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N. S. Narayanaswamy

Publications and source records attributed to N. S. Narayanaswamy.

At least 19 recordsLinked to original sources
arXiv2025

A faster algorithm for Vertex Cover parameterized by solution size

We describe a new algorithm for vertex cover with runtime $O^*(1.25284^k)$, where $k$ is the size of the desired solution and $O^*$ hides polynomial factors in the input size. This improves over previous runtime of $O^*(1.2738^k)$ due to Chen, Kanj, & Xia (2010) standing for more than a decade. The key to our algorithm is to use a potential function which simultaneously tracks $k$ as well as the optimal value $λ$ of the vertex cover LP relaxation. This approach also allows us to make use of prior algorithms for Maximum Independent Set in bounded-degree graphs and Above-Guarantee Vertex Cover. The main step in the algorithm is to branch on high-degree vertices, while ensuring that both $k$ and $μ= k - λ$ are decreased at each step. There can be local obstructions in the graph that prevent $μ$ from decreasing in this process; we develop a number of novel branching steps to handle these situations.

cs.DS↗
arXiv2023

Exactly Hittable Interval Graphs

Given a set system $\mathcal{X} = \{\mathcal{U},\mathcal{S}\}$, where $\mathcal{U}$ is a set of elements and $\mathcal{S}$ is a set of subsets of $\mathcal{U}$, an exact hitting set $\mathcal{U}'$ is a subset of $\mathcal{U}$ such that each subset in $\mathcal{S}$ contains exactly one element in $\mathcal{U}'$. We refer to a set system as exactly hittable if it has an exact hitting set. In this paper, we study interval graphs which have intersection models that are exactly hittable. We refer to these interval graphs as exactly hittable interval graphs (EHIG). We present a forbidden structure characterization for EHIG. We also show that the class of proper interval graphs is a strict subclass of EHIG. Finally, we give an algorithm that runs in polynomial time to recognize graphs belonging to the class of EHIG.

cs.DS↗
arXiv2022

On the Complexity of Connected $(s,t)$-Vertex Separator

We show that minimum connected $(s,t)$-vertex separator ($(s,t)$-CVS) is $Ω(log^{2-ε}n)$-hard for any $ε>0$ unless NP has quasi-polynomial Las-Vegas algorithms. i.e., for any $ε>0$ and for some $δ>0$, $(s,t)$-CVS is unlikely to have $δ.log^{2-ε}n$-approximation algorithm. We show that $(s,t)$-CVS is NP-complete on graphs with chordality at least 5 and present a polynomial-time algorithm for $(s,t)$-CVS on bipartite chordality 4 graphs. We also present a $\lceil\frac{c}{2}\rceil$-approximation algorithm for $(s,t)$-CVS on graphs with chordality $c$. Finally, from the parameterized setting, we show that $(s,t)$-CVS parameterized above the $(s,t)$-vertex connectivity is $W[2]$-hard.

cs.DM↗
arXiv2021

Parameterized Complexity of Minimum Membership Dominating Set

Given a graph $G=(V,E)$ and an integer $k$, the Minimum Membership Dominating Set (MMDS) problem seeks to find a dominating set $S \subseteq V$ of $G$ such that for each $v \in V$, $|N[v] \cap S|$ is at most $k$. We investigate the parameterized complexity of the problem and obtain the following results about MMDS: W[1]-hardness of the problem parameterized by the pathwidth (and thus, treewidth) of the input graph. W[1]-hardness parameterized by $k$ on split graphs. An algorithm running in time $2^{\mathcal{O}(\textbf{vc})} |V|^{\mathcal{O}(1)}$, where $\textbf{vc}$ is the size of a minimum-sized vertex cover of the input graph. An ETH-based lower bound showing that the algorithm mentioned in the previous item is optimal.

