arXiv subjects

Sudeshna Kolay

Publications and source records attributed to Sudeshna Kolay.

At least 19 recordsLinked to original sources

arXiv2026

Parameterized Complexity of Finding a Maximum Common Vertex Subgraph Without Isolated Vertices

In this paper, we study the Maximum Common Vertex Subgraph problem: Given two input graphs $G_1,G_2$ and a non-negative integer $h$, is there a common subgraph $H$ on at least $h$ vertices such that there is no isolated vertex in $H$. In other words, each connected component of $H$ has at least $2$ vertices. This problem naturally arises in graph theory along with other variants of the well-studied Maximum Common Subgraph problem and also has applications in computational social choice. We show that this problem is NP-hard and provide an FPT algorithm when parameterized by $h$. Next, we conduct a study of the problem on common structural parameters like vertex cover number, maximum degree, treedepth, pathwidth and treewidth of one or both input graphs. We derive a complete dichotomy of parameterized results for our problem with respect to individual parameterizations as well as combinations of parameterizations from the above structural parameters. This provides us with a deep insight into the complexity theoretic and parameterized landscape of this problem.

cs.CC↗

arXiv2026

On Euler Paths and the Maximum Degree Growth of Iterated Higher Order Line Graphs

Given a simple graph $G$, its line graph, denoted by $L(G)$, is obtained by representing each edge of $G$ as a vertex, with two vertices in $L(G)$ adjacent whenever the corresponding edges in $G$ share a common endpoint. By applying the line graph operation repeatedly, we obtain higher order line graphs, denoted by $L^{r}(G)$. In other words, $L^{0}(G) = G$, and for any integer $r \ge 1$, $L^{r}(G) = L(L^{r-1}(G))$. Given a graph $G$ on $n$ vertices, we wish to efficiently find out (i) if $L^k(G)$ has an Euler path, (ii) the value of $Δ(L^k(G))$. Note that the size of a higher order line graph could be much larger than that of $G$. For the first question, we show that for a graph $G$ with $n$ vertices and $m$ edges the largest $k$ where $L^k(G)$ has an Euler path satisfies $k = \mathcal O(nm)$. We also design an $\mathcal{O}(n^2m)$-time algorithm to output all $k$ such that $L^k(G)$ has an Euler path. For the second question, we study the growth of maximum degree of $L^k(G)$, $k \ge 0$. It is easy to calculate $Δ(L^k(G))$ when $G$ is a path, cycle or a claw. Any other connected graph is called a prolific graph and we denote the set of all prolific graphs by $\mathcal G$. We extend the works of Hartke and Higgins to show that for any prolific graph $G$, there exists a constant rational number $dgc(G)$ and an integer $k_0$ such that for all $k \ge k_0$, $Δ(L^k(G)) = dgc(G) \cdot 2^{k-4} + 2$. We show that $\{dgc(G) \mid G \in \mathcal G\}$ has first, second, third, fourth and fifth minimums, namely, $c_1 = 3$, $c_2 = 4$, $c_3 = 5.5$, $c_4 = 6$ and $c_5=7$; the third minimum stands out surprisingly from the other four. Moreover, for $i \in \{1, 2, 3, 4\}$, we provide a complete characterization of $\mathcal G_i = \{dgc(G) = c_i \mid G \in \mathcal G \}$. Apart from this, we show that the set $\{dgc(G) \mid G \in \mathcal G, 7 < dgc(G) < 8\}$ is countably infinite.

math.CO↗

arXiv2026

Minimizing Envy and Maximizing Happiness in Graphical House Allocation

We study the house allocation problem in a setting where agents are connected by a graph representing friendships. In this model, two agents can only envy each other if they are neighbors (i.e., friends) in the graph. Each agent has a set of preferred (liked) houses and dislikes the rest. An agent $a$ is said to envy a friend $b$ if $a$ is not assigned any house she likes, while $b$ is allocated a house that $a$ likes. This framework is known as graphical house allocation. Within this framework, we investigate two central problems. The first problem is to compute a house allocation that minimizes the number of envious agents. Multiple such allocations may exist that achieve the same minimum level of envy. Among all allocations that minimize envy, the second problem aims to find one that maximizes the number of agents who receive one of their preferred houses. We present a detailed complexity-theoretic analysis of these problems. In particular, we show that both problems can be solved in polynomial time when each agent prefers at most one house. However, both become NP-hard even when agents are allowed to prefer at most two houses, thereby highlighting the tight boundary between tractability and intractability. Additionally, we design exact algorithms for both problems under certain structural conditions on the agent graph, such as when the graph is sparse, has a small balanced separator, or admits a small vertex cover. These algorithms are significantly faster than the naive brute-force approach.

