchy when it has a succinct (binary) representation. Our tools are drawn from elementary analysis and numerical analysis, and include the Newton-Raphson method. The proof of our main result is entirely elementary, preferring to use the elementary Liouville's theorem over the much deeper Roth's theorem for algebraic numbers. We leave the possibility of proving non-trivial lower bounds for the problem of computing the bits of an algebraic number given the bit position in binary, as our main open question. In this direction we show very limited progress by proving a lower bound for rationals.

cs.CC↗
arXiv2011

Planarity Testing Revisited

Planarity Testing is the problem of determining whether a given graph is planar while planar embedding is the corresponding construction problem. The bounded space complexity of these problems has been determined to be exactly Logspace by Allender and Mahajan with the aid of Reingold's result. Unfortunately, the algorithm is quite daunting and generalizing it to say, the bounded genus case seems a tall order. In this work, we present a simple planar embedding algorithm running in logspace. We hope this algorithm will be more amenable to generalization. The algorithm is based on the fact that 3-connected planar graphs have a unique embedding, a variant of Tutte's criterion on conflict graphs of cycles and an explicit change of cycle basis.% for planar graphs. We also present a logspace algorithm to find obstacles to planarity, viz. a Kuratowski minor, if the graph is non-planar. To the best of our knowledge this is the first logspace algorithm for this problem.

cs.CC↗
arXiv2010

Some Tractable Win-Lose Games

Determining a Nash equilibrium in a $2$-player non-zero sum game is known to be PPAD-hard (Chen and Deng (2006), Chen, Deng and Teng (2009)). The problem, even when restricted to win-lose bimatrix games, remains PPAD-hard (Abbott, Kane and Valiant (2005)). However, there do exist polynomial time tractable classes of win-lose bimatrix games - such as, very sparse games (Codenotti, Leoncini and Resta (2006)) and planar games (Addario-Berry, Olver and Vetta (2007)). We extend the results in the latter work to $K_{3,3}$ minor-free games and a subclass of $K_5$ minor-free games. Both these classes of games strictly contain planar games. Further, we sharpen the upper bound to unambiguous logspace, a small complexity class contained well within polynomial time. Apart from these classes of games, our results also extend to a class of games that contain both $K_{3,3}$ and $K_5$ as minors, thereby covering a large and non-trivial class of win-lose bimatrix games. For this class, we prove an upper bound of nondeterministic logspace, again a small complexity class within polynomial time. Our techniques are primarily graph theoretic and use structural characterizations of the considered minor-closed families.

cs.GT↗
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.

cs.CC↗
arXiv2010

Log-space Algorithms for Paths and Matchings in k-trees

Reachability and shortest path problems are NL-complete for general graphs. They are known to be in L for graphs of tree-width 2 [JT07]. However, for graphs of tree-width larger than 2, no bound better than NL is known. In this paper, we improve these bounds for k-trees, where k is a constant. In particular, the main results of our paper are log-space algorithms for reachability in directed k-trees, and for computation of shortest and longest paths in directed acyclic k-trees. Besides the path problems mentioned above, we also consider the problem of deciding whether a k-tree has a perfect macthing (decision version), and if so, finding a perfect match- ing (search version), and prove that these two problems are L-complete. These problems are known to be in P and in RNC for general graphs, and in SPL for planar bipartite graphs [DKR08]. Our results settle the complexity of these problems for the class of k-trees. The results are also applicable for bounded tree-width graphs, when a tree-decomposition is given as input. The technique central to our algorithms is a careful implementation of divide-and-conquer approach in log-space, along with some ideas from [JT07] and [LMR07].

cs.CC↗
arXiv2009

A Log-space Algorithm for Canonization of Planar Graphs

Graph Isomorphism is the prime example of a computational problem with a wide difference between the best known lower and upper bounds on its complexity. We bridge this gap for a natural and important special case, planar graph isomorphism, by presenting an upper bound that matches the known logspace hardness [Lindell'92]. In fact, we show the formally stronger result that planar graph canonization is in logspace. This improves the previously known upper bound of AC1 [MillerReif'91]. Our algorithm first constructs the biconnected component tree of a connected planar graph and then refines each biconnected component into a triconnected component tree. The next step is to logspace reduce the biconnected planar graph isomorphism and canonization problems to those for 3-connected planar graphs, which are known to be in logspace by [DattaLimayeNimbhorkar'08]. This is achieved by using the above decomposition, and by making significant modifications to Lindell's algorithm for tree canonization, along with changes in the space complexity analysis. The reduction from the connected case to the biconnected case requires further new ideas, including a non-trivial case analysis and a group theoretic lemma to bound the number of automorphisms of a colored 3-connected planar graph. This lemma is crucial for the reduction to work in logspace.

cs.CC↗
arXiv2008

Deterministically Isolating a Perfect Matching in Bipartite Planar Graphs

We present a deterministic way of assigning small (log bit) weights to the edges of a bipartite planar graph so that the minimum weight perfect matching becomes unique. The isolation lemma as described in (Mulmuley et al. 1987) achieves the same for general graphs using a randomized weighting scheme, whereas we can do it deterministically when restricted to bipartite planar graphs. As a consequence, we reduce both decision and construction versions of the matching problem to testing whether a matrix is singular, under the promise that its determinant is 0 or 1, thus obtaining a highly parallel SPL algorithm for bipartite planar graphs. This improves the earlier known bounds of non-uniform SPL by (Allender et al. 1999) and $NC^2$ by (Miller and Naor 1995, Mahajan and Varadarajan 2000). It also rekindles the hope of obtaining a deterministic parallel algorithm for constructing a perfect matching in non-bipartite planar graphs, which has been open for a long time. Our techniques are elementary and simple.

cs.DS↗