arXiv subjects

Prafullkumar Tale

Publications and source records attributed to Prafullkumar Tale.

At least 19 recordsLinked to original sources

arXiv2026

On the Hardness of Strong Metric Dimension

Let $G$ be a connected simple undirected graph. A vertex $w$ is said to \emph{strongly resolve} a pair of distinct vertices $u, v \in V(G)$ if either there exists an isometric path (i.e.~a shortest path) from $w$ to $u$ that contains $v$, or there exists an isometric path from $w$ to $v$ that contains $u$. A subset $S \subseteq V(G)$ is said to \emph{strongly resolve} $G$ if every pair of distinct vertices of $G$ is strongly resolved by at least one vertex in $S$. In the \textsc{Strong Metric Dimension} problem, the input consists of a graph $G$ and a positive integer $k$, and the objective is to determine whether there exists a subset $S \subseteq V(G)$ of size at most $k$ that strongly resolves $G$. In this article, we show that \textsc{Strong Metric Dimension} is \NP-complete even on $(i)$ graphs of diameter two, and $(ii)$ graphs of constant pathwidth and constant feedback vertex set number.

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arXiv2026

Structural parameterizations of Geodetic Set on directed (acyclic) graphs

In DIRECTED GEODETIC SET, we are given a (directed) graph and seek a small solution set $S \subseteq V(G)$ such that every vertex lies on a shortest directed path between two vertices in $S$. It is known that the problem is W[2]-hard when parameterized by the solution size $k$, even on directed acyclic graphs (DAGs). Our first result is a kernel of size $2^{O(vcn)}$ for DIRECTED GEODETIC SET on general digraphs, where $vcn$ denotes the vertex cover number of the underlying (undirected) graph. This implies an algorithm running in time $2^{O(vcn^2)} \cdot n^{O(1)}$. Furthermore, we prove that, assuming the ETH, the problem does not admit an algorithm running in time $2^{o(vcn^2)} \cdot n^{O(1)}$. Next, we show that on general digraphs, DIRECTED GEODETIC SET admits a natural kernel of size $(k\Delta)^{O(rdiam)}$, where $\Delta$ is the maximum degree and $rdiam$ denotes the reachability diameter of the digraph (a natural analogue of diameter of undirected graphs). This yields an algorithm running in time $(k\Delta)^{O(rdiam \cdot k)}\cdot n^{O(1)}$. We further prove that, assuming the ETH, the problem does not admit an algorithm running in time $(k\Delta)^{o(rdiam \cdot k)} \cdot n^{O(1)}$. Finally, we justify the necessity of combining parameters by establishing the following hardness results for DIRECTED GEODETIC SET: - It is W[2]-hard parameterized by $k$, even on digraphs of maximum degree 3. - It is para-NP-hard parameterized by maximum degree and reachability diameter. One can infer that the problem remains W[2]-hard when parameterized by k, even on graphs of reachability diameter 3 from Ara\'ujo and Arraes [DAM 2022]. All our conditional lower bounds and hardness results hold even when the input digraph is restricted to be a DAG.

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arXiv2026

Algorithms and Hardness for Geodetic Set on Tree-like Digraphs

In the GEODETIC SET problem, an input is a (di)graph $G$ and integer $k$, and the objective is to decide whether there exists a vertex subset $S$ of size $k$ such that any vertex in $V(G)\setminus S$ lies on a shortest (directed) path between two vertices in $S$. The problem has been studied on undirected and directed graphs from both algorithmic and graph-theoretical perspectives. We focus on directed graphs and prove that GEODETIC SET admits a polynomial-time algorithm on ditrees, that is, digraphs with possible 2-cycles when the underlying undirected graph is a tree (after deleting possible parallel edges). This positive result naturally leads us to investigate cases where the underlying undirected graph is "close to a tree". Towards this, we show that GEODETIC SET on digraphs without 2-cycles and whose underlying undirected graph has feedback edge set number $\textsf{fen}$, can be solved in time $2^{\mathcal{O}(\textsf{fen})} \cdot n^{\mathcal{O}(1)}$, where $n$ is the number of vertices. To complement this, we prove that the problem remains NP-hard on DAGs (which do not contain 2-cycles) even when the underlying undirected graph has constant feedback vertex set number and constant pathwidth. Our last result significantly strengthens the result of Ara\'ujo and Arraes [Discrete Applied Mathematics, 2022] that the problem is NP-hard on DAGs when the underlying undirected graph is either bipartite, cobipartite or split.