cs.DS↗

arXiv2026

Universal Solvability for Robot Motion Planning on Graphs

We study the Universal Solvability of Robot Motion Planning on Graphs (USolR) problem: given an undirected graph $G = (V, E)$ and $p$ robots, determine whether any arbitrary configuration of the robots can be transformed into any other arbitrary configuration via a sequence of valid, collision-free moves. We design a canonical accumulation procedure that maps arbitrary configurations to configurations that occupy a fixed subset of vertices, enabling us to analyze configuration reachability in terms of equivalence classes. We prove that in instances that are not universally solvable, at least half of all configurations are unreachable from a given one, and leverage this to design an efficient randomized algorithm with one-sided error, which can be derandomized with a blow-up in the running time by a factor of $p$. Further, we optimize our deterministic algorithm by using the structure of the input graph $G = (V, E)$, achieving a running time of $\mathcal{O}(p \cdot (|V| + |E|))$ in sparse graphs and $\mathcal{O}(|V| + |E|)$ in dense graphs. Finally, we consider the Graph Edge Augmentation for Universal Solvability (EAUS) problem, where given a connected graph $G$ that is not universally solvable for $p$ robots, the question is to check if for a given budget $b$, at most $b$ edges can be added to $G$ to make it universally solvable for $p$ robots. We provide an upper bound of $p - 2$ on $b$ for general graphs. On the other hand, we also provide examples of graphs that require $Θ(p)$ edges to be added. We further study the Graph Vertex and Edge Augmentation for Universal Solvability (VEAUS) problem, where $a$ vertices and $b$ edges can be added, and we provide lower bounds on $a$ and $b$.

cs.CC↗

arXiv2025

Knapsack on Graphs with Relaxed Neighborhood Constraints

In the knapsack problems with neighborhood constraints that were studied before, the input is a graph $\mathcal{G}$ on a set $\mathcal{V}$ of items, each item $v \in \mathcal{V}$ has a weight $w_v$ and profit $p_v$, the size $s$ of the knapsack, and the demand $d$. The goal is to compute if there exists a feasible solution whose total weight is at most $s$ and total profit is at most $d$. Here, feasible solutions are all subsets $\mathcal{S}$ of the items such that, for every item in $\mathcal{S}$, at least one of its neighbors in $\mathcal{G}$ is also in $\mathcal{S}$ for \hor, and all its neighbors in $\mathcal{G}$ are also in $\mathcal{S}$ for \hand~\cite{borradaile2012knapsack}. We study a relaxation of the above problems. Specifically, we allow all possible subsets of items to be feasible solutions. However, only those items for which we pick at least one or all of its neighbor (out-neighbor for directed graph) contribute to profit whereas every item picked contribute to the weight; we call the corresponding problems \sor and \sand. We show that both \sor and \sand are strongly \NPC even on undirected graphs. Regarding parameterized complexity, we show both \sor and \hor are \WTH parameterized by the size $s$ of the knapsack size. Interestingly, both \sand and \hand are \WOH parameterized by knapsack size, $s$ plus profit demand, $d$ and also parameterized by solution size, $b$. For \sor and \hor, we present a randomized color-coding-based pseudo-\FPT algorithm, parameterized by the solution size $b$, and consequently by the demand $d$. We then consider the treewidth of the input graph as our parameter and design pseudo fixed-parameter tractable (\FPT) algorithm parameterized by treewidth, $\text{tw}$ for all variants. Finally, we present an additive $1$ approximation for \sor when both the weight and profit of every vertex is $1$.

cs.DS↗

arXiv2024

Efficient Exact Algorithms for Minimum Covering of Orthogonal Polygons with Squares