cs.DS↗
arXiv2021

Budgeted Dominating Sets in Uncertain Graphs

We study the {\em Budgeted Dominating Set} (BDS) problem on uncertain graphs, namely, graphs with a probability distribution $p$ associated with the edges, such that an edge $e$ exists in the graph with probability $p(e)$. The input to the problem consists of a vertex-weighted uncertain graph $\G=(V, E, p, ω)$ and an integer {\em budget} (or {\em solution size}) $k$, and the objective is to compute a vertex set $S$ of size $k$ that maximizes the expected total domination (or total weight) of vertices in the closed neighborhood of $S$. We refer to the problem as the {\em Probabilistic Budgeted Dominating Set}~(PBDS) problem and present the following results. \begin{enumerate} \dnsitem We show that the PBDS problem is NP-complete even when restricted to uncertain {\em trees} of diameter at most four. This is in sharp contrast with the well-known fact that the BDS problem is solvable in polynomial time in trees. We further show that PBDS is \wone-hard for the budget parameter $k$, and under the {\em Exponential time hypothesis} it cannot be solved in $n^{o(k)}$ time. \item We show that if one is willing to settle for $(1-ε)$ approximation, then there exists a PTAS for PBDS on trees. Moreover, for the scenario of uniform edge-probabilities, the problem can be solved optimally in polynomial time. \item We consider the parameterized complexity of the PBDS problem, and show that Uni-PBDS (where all edge probabilities are identical) is \wone-hard for the parameter pathwidth. On the other hand, we show that it is FPT in the combined parameters of the budget $k$ and the treewidth. \item Finally, we extend some of our parameterized results to planar and apex-minor-free graphs. \end{enumerate}

cs.DS↗
arXiv2021

Perfect Resolution of Strong Conflict-Free Colouring of Interval Hypergraphs

The $k$-Strong Conflict-Free ($k$-SCF, in short) colouring problem seeks to find a colouring of the vertices of a hypergraph $H$ using minimum number of colours so that in every hyperedge $e$ of $H$, there are at least $\min\{|e|,k\}$ vertices whose colour is different from that of all other vertices in $e$. In the case of interval hypergraphs, we present an exact $P$-time algorithm for the $k$-SCF problem thus solving an open problem posed by Cheilaris et al. (2014). We achieve our results by showing that for any hypergraph a $k$-SCF colouring is a proper colouring of a related simple graph which we refer to as a co-occurrence graph. We then show that a co-occurrence graph is obtained by identifying an induced subgraph of a second simple graph that we introduce, which we refer to as the conflict graph. For interval hypergraphs, we show that each co-occurrence graph and the conflict graph are perfect graphs. This property plays a crucial role in our polynomial time algorithm. Secondly, we show that for an interval hypergraph, the $1$-SCF colouring number is the minimum partition of its intervals into sets such that each set has an exact hitting set (a hitting set in which each interval is hit exactly once).

math.CO↗
arXiv2020

Trade-offs in dynamic coloring for bipartite and general graphs

We present trade-offs in the incremental and fully dynamic settings to maintian a proper coloring. For any fully dynamic $2$-coloring algorithm, the maximum of the update time, number of recolorings, and query time is $Ω(\log n)$. We present a deterministic fully dynamic $2$-coloring algorithm with $O(\log^2 n)$ amortized update time, $O(\log n)$ amortized query time, and one recoloring in the worst case. For any incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, the amortized update time and the amortized number of recolorings is $Ω(\log n)$. For such an algorithm, in the worst case the update time and the number of recolorings is $Ω(n)$. We then design a deterministic incremental $2$-coloring algorithm which explicitly maintains the color of every vertex after each update, with amortized $O(\log n)$ update time and amortized $O(\log n)$ many recolorings. Further, in the worst case the update time and the number of recolorings is $O(n)$. Further, we present a deterministic incremental $(1+2 \log n)$-coloring algorithm which explicitly maintains the color of every vertex after each update, with $O(α(n))$ amortized update time, at most one recoloring and $O(1)$ query time. We then show a deterministic incremental $2$-coloring algorithm which does not maintain color of every vertex after each update, with amortized $O(α(n))$ update time, amortized $O(α(n))$ recolorings, and amortized $O(α(n))$ query time. For general graphs and graphs of bounded arboricity $γ$ and maximum degree $Δ$ we present a deterministic $(Δ+1)$-coloring algorithm with $O(\sqrt{m})$ update time, $O(1)$ query time, and one recoloring. Finally, we show a deterministic $(Δ+1)$-coloring algorithm with amortized $O(γ+ \log{n})$ update time, $O(1)$ query time, and one recoloring.