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arXiv2025

The Parameterized Complexity of Computing the VC-Dimension

The VC-dimension is a well-studied and fundamental complexity measure of a set system (or hypergraph) that is central to many areas of machine learning. We establish several new results on the complexity of computing the VC-dimension. In particular, given a hypergraph $\mathcal{H}=(\mathcal{V},\mathcal{E})$, we prove that the naive $2^{\mathcal{O}(|\mathcal{V}|)}$-time algorithm is asymptotically tight under the Exponential Time Hypothesis (ETH). We then prove that the problem admits a $1$-additive fixed-parameter approximation algorithm when parameterized by the maximum degree of $\mathcal{H}$ and a fixed-parameter algorithm when parameterized by its dimension, and that these are essentially the only such exploitable structural parameters. Lastly, we consider a generalization of the problem, formulated using graphs, which captures the VC-dimension of both set systems and graphs. We design a $2^{\mathcal{O}(\rm{tw}\cdot \log \rm{tw})}\cdot |V|$-time algorithm for any graph $G=(V,E)$ of treewidth $\rm{tw}$ (which, for a set system, applies to the treewidth of its incidence graph). This is in contrast with closely related problems that require a double-exponential dependency on the treewidth (assuming the ETH).

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arXiv2025

A Finer View of the Parameterized Landscape of Labeled Graph Contractions

We study the \textsc{Labeled Contractibility} problem, where the input consists of two vertex-labeled graphs $G$ and $H$, and the goal is to determine whether $H$ can be obtained from $G$ via a sequence of edge contractions. Lafond and Marchand~[WADS 2025] initiated the parameterized complexity study of this problem, showing it to be $\W[1]$-hard when parameterized by the number $k$ of allowed contractions. They also proved that the problem is fixed-parameter tractable when parameterized by the tree-width $\tw$ of $G$, via an application of Courcelle's theorem resulting in a non-constructive algorithm. In this work, we present a constructive fixed-parameter algorithm for \textsc{Labeled Contractibility} with running time $2^{\mathcal{O}(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}$. We also prove that unless the Exponential Time Hypothesis (\ETH) fails, it does not admit an algorithm running in time $2^{o(\tw^2)} \cdot |V(G)|^{\mathcal{O}(1)}$. This result adds \textsc{Labeled Contractibility} to a small list of problems that admit such a lower bound and matching algorithm. We further strengthen existing hardness results by showing that the problem remains \NP-complete even when both input graphs have bounded maximum degree. We also investigate parameterizations by $(k + \delta(G))$ where $\delta(G)$ denotes the degeneracy of $G$, and rule out the existence of subexponential-time algorithms. This answers question raised in Lafond and Marchand~[WADS 2025]. We additionally provide an improved \FPT\ algorithm with better dependence on $(k + \delta(G))$ than previously known. Finally, we analyze a brute-force algorithm for \textsc{Labeled Contractibility} with running time $|V(H)|^{\mathcal{O}(|V(G)|)}$, and show that this running time is optimal under \ETH.