Let $P$ be an orthogonal polygon of $n$ vertices, without holes. The Orthogonal Polygon Covering with Squares (OPCS) problem takes as input such an orthogonal polygon $P$ with integral vertex coordinates, and asks to find the minimum number of axis-parallel squares whose union is $P$ itself. [Aupperle et. al, 1988] provide an $\mathcal O(N^{1.5})$-time algorithm for OPCS, where $N$ is the number of integral lattice points lying in $P$. In their paper, designing algorithms for OPCS with a running time polynomial in $n$, was stated as an open question; $N$ can be arbitrarily larger than $n$. Output sensitive algorithms were known due to [Bar-Yehuda and Ben-Chanoch, 1994], but these fail to address the open question, as the output can be arbitrarily larger than $n$. We address this open question by designing a polynomial-time exact algorithm for OPCS with a worst-case running time of $\mathcal O(n^{10})$. We also consider the following structural parameterized version of the problem. Let a knob be a polygon edge whose both endpoints are convex polygon vertices. Given an input orthogonal polygon without holes that has $n$ vertices and at most $k$ knobs, we design an algorithm for OPCS with a worst-case running time $\mathcal O(n^2 + k^{10} \cdot n)$. This algorithm is more efficient than the former, whenever $k = o(n^{9/10})$. The problem of Orthogonal Polygon with Holes Covering with Squares (OPCSH) is also studied by [Aupperle et. al, 1988], where the input polygon could have holes. They claim a proof that OPCSH is NP-complete even when the input is the $N$ lattice points inside the polygon. We think there is an error in their proof, where an incorrect reduction from Planar 3-CNF is shown. We provide a correct reduction with a novel construction of one of the gadgets, and show how this leads to a correct proof of NP-completeness of OPCSH.

cs.CG↗

arXiv2024

Knapsack with Vertex Cover, Set Cover, and Hitting Set

Given an undirected graph $\mathcal{G}=(\mathcal{V},\mathcal{E})$, with vertex weights $(w(u))_{u\in\mathcal{V}}$, vertex values $(α(u))_{u\in\mathcal{V}}$, a knapsack size $s$, and a target value $d$, the \vcknapsack problem is to determine if there exists a subset $\mathcal{U}\subseteq\mathcal{V}$ of vertices such that $\mathcal{U}$ forms a vertex cover, $w(\mathcal{U})=\sum_{u\in\mathcal{U}} w(u) \le s$, and $α(\mathcal{U})=\sum_{u\in\mathcal{U}} α(u) \ge d$. In this paper, we closely study the \vcknapsack problem and its variations, such as \vcknapsackbudget, \minimalvcknapsack, and \minimumvcknapsack, for both general graphs and trees. We first prove that the \vcknapsack problem belongs to the complexity class \NPC and then study the complexity of the other variations. We generalize the problem to \setc and \hs versions and design polynomial time $H_g$-factor approximation algorithm for the \setckp problem and d-factor approximation algorithm for \hstp using primal dual method. We further show that \setcks and \hsmb are hard to approximate in polynomial time. Additionally, we develop a fixed parameter tractable algorithm running in time $8^{\mathcal{O}({\rm tw})}\cdot n\cdot {\sf min}\{s,d\}$ where ${\rm tw},s,d,n$ are respectively treewidth of the graph, the size of the knapsack, the target value of the knapsack, and the number of items for the \minimalvcknapsack problem.

cs.DS↗

arXiv2024

Knapsack: Connectedness, Path, and Shortest-Path

We study the knapsack problem with graph theoretic constraints. That is, we assume that there exists a graph structure on the set of items of knapsack and the solution also needs to satisfy certain graph theoretic properties on top of knapsack constraints. In particular, we need to compute in the connected knapsack problem a connected subset of items which has maximum value subject to the size of knapsack constraint. We show that this problem is strongly NP-complete even for graphs of maximum degree four and NP-complete even for star graphs. On the other hand, we develop an algorithm running in time $O\left(2^{tw\log tw}\cdot\text{poly}(\min\{s^2,d^2\})\right)$ where $tw,s,d$ are respectively treewidth of the graph, size, and target value of the knapsack. We further exhibit a $(1-ε)$ factor approximation algorithm running in time $O\left(2^{tw\log tw}\cdot\text{poly}(n,1/ε)\right)$ for every $ε>0$. We show similar results for several other graph theoretic properties, namely path and shortest-path under the problem names path-knapsack and shortestpath-knapsack. Our results seems to indicate that connected-knapsack is computationally hardest followed by path-knapsack and shortestpath-knapsack.