cs.DS↗
arXiv2020

Dynamic Data Structures for Interval Coloring

We consider the dynamic graph coloring problem restricted to the class of interval graphs. At each update step the algorithm is presented with an interval to be colored, or a previously colored interval to delete. The goal of the algorithm is to efficiently maintain a proper coloring of the intervals with as few colors as possible by an online algorithm. In the incremental model, each update step presents the algorithm with an interval to be colored. The problem is closely connected to the online vertex coloring problem of interval graphs for which the Kierstead-Trotter (KT) algorithm achieves the best possible competitive ratio. We first show that a sub-quadratic time direct implementation of the KT-algorithm is unlikely to exist conditioned on the correctness of the Online Boolean Matrix Vector multiplication conjecture due to Henzinger et al. \cite{DBLP:conf/stoc/HenzingerKNS15}. We then design an incremental algorithm that is subtly different from the KT-algorithm and uses at most $3 ω- 2$ colors, where $ω$ is the maximum clique in the interval graph associated with the set of intervals. Our incremental data structure maintains a proper coloring in amortized $O(\log n + Δ)$ update time where $n$ is the total number of intervals inserted and $Δ$ is the maximum degree of a vertex in the interval graph. We then consider the fully dynamic framework involving insertions and deletions. On each update, our aim is to maintain a $3 ω- 2$ coloring of the remaining set of intervals, where $ω$ is the maximum clique in the interval graph associated with the remaining set of intervals. Our fully dynamic algorithm supports insertion of an interval in $O(\log n + Δ\log ω)$ worst case update time and deletion of an interval in $O(Δ^2 \log n)$ worst case update time.

cs.DS↗
arXiv2019

Conflict-Free Colouring using Maximum Independent Set and Minimum Colouring

Given a hypergraph $H$, the conflict-free colouring problem is to colour vertices of $H$ using minimum colours so that each hyperedge in $H$ sees a unique colour. We present a polynomial time reduction from the conflict-free colouring problem in hypergraphs to the maximum independent set problem in a class of simple graphs, which we refer to as \textit{conflict graphs}. We also present another characterization of the conflict-free colouring number in terms of the chromatic number of graphs in an associated family of simple graphs, which we refer to as \textit{co-occurrence graphs}. We present perfectness results for co-occurrence graphs and a special case of conflict graphs. Based on these results and a linear program that returns an integer solution in polynomial time, we obtain a polynomial time algorithm to compute a minimum conflict-free colouring of interval hypergraphs, thus solving an open problem due to Cheilaris et al.\cite{CPLGARSS2014}. Finally, we use the co-occurrence graph characterization to prove that for an interval hypergraph, the conflict-free colouring number is the minimum partition of its intervals into sets such that each set has an exact hitting set (a hitting set in which each interval is hit exactly once).

cs.DS↗
arXiv2018

On the Complexity Landscape of Connected f -Factor Problems

Let G be an undirected simple graph having n vertices and let f be a function defined to be f:V(G) -> {0,..., n-1}. An f-factor of G is a spanning subgraph H such that degree of a vertex v in H is f(v) for every vertex v in V(G). The subgraph H is called a connected f-factor if, in addition, H is connected. A classical result of Tutte(1954) is the polynomial time algorithm to check whether a given graph has a specified f-factor. However, checking for the presence of a connected f-factor is easily seen to generalize HAMILTONIAN CYCLE and hence is NP-complete. In fact, the CONNECTED f-FACTOR problem remains NP-complete even when we restrict f(v) to be at least n^e for each vertex v and 0 1, the problem is NP-intermediate.