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arXiv2025

Parameterized complexity of isometric path partition: treewidth and diameter

We investigate the parameterized complexity of the Isometric Path Partition problem when parameterized by the treewidth ($\mathrm{tw}$) of the input graph, arguably one of the most widely studied parameters. Courcelle's theorem shows that graph problems that are expressible as MSO formulas of constant size admit FPT algorithms parameterized by the treewidth of the input graph. This encompasses many natural graph problems. However, many metric-based graph problems, where the solution is defined using some metric-based property of the graph (often the distance) are not expressible as MSO formulas of constant size. These types of problems, Isometric Path Partition being one of them, require individual attention and often draw the boundary for the success story of parameterization by treewidth. We prove that Isometric Path Partition is $W[1]$-hard when parameterized by treewidth (in fact, even pathwidth), answering the question by Dumas et al. [SIDMA, 2024], Fernau et al. [CIAC, 2023], and confirming the aforementioned tendency. We complement this hardness result by designing a tailored dynamic programming algorithm running in $n^{O(\mathrm{tw})}$ time. This dynamic programming approach also results in an algorithm running in time $\textrm{diam}^{O(\mathrm{tw}^2)} \cdot n^{O(1)}$, where $\textrm{diam}$ is the diameter of the graph. Note that the dependency on treewidth is unusually high, as most problems admit algorithms running in time $2^{O(\mathrm{tw})}\cdot n^{O(1)}$ or $2^{O(\mathrm{tw} \log (\mathrm{tw}))}\cdot n^{O(1)}$. However, we rule out the possibility of a significantly faster algorithm by proving that Isometric Path Partition does not admit an algorithm running in time $\textrm{diam}^{o(\mathrm{tw}^2/(\log^3(\mathrm{tw})))} \cdot n^{O(1)}$, unless the Randomized-ETH fails.

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arXiv2025

Path Contraction Faster than $2^n$

A graph $G$ is contractible to a graph $H$ if there is a set $X \subseteq E(G)$, such that $G/X$ is isomorphic to $H$. Here, $G/X$ is the graph obtained from $G$ by contracting all the edges in $X$. For a family of graphs $\cal F$, the $\mathcal{F}$-\textsc{Contraction} problem takes as input a graph $G$ on $n$ vertices, and the objective is to output the largest integer $t$, such that $G$ is contractible to a graph $H \in {\cal F}$, where $|V(H)|=t$. When $\cal F$ is the family of paths, then the corresponding $\mathcal{F}$-\textsc{Contraction} problem is called \textsc{Path Contraction}. The problem \textsc{Path Contraction} admits a simple algorithm running in time $2^{n}\cdot n^{\mathcal{O}(1)}$. In spite of the deceptive simplicity of the problem, beating the $2^{n}\cdot n^{\mathcal{O}(1)}$ bound for \textsc{Path Contraction} seems quite challenging. In this paper, we design an exact exponential time algorithm for \textsc{Path Contraction} that runs in time $1.99987^n\cdot n^{\mathcal{O}(1)}$. We also define a problem called \textsc{$3$-Disjoint Connected Subgraphs}, and design an algorithm for it that runs in time $1.88^n\cdot n^{\mathcal{O}(1)}$. The above algorithm is used as a sub-routine in our algorithm for {\sc Path Contraction}

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arXiv2025

A Single Exponential-Time FPT Algorithm for Cactus Contraction

For a collection $\mathcal{F}$ of graphs, the $\mathcal{F}$-\textsc{Contraction} problem takes a graph $G$ and an integer $k$ as input and decides if $G$ can be modified to some graph in $\mathcal{F}$ using at most $k$ edge contractions. The $\mathcal{F}$-\textsc{Contraction} problem is \NP-Complete for several graph classes $\mathcal{F}$. Heggerners et al. [Algorithmica, 2014] initiated the study of $\mathcal{F}$-\textsc{Contraction} in the realm of parameterized complexity. They showed that it is \FPT\ if $\mathcal{F}$ is the set of all trees or the set of all paths. In this paper, we study $\mathcal{F}$-\textsc{Contraction} where $\mathcal{F}$ is the set of all cactus graphs and show that we can solve it in $2^{\calO(k)} \cdot |V(G)|^{\OO(1)}$ time.

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arXiv2025

Geodetic Set on Graphs of Constant Pathwidth and Feedback Vertex Set Number

In the \textsc{Geodetic Set} problem, the input consists of a graph $G$ and a positive integer $k$. The goal is to determine whether there exists a subset $S$ of vertices of size $k$ such that every vertex in the graph is included in a shortest path between two vertices in $S$. Kellerhals and Koana [IPEC 2020; J. Graph Algorithms Appl 2022] proved that the problem is $\W[1]$-hard when parameterized by the pathwidth and the feedback vertex set number of the input graph. They posed the question of whether the problem admits an $\XP$ algorithm when parameterized by the combination of these two parameters. We answer this in negative by proving that the problem remains \NP-hard on graphs of constant pathwidth and feedback vertex set number.