cs.DS↗

arXiv2023

Efficient Algorithms for Euclidean Steiner Minimal Tree on Near-Convex Terminal Sets

The Euclidean Steiner Minimal Tree problem takes as input a set $\mathcal P$ of points in the Euclidean plane and finds the minimum length network interconnecting all the points of $\mathcal P$. In this paper, in continuation to the works of Du et al. and Weng et al., we study Euclidean Steiner Minimal Tree when $\mathcal P$ is formed by the vertices of a pair of regular, concentric and parallel $n$-gons. We restrict our attention to the cases where the two polygons are not very close to each other. In such cases, we show that Euclidean Steiner Minimal Tree is polynomial-time solvable, and we describe an explicit structure of a Euclidean Steiner minimal tree for $\mathcal P$. We also consider point sets $\mathcal P$ of size $n$ where the number of input points not on the convex hull of $\mathcal P$ is $f(n) \leq n$. We give an exact algorithm with running time $2^{\mathcal{O}(f(n)\log n)}$ for such input point sets $\mathcal P$. Note that when $f(n) = \mathcal{O}(\frac{n}{\log n})$, our algorithm runs in single-exponential time, and when $f(n) = o(n)$ the running time is $2^{o(n\log n)}$ which is better than the known algorithm stated in Hwang et al. We know that no FPTAS exists for Euclidean Steiner Minimal Tree unless P=NP, as shown by Garey et al. On the other hand FPTASes exist for Euclidean Steiner Minimal Tree on convex point sets, as given by Scott Provan. In this paper, we show that if the number of input points in $\mathcal P$ not belonging to the convex hull of $\mathcal P$ is $\mathcal{O}(\log n)$, then an FPTAS exists for Euclidean Steiner Minimal Tree. In contrast, we show that for any $ε\in (0,1]$, when there are $Ω(n^ε)$ points not belonging to the convex hull of the input set, then no FPTAS can exist for Euclidean Steiner Minimal Tree unless P=NP.

cs.CG↗

arXiv2023

Almost optimal query algorithm for hitting set using a subset query

Given access to the hypergraph through a subset query oracle in the query model, we give sublinear time algorithms for Hitting-Set with almost tight parameterized query complexity. In parameterized query complexity, we estimate the number of queries to the oracle based on the parameter $k$, the size of the Hitting-Set. The subset query oracle we use in this paper is called Generalized $d$-partite Independent Set query oracle (GPIS) and it was introduced by Bishnu et al. (ISAAC'18). GPIS is a generalization to hypergraphs of the Bipartite Independent Set query oracle (BIS) introduced by Beame et al. (ITCS'18 and TALG'20) for estimating the number of edges in graphs. Formally, GPIS is defined as follows: GPIS oracle for a $d$-uniform hypergraph $\mathcal{H}$ takes as input $d$ pairwise disjoint non-empty subsets $A_1, \ldots, A_d$ of vertices in $\cal H$ and answers whether there is a hyperedge in $\mathcal{H}$ that intersects each set $A_i$, where $i \in \{1, \, 2, \, \ldots, d\}$. } For $d=2$, the GPIS oracle is nothing but BIS oracle. We show that $d$-Hitting-Set, the hitting set problem for $d$-uniform hypergraphs, can be solved using $\widetilde{\mathcal{O}}_d(k^{d} \log n)$ GPIS queries. Additionally, we also showed that $d$-Decesion-Hitting-Set, the decision version of $d$-Hitting-Set can be solved with $\widetilde{\mathcal{O}}_d\left( \min \left\{ k^d\log n, k^{2d^2} \right\} \right)$ {\sc GPIS} queries. We complement these parameterized upper bounds with an almost matching parameterized lower bound that states that any algorithm that solves $d$-Decesion-Hitting-Set requires $Ω\left( \binom{k+d}{d} \right)$ GPIS queries.