cs.CC↗
arXiv2018

Fully dynamic $3/2$ approximate maximum cardinality matching in $O(\sqrt{n})$ update time

We present a randomized algorithm to maintain a maximal matching without 3 length augmenting paths in the fully dynamic setting. Consequently, we maintain a $3/2$ approximate maximum cardinality matching. Our algorithm takes expected amortized $O(\sqrt{n})$ time where $n$ is the number of vertices in the graph when the update sequence is generated by an oblivious adversary. Over any sequence of $t$ edge insertions and deletions presented by an oblivious adversary, the total update time of our algorithm is $O(t\sqrt{n})$ in expectation and $O(t\sqrt{n} + n \log n)$ with high probability. To the best of our knowledge, our algorithm is the first one to maintain an approximate matching in which all augmenting paths are of length at least $5$ in $o(\sqrt{m})$ update time.

cs.DS↗
arXiv2016

Hitting Set for hypergraphs of low VC-dimension

We study the complexity of the Hitting Set problem in set systems (hypergraphs) that avoid certain sub-structures. In particular, we characterize the classical and parameterized complexity of the problem when the Vapnik-Chervonenkis dimension (VC-dimension) of the input is small. VC-dimension is a natural measure of complexity of set systems. Several tractable instances of Hitting Set with a geometric or graph-theoretical flavor are known to have low VC-dimension. In set systems of bounded VC-dimension, Hitting Set is known to admit efficient and almost optimal approximation algorithms (Brönnimann and Goodrich, 1995; Even, Rawitz, and Shahar, 2005; Agarwal and Pan, 2014). In contrast to these approximation-results, a low VC-dimension does not necessarily imply tractability in the parameterized sense. In fact, we show that Hitting Set is W[1]-hard already on inputs with VC-dimension 2, even if the VC-dimension of the dual set system is also 2. Thus, Hitting Set is very unlikely to be fixed-parameter tractable even in this arguably simple case. This answers an open question raised by King in 2010. For set systems whose (primal or dual) VC-dimension is 1, we show that Hitting Set is solvable in polynomial time. To bridge the gap in complexity between the classes of inputs with VC-dimension 1 and 2, we use a measure that is more fine-grained than VC-dimension. In terms of this measure, we identify a sharp threshold where the complexity of Hitting Set transitions from polynomial-time-solvable to NP-hard. The tractable class that lies just under the threshold is a generalization of Edge Cover, and thus extends the domain of polynomial-time tractability of Hitting Set.

cs.DS↗
arXiv2016

Faster Randomized Branching Algorithms for $r$-SAT

The problem of determining if an $r$-CNF boolean formula $F$ over $n$ variables is satisifiable reduces to the problem of determining if $F$ has a satisfying assignment with a Hamming distance of at most $d$ from a fixed assignment $α$. This problem is also a very important subproblem in Schoning's local search algorithm for $r$-SAT. While Schoning described a randomized algorithm solves this subproblem in $O((r-1)^d)$ time, Dantsin et al. presented a deterministic branching algorithm with $O^*(r^d)$ running time. In this paper we present a simple randomized branching algorithm that runs in time $O^*({(\frac{r+1}{2})}^d)$. As a consequence we get a randomized algorithm for $r$-SAT that runs in $O^*({(\frac{2(r+1)}{r+3})}^n)$ time. This algorithm matches the running time of Schoning's algorithm for 3-SAT and is an improvement over Schoning's algorithm for all $r \geq 4$. For $r$-uniform hitting set parameterized by solution size $k$, we describe a randomized FPT algorithm with a running time of $O^*({(\frac{r+1}{2})}^k)$. For the above LP guarantee parameterization of vertex cover, we have a randomized FPT algorithm to find a vertex cover of size $k$ in a running time of $O^*(2.25^{k-vc^*})$, where $vc^*$ is the LP optimum of the natural LP relaxation of vertex cover. In both the cases, these randomized algorithms have a better running time than the current best deterministic algorithms.