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arXiv2025

Revisiting Token Sliding on Chordal Graphs

In this article, we revisit the complexity of the reconfiguration of independent sets under the token sliding rule on chordal graphs. In the \textsc{Token Sliding-Connectivity} problem, the input is a graph $G$ and an integer $k$, and the objective is to determine whether the reconfiguration graph $TS_k(G)$ of $G$ is connected. The vertices of $TS_k(G)$ are $k$-independent sets of $G$, and two vertices are adjacent if and only if one can transform one of the two corresponding independent sets into the other by sliding a vertex (also called a \emph{token}) along an edge. Bonamy and Bousquet [WG'17] proved that the \textsc{Token Sliding-Connectivity} problem is polynomial-time solvable on interval graphs but \NP-hard on split graphs. In light of these two results, the authors asked: can we decide the connectivity of $TS_k(G)$ in polynomial time for chordal graphs with \emph{maximum clique-tree degree} $d$? We answer this question in the negative and prove that the problem is \para-\NP-hard when parameterized by $d$. More precisely, the problem is \NP-hard even when $d = 4$. We then study the parameterized complexity of the problem for a larger parameter called \emph{leafage} and prove that the problem is \co-\W[1]-hard. We prove similar results for a closely related problem called \textsc{Token Sliding-Reachability}. In this problem, the input is a graph $G$ with two of its $k$-independent sets $I$ and $J$, and the objective is to determine whether there is a sequence of valid token sliding moves that transform $I$ into $J$.

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arXiv2024

Robust Contraction Decomposition for Minor-Free Graphs and its Applications

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

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arXiv2024

Structural Parameterization of Locating-Dominating Set and Test Cover

We investigate structural parameterizations of two identification problems: LOCATING-DOMINATING SET and TEST COVER. In the first problem, an input is a graph $G$ on $n$ vertices and an integer $k$, and one asks if there is a subset $S$ of $k$ vertices such that any two distinct vertices not in $S$ are dominated by distinct subsets of $S$. In the second problem, an input is a set of items $U$, a set of subsets $\mathcal{F}$ of $U$ called $tests$ and an integer $k$, and one asks if there is a set $S$ of at most $k$ tests such that any two items belong to distinct subsets of tests of $S$. These two problems are "identification" analogues of DOMINATING SET and SET COVER, respectively. Chakraborty et al. [ISAAC 2024] proved that both the problems admit conditional double-exponential lower bounds and matching algorithms when parameterized by treewidth of the input graph. We continue this line of investigation and consider parameters larger than treewidth, like vertex cover number and feedback edge set number. We design a nontrivial dynamic programming scheme to solve TEST COVER in "slightly super-exponential" time $2^{O(|U|\log |U|)}(|U|+|\mathcal{F}|)^{O(1)}$ in the number $|U|$ of items and LOCATING-DOMINATING SET in time $2^{O(\textsf{vc} \log \textsf{vc})} \cdot n^{O(1)}$, where $\textsf{vc}$ is the vertex cover number and $n$ is the order of the graph. This shows that the lower bound results with respect to treewidth from Chakraborty et al. [ISAAC 2024] cannot be extended to vertex cover number. We also show that, parameterized by feedback edge set number, LOCATING-DOMINATING SET admits a linear kernel thereby answering an open question in [Cappelle et al., LAGOS 2021]. Finally, we show that neither LOCATING-DOMINATING SET nor TEST COVER is likely to admit a compression algorithm returning an input with a subquadratic number of bits, unless $\textsf{NP} \subseteq \textsf{coNP}/poly$.