cs.DS↗

arXiv2023

An ETH-Tight Exact Algorithm for Euclidean TSP

We study exact algorithms for Euclidean TSP in $\mathbb{R}^d$. In the early 1990s algorithms with $n^{O(\sqrt{n})}$ running time were presented for the planar case, and some years later an algorithm with $n^{O(n^{1-1/d})}$ running time was presented for any $d\geq 2$. Despite significant interest in subexponential exact algorithms over the past decade, there has been no progress on Euclidean TSP, except for a lower bound stating that the problem admits no $2^{O(n^{1-1/d-ε})}$ algorithm unless ETH fails. Up to constant factors in the exponent, we settle the complexity of Euclidean TSP by giving a $2^{O(n^{1-1/d})}$ algorithm and by showing that a $2^{o(n^{1-1/d})}$ algorithm does not exist unless ETH fails.

cs.CG↗

arXiv2020

Parameterized Study of Steiner Tree on Unit Disk Graphs

We study the Steiner Tree problem on unit disk graphs. Given a $n$ vertex unit disk graph $G$, a subset $R\subseteq V(G)$ of $t$ vertices and a positive integer $k$, the objective is to decide if there exists a tree $T$ in $G$ that spans over all vertices of $R$ and uses at most $k$ vertices from $V\setminus R$. The vertices of $R$ are referred to as terminals and the vertices of $V(G)\setminus R$ as Steiner vertices. First, we show that the problem is NP-Hard. Next, we prove that the Steiner Tree problem on unit disk graphs can be solved in $n^{O(\sqrt{t+k})}$ time. We also show that the Steiner Tree problem on unit disk graphs parameterized by $k$ has an FPT algorithm with running time $2^{O(k)}n^{O(1)}$. In fact, the algorithms are designed for a more general class of graphs, called clique-grid graphs. We mention that the algorithmic results can be made to work for the Steiner Tree on disk graphs with bounded aspect ratio. Finally, we prove that the Steiner Tree on disk graphs parameterized by $k$ is W[1]-hard.

cs.CG↗

arXiv2019

Structural Parameterization for Graph Deletion Problems over Data Streams

The study of parameterized streaming complexity on graph problems was initiated by Fafianie et al. (MFCS'14) and Chitnis et al. (SODA'15 and SODA'16). Simply put, the main goal is to design streaming algorithms for parameterized problems such that $O\left(f(k)\log^{O(1)}n\right)$ space is enough, where $f$ is an arbitrary computable function depending only on the parameter $k$. However, in the past few years, very few positive results have been established. Most of the graph problems that do have streaming algorithms of the above nature are ones where localized checking is required, like Vertex Cover or Maximum Matching parameterized by the size $k$ of the solution we are seeking. Many important parameterized problems that form the backbone of traditional parameterized complexity are known to require $Ω(n)$ bits for any streaming algorithm; e.g., Feedback Vertex Set, Even/Odd Cycle Transversal, Triangle Deletion or the more general ${\cal F}$-Subgraph Deletion when parameterized by solution size $k$. Our main conceptual contribution is to overcome the obstacles to efficient parameterized streaming algorithms by utilizing the power of parameterization. To the best of our knowledge, this is the first work in parameterized streaming complexity that considers structural parameters instead of the solution size as a parameter. We focus on the vertex cover size $K$ as the parameter for the parameterized graph deletion problems we consider. At the same time, most of the previous work in parameterized streaming complexity was restricted to the EA (edge arrival) or DEA (dynamic edge arrival) models. In this work, we consider the above mentioned graph deletion problems in the four most well-studied streaming models, i.e., the EA, DEA, VA (vertex arrival) and AL (adjacency list) models.