cs.DS↗
arXiv2015

A Classification of Connected f -factor Problems inside NP

Given an undirected graph G = (V, E) with n vertices, and a function f : V -> N, we consider the problem of finding a connected f -factor in G. In this work we design an algorithm to check for the existence of a connected f -factor, for the case where f (v) >= n/g(n), for all v in V and g(n) is polylogarithmic in n. The running time of our algorithm is O(n^{2g(n)}. As a consequence of this algorithm we conclude that the complexity of connected f -factor for the case we consider is unlikely to be NP-Complete unless the Exponential Time Hypothesis (ETH) is false. Secondly, under the assumption of the ETH, we show that it is also unlikely to be in P for g(n) in O((log n)^{1+eps} ) for any eps> 0. Therefore, our results show that for all eps> 0, connected f -factor for f (v) >= n/O(log n)^{1+eps}) is in NP-Intermediate unless the ETH is false. Further, for any constant c > 0, when g(n) = c, our algorithm for connected f -factor runs in polynomial time. Finally, we extend our algorithm to compute a minimum weight connected f -factor in edge weighted graphs in the same asymptotic time bounds.

cs.CC↗
arXiv2014

On Minimum Average Stretch Spanning Trees in Polygonal 2-trees

A spanning tree of an unweighted graph is a minimum average stretch spanning tree if it minimizes the ratio of sum of the distances in the tree between the end vertices of the graph edges and the number of graph edges. We consider the problem of computing a minimum average stretch spanning tree in polygonal 2-trees, a super class of 2-connected outerplanar graphs. For a polygonal 2-tree on $n$ vertices, we present an algorithm to compute a minimum average stretch spanning tree in $O(n \log n)$ time. This algorithm also finds a minimum fundamental cycle basis in polygonal 2-trees.

cs.DS↗
arXiv2013

Tree t-spanners in Outerplanar Graphs via Supply Demand Partition

A tree t-spanner of an unweighted graph G is a spanning tree T such that for every two vertices their distance in T is at most t times their distance in G. Given an unweighted graph G and a positive integer t as input, the tree t-spanner problem is to compute a tree t-spanner of G if one exists. This decision problem is known to be NP-complete even in the restricted class of unweighted planar graphs. We present a linear-time reduction from tree t-spanner in outerplanar graphs to the supply-demand tree partition problem. Based on this reduction, we obtain a linear-time algorithm to solve tree t-spanner in outerplanar graphs. Consequently, we show that the minimum value of t for which an input outerplanar graph on n vertices has a tree t-spanner can be found in O(n log n) time.

cs.DS↗
arXiv2013

Approximability of Connected Factors

Finding a d-regular spanning subgraph (or d-factor) of a graph is easy by Tutte's reduction to the matching problem. By the same reduction, it is easy to find a minimal or maximal d-factor of a graph. However, if we require that the d-factor is connected, these problems become NP-hard - finding a minimal connected 2-factor is just the traveling salesman problem (TSP). Given a complete graph with edge weights that satisfy the triangle inequality, we consider the problem of finding a minimal connected $d$-factor. We give a 3-approximation for all $d$ and improve this to an (r+1)-approximation for even d, where r is the approximation ratio of the TSP. This yields a 2.5-approximation for even d. The same algorithm yields an (r+1)-approximation for the directed version of the problem, where r is the approximation ratio of the asymmetric TSP. We also show that none of these minimization problems can be approximated better than the corresponding TSP. Finally, for the decision problem of deciding whether a given graph contains a connected d-factor, we extend known hardness results.

cs.DS↗