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arXiv2024

Metric Dimension and Geodetic Set Parameterized by Vertex Cover

For a graph $G$, a subset $S\subseteq V(G)$ is called a resolving set of $G$ if, for any two vertices $u,v\in V(G)$, there exists a vertex $w\in S$ such that $d(w,u)\neq d(w,v)$. The Metric Dimension problem takes as input a graph $G$ on $n$ vertices and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. In another metric-based graph problem, Geodetic Set, the input is a graph $G$ and an integer $k$, and the objective is to determine whether there exists a subset $S\subseteq V(G)$ of size at most $k$ such that, for any vertex $u \in V(G)$, there are two vertices $s_1, s_2 \in S$ such that $u$ lies on a shortest path from $s_1$ to $s_2$. These two classical problems turn out to be intractable with respect to the natural parameter, i.e., the solution size, as well as most structural parameters, including the feedback vertex set number and pathwidth. Some of the very few existing tractable results state that they are both FPT with respect to the vertex cover number $vc$. More precisely, we observe that both problems admit an FPT algorithm running in time $2^{\mathcal{O}(vc^2)}\cdot n^{\mathcal{O}(1)}$, and a kernelization algorithm that outputs a kernel with $2^{\mathcal{O}(vc)}$ vertices. We prove that unless the Exponential Time Hypothesis fails, Metric Dimension and Geodetic Set, even on graphs of bounded diameter, neither admit an FPT algorithm running in time $2^{o(vc^2)}\cdot n^{\mathcal(1)}$, nor a kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(vc)}$ vertices. The versatility of our technique enables us to apply it to both these problems. We only know of one other problem in the literature that admits such a tight lower bound. Similarly, the list of known problems with exponential lower bounds on the number of vertices in kernelized instances is very short.

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arXiv2024

Revisiting Path Contraction and Cycle Contraction

The Path Contraction and Cycle Contraction problems take as input an undirected graph $G$ with $n$ vertices, $m$ edges and an integer $k$ and determine whether one can obtain a path or a cycle, respectively, by performing at most $k$ edge contractions in $G$. We revisit these NP-complete problems and prove the following results. Path Contraction admits an algorithm running in $\mathcal{O}^*(2^{k})$ time. This improves over the current algorithm known for the problem [Algorithmica 2014]. Cycle Contraction admits an algorithm running in $\mathcal{O}^*((2 + \epsilon_{\ell})^k)$ time where $0 < \epsilon_{\ell} \leq 0.5509$ is inversely proportional to $\ell = n - k$. Central to these results is an algorithm for a general variant of Path Contraction, namely, Path Contraction With Constrained Ends. We also give an $\mathcal{O}^*(2.5191^n)$-time algorithm to solve the optimization version of Cycle Contraction. Next, we turn our attention to restricted graph classes and show the following results. Path Contraction on planar graphs admits a polynomial-time algorithm. Path Contraction on chordal graphs does not admit an algorithm running in time $\mathcal{O}(n^{2-\epsilon} \cdot 2^{o(tw)})$ for any $\epsilon > 0$, unless the Orthogonal Vectors Conjecture fails. Here, $tw$ is the treewidth of the input graph. The second result complements the $\mathcal{O}(nm)$-time, i.e., $\mathcal{O}(n^2 \cdot tw)$-time, algorithm known for the problem [Discret. Appl. Math. 2014].

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arXiv2024

Conflict and Fairness in Resource Allocation

In the standard model of fair allocation of resources to agents, every agent has some utility for every resource, and the goal is to assign resources to agents so that the agents' welfare is maximized. Motivated by job scheduling, interest in this problem dates back to the work of Deuermeyer et al. [SIAM J. on Algebraic Discrete Methods'82]. Recent works consider the compatibility between resources and assign only mutually compatible resources to an agent. We study a fair allocation problem in which we are given a set of agents, a set of resources, a utility function for every agent over a set of resources, and a {\it conflict graph} on the set of resources (where an edge denotes incompatibility). The goal is to assign resources to the agents such that $(i)$ the set of resources allocated to an agent are compatible with each other, and $(ii)$ the minimum satisfaction of an agent is maximized, where the satisfaction of an agent is the sum of the utility of the assigned resources. Chiarelli et al. [Algorithmica'22] explore this problem from the classical complexity perspective to draw the boundary between the cases that are polynomial-time solvable and those that are \NP-hard. In this article, we study the parameterized complexity of the problem (and its variants) by considering several natural and structural parameters.