cs.CC↗

arXiv2019

Parameterized Complexity of Conflict-free Graph Coloring

Given a graph G, a q-open neighborhood conflict-free coloring or q-ONCF-coloring is a vertex coloring $c:V(G) \rightarrow \{1,2,\ldots,q\}$ such that for each vertex $v \in V(G)$ there is a vertex in $N(v)$ that is uniquely colored from the rest of the vertices in $N(v)$. When we replace $N(v)$ by the closed neighborhood $N[v]$, then we call such a coloring a q-closed neighborhood conflict-free coloring or simply q-CNCF-coloring. In this paper, we study the NP-hard decision questions of whether for a constant q an input graph has a q-ONCF-coloring or a q-CNCF-coloring. We will study these two problems in the parameterized setting. First of all, we study running time bounds on FPT-algorithms for these problems, when parameterized by treewidth. We improve the existing upper bounds, and also provide lower bounds on the running time under ETH and SETH. Secondly, we study the kernelization complexity of both problems, using vertex cover as the parameter. We show that both $(q \geq 2)$-ONCF-coloring and $(q \geq 3)$-CNCF-coloring cannot have polynomial kernels when parameterized by the size of a vertex cover unless $NP \in coNP/poly$. However, we obtain a polynomial kernel for 2-CNCF-coloring parameterized by vertex cover. We conclude with some combinatorial results. Denote $χ_{ON}(G)$ and $χ_{CN}(G)$ to be the minimum number of colors required to ONCF-color and CNCF-color G, respectively. Upper bounds on $χ_{CN}(G)$ with respect to structural parameters like minimum vertex cover size, minimum feedback vertex set size and treewidth are known. To the best of our knowledge only an upper bound on $χ_{ON}(G)$ with respect to minimum vertex cover size was known. We provide tight bounds for $χ_{ON}(G)$ with respect to minimum vertex cover size. Also, we provide the first upper bounds on $χ_{ON}(G)$ with respect to minimum feedback vertex set size and treewidth.

cs.CC↗

arXiv2018

FPT algorithms for embedding into low complexity graphic metrics

The Metric Embedding problem takes as input two metric spaces $(X,D_X)$ and $(Y,D_Y)$, and a positive integer $d$. The objective is to determine whether there is an embedding $F:X \rightarrow Y$ such that $d_{F} \leq d$, where $d_{F}$ denotes the distortion of the map $F$. Such an embedding is called a distortion $d$ embedding. The bijective Metric Embedding problem is a special case of the Metric Embedding problem where $|X| = |Y|$. In parameterized complexity, the Metric Embedding problem, in full generality, is known to be W-hard and therefore, not expected to have an FPT algorithm. In this paper, we consider the Gen-Graph Metric Embedding problem, where the two metric spaces are graph metrics. We explore the extent of tractability of the problem in the parameterized complexity setting. We determine whether an unweighted graph metric $(G,D_G)$ can be embedded, or bijectively embedded, into another unweighted graph metric $(H,D_H)$, where the graph $H$ has low structural complexity. For example, $H$ is a cycle, or $H$ has bounded treewidth or bounded connected treewidth. The parameters for the algorithms are chosen from the upper bound $d$ on distortion, bound $Δ$ on the maximum degree of $H$, treewidth $α$ of $H$, and the connected treewidth $α_{c}$ of $H$. Our general approach to these problems can be summarized as trying to understand the behavior of the shortest paths in $G$ under a low distortion embedding into $H$, and the structural relation the mapping of these paths has to shortest paths in $H$.