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arXiv2024

Double Exponential Lower Bound for Telephone Broadcast

Consider the Telephone Broadcast problem in which an input is a connected graph $G$ on $n$ vertices, a source vertex $s \in V(G)$, and a positive integer $t$. The objective is to decide whether there is a broadcast protocol from $s$ that ensures that all the vertices of $G$ get the message in at most $t$ rounds. We consider the broadcast protocol where, in a round, any node aware of the message can forward it to at most one of its neighbors. As the number of nodes aware of the message can at most double at each round, for a non-trivial instance we have $n \le 2^t$. Hence, the brute force algorithm that checks all the permutations of the vertices runs in time $2^{2^{\calO(t)}} \cdot n^{\calO(1)}$. As our first result, we prove this simple algorithm is the best possible in the following sense. Telephone Broadcast does not admit an algorithm running in time $2^{2^{o(t)}} \cdot n^{\calO(1)}$, unless the \ETH\ fails. To the best of our knowledge, this is only the fourth example of \NP-Complete problem that admits a double exponential lower bound when parameterized by the solution size. It also resolves the question by Fomin, Fraigniaud, and Golovach [WG 2023]. In the same article, the authors asked whether the problem is \FPT\ when parameterized by the feedback vertex set number of the graph. We answer this question in the negative. Telephone Broadcast, when restricted to graphs of the feedback vertex number one, and hence treewidth of two, is \NP-\complete. We find this a relatively rare example of problems that admit a polynomial-time algorithm on trees but is \NP-\complete\ on graphs of treewidth two.

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arXiv2024

Tight (Double) Exponential Bounds for Identification Problems: Locating-Dominating Set and Test Cover

We investigate fine-grained algorithmic aspects of identification problems in graphs and set systems, with a focus on Locating-Dominating Set and Test Cover. We prove the (tight) conditional lower bounds for these problems when parameterized by treewidth and solution as. Formally, \textsc{Locating-Dominating Set} (respectively, \textsc{Test Cover}) parameterized by the treewidth of the input graph (respectively, of the natural auxiliary graph) does not admit an algorithm running in time $2^{2^{o(tw)}} \cdot poly(n)$ (respectively, $2^{2^{o(tw)}} \cdot poly(|U| + |\mathcal{F}|))$. This result augments the small list of NP-Complete problems that admit double exponential lower bounds when parameterized by treewidth. Then, we first prove that \textsc{Locating-Dominating Set} does not admit an algorithm running in time $2^{o(k^2)} \cdot poly(n)$, nor a polynomial time kernelization algorithm that reduces the solution size and outputs a kernel with $2^{o(k)}$ vertices, unless the \ETH\ fails. To the best of our knowledge, \textsc{Locating-Dominating Set} is the first problem that admits such an algorithmic lower-bound (with a quadratic function in the exponent) when parameterized by the solution size. Finally, we prove that \textsc{Test Cover} does not admit an algorithm running in time $2^{2^{o(k)}} \cdot poly(|U| + |\mathcal{F}|)$. This is also a rare example of the problem that admits a double exponential lower bound when parameterized by the solution size. We also present algorithms whose running times match the above lower bounds.

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arXiv2023

Parameterized Complexity of Biclique Contraction and Balanced Biclique Contraction

In this work, we initiate the complexity study of Biclique Contraction and Balanced Biclique Contraction. In these problems, given as input a graph G and an integer k, the objective is to determine whether one can contract at most k edges in G to obtain a biclique and a balanced biclique, respectively. We first prove that these problems are NP-complete even when the input graph is bipartite. Next, we study the parameterized complexity of these problems and show that they admit single exponential-time FPT algorithms when parameterized by the number k of edge contractions. Then, we show that Balanced Biclique Contraction admits a quadratic vertex kernel while Biclique Contraction does not admit any polynomial compression (or kernel) under standard complexity-theoretic assumptions. We also give faster FPT algorithms for contraction to restricted bicliques.

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