cs.CG↗

arXiv2015

Parameterized Algorithms on Perfect Graphs for deletion to $(r,\ell)$-graphs

For fixed integers $r,\ell \geq 0$, a graph $G$ is called an {\em $(r,\ell)$-graph} if the vertex set $V(G)$ can be partitioned into $r$ independent sets and $\ell$ cliques. The class of $(r, \ell)$ graphs generalizes $r$-colourable graphs (when $\ell =0)$ and hence not surprisingly, determining whether a given graph is an $(r, \ell)$-graph is \NP-hard even when $r \geq 3$ or $\ell \geq 3$ in general graphs. When $r$ and $\ell$ are part of the input, then the recognition problem is NP-hard even if the input graph is a perfect graph (where the {\sc Chromatic Number} problem is solvable in polynomial time). It is also known to be fixed-parameter tractable (FPT) on perfect graphs when parameterized by $r$ and $\ell$. I.e. there is an $f(r+\ell) \cdot n^{\Oh(1)}$ algorithm on perfect graphs on $n$ vertices where $f$ is some (exponential) function of $r$ and $\ell$. In this paper, we consider the parameterized complexity of the following problem, which we call {\sc Vertex Partization}. Given a perfect graph $G$ and positive integers $r,\ell,k$ decide whether there exists a set $S\subseteq V(G)$ of size at most $k$ such that the deletion of $S$ from $G$ results in an $(r,\ell)$-graph. We obtain the following results: \begin{enumerate} \item {\sc Vertex Partization} on perfect graphs is FPT when parameterized by $k+r+\ell$. \item The problem does not admit any polynomial sized kernel when parameterized by $k+r+\ell$. In other words, in polynomial time, the input graph can not be compressed to an equivalent instance of size polynomial in $k+r+\ell$. In fact, our result holds even when $k=0$. \item When $r,\ell$ are universal constants, then {\sc Vertex Partization} on perfect graphs, parameterized by $k$, has a polynomial sized kernel. \end{enumerate}

cs.DS↗

arXiv2015

Multivariate Complexity Analysis of Geometric {\sc Red Blue Set Cover}

We investigate the parameterized complexity of GENERALIZED RED BLUE SET COVER (Gen-RBSC), a generalization of the classic SET COVER problem and the more recently studied RED BLUE SET COVER problem. Given a universe $U$ containing $b$ blue elements and $r$ red elements, positive integers $k_\ell$ and $k_r$, and a family $\F$ of $\ell$ sets over $U$, the \srbsc\ problem is to decide whether there is a subfamily $\F'\subseteq \F$ of size at most $k_\ell$ that covers all blue elements, but at most $k_r$ of the red elements. This generalizes SET COVER and thus in full generality it is intractable in the parameterized setting. In this paper, we study a geometric version of this problem, called Gen-RBSC-lines, where the elements are points in the plane and sets are defined by lines. We study this problem for an array of parameters, namely, $k_\ell, k_r, r, b$, and $\ell$, and all possible combinations of them. For all these cases, we either prove that the problem is W-hard or show that the problem is fixed parameter tractable (FPT). In particular, on the algorithmic side, our study shows that a combination of $k_\ell$ and $k_r$ gives rise to a nontrivial algorithm for Gen-RBSC-lines. On the hardness side, we show that the problem is para-NP-hard when parameterized by $k_r$, and W[1]-hard when parameterized by $k_\ell$. Finally, for the combination of parameters for which Gen-RBSC-lines admits FPT algorithms, we ask for the existence of polynomial kernels. We are able to provide a complete kernelization dichotomy by either showing that the problem admits a polynomial kernel or that it does not contain a polynomial kernel unless $\CoNP \subseteq \NP/\mbox{poly}$.

cs.DS↗

arXiv2015

A faster FPT Algorithm and a smaller Kernel for Block Graph Vertex Deletion

A graph $G$ is called a \emph{block graph} if each maximal $2$-connected component of $G$ is a clique. In this paper we study the Block Graph Vertex Deletion from the perspective of fixed parameter tractable (FPT) and kernelization algorithm. In particular, an input to Block Graph Vertex Deletion consists of a graph $G$ and a positive integer $k$ and the objective to check whether there exists a subset $S \subseteq V(G)$ of size at most $k$ such that the graph induced on $V(G)\setminus S$ is a block graph. In this paper we give an FPT algorithm with running time $4^kn^{O(1)}$ and a polynomial kernel of size $O(k^4)$ for Block Graph Vertex Deletion. The running time of our FPT algorithm improves over the previous best algorithm for the problem that ran in time $10^kn^{O(1)}$ and the size of our kernel reduces over the previously known kernel of size $O(k^9)$. Our results are based on a novel connection between Block Graph Vertex Deletion and the classical {\sc Feedback Vertex Set} problem in graphs without induced $C_4$ and $K_4-e$. To achieve our results we also obtain an algorithm for {\sc Weighted Feedback Vertex Set} running in time $3.618^kn^{O(1)}$ and improving over the running time of previously known algorithm with running time $5^kn^{O(1)}$.

cs.DS↗