Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

Kernelization is a concept that enables the formal mathematical analysis of data reduction through the framework of parameterized complexity. Intensive research into the Vertex Cover problem has shown that there is a preprocessing algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G\u27,k\u27) in polynomial time with the guarantee that G\u27 has at most 2k\u27 vertices (and thus O((k\u27)^2) edges) with k\u27 <= k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number VC(G) since fvs(G) <= VC(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G\u27,X\u27,k\u27) such that k\u27 <= k, |X\u27| <= |X| and most importantly |V(G\u27)| <= 2k and |V(G\u27)| in O(|X\u27|^3). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have polynomial kernel when parameterized by fvs(G) unless the polynomial hierarchy collapses to the third level (PH=Sigma_3^p). Our work is one of the first examples of research in kernelization using a non-standard parameter, and shows that this approach can yield interesting computational insights. To obtain our results we make extensive use of the combinatorial structure of independent sets in forests

Vertex Cover Kernelization Revisited: Upper and Lower Bounds for a Refined Parameter

An important result in the study of polynomial-time preprocessing shows that there is an algorithm which given an instance (G,k) of Vertex Cover outputs an equivalent instance (G',k') in polynomial time with the guarantee that G' has at most 2k' vertices (and thus O((k')^2) edges) with k' <= k. Using the terminology of parameterized complexity we say that k-Vertex Cover has a kernel with 2k vertices. There is complexity-theoretic evidence that both 2k vertices and Theta(k^2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size fvs(G) of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number vc(G) since fvs(G) <= vc(G) and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in fvs(G): an instance (G,X,k) of Vertex Cover, where X is a feedback vertex set for G, can be transformed in polynomial time into an equivalent instance (G',X',k') such that |V(G')| <= 2k and |V(G')| <= O(|X'|^3). A similar result holds when the feedback vertex set X is not given along with the input. In sharp contrast we show that the Weighted Vertex Cover problem does not have a polynomial kernel when parameterized by the cardinality of a given vertex cover of the graph unless NP is in coNP/poly and the polynomial hierarchy collapses to the third level.Comment: Published in "Theory of Computing Systems" as an Open Access publicatio

A survey of parameterized algorithms and the complexity of edge modification

The survey is a comprehensive overview of the developing area of parameterized algorithms for graph modification problems. It describes state of the art in kernelization, subexponential algorithms, and parameterized complexity of graph modification. The main focus is on edge modification problems, where the task is to change some adjacencies in a graph to satisfy some required properties. To facilitate further research, we list many open problems in the area.publishedVersio

On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT

Hitting Set is a classic problem in combinatorial optimization. Its input consists of a set system F over a finite universe U and an integer t; the question is whether there is a set of t elements that intersects every set in F. The Hitting Set problem parameterized by the size of the solution is a well-known W[2]-complete problem in parameterized complexity theory. In this paper we investigate the complexity of Hitting Set under various structural parameterizations of the input. Our starting point is the folklore result that Hitting Set is polynomial-time solvable if there is a tree T on vertex set U such that the sets in F induce connected subtrees of T. We consider the case that there is a treelike graph with vertex set U such that the sets in F induce connected subgraphs; the parameter of the problem is a measure of how treelike the graph is. Our main positive result is an algorithm that, given a graph G with cyclomatic number k, a collection P of simple paths in G, and an integer t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of size t that hits all paths in P. It is based on a connection to the 2-SAT problem in multiple valued logic. For other parameterizations we derive W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was corrected in this update.

Parameterized Complexity of Vertex Splitting to Pathwidth at most 1

Motivated by the planarization of 2-layered straight-line drawings, we consider the problem of modifying a graph such that the resulting graph has pathwidth at most 1. The problem Pathwidth-One Vertex Explosion (POVE) asks whether such a graph can be obtained using at most $k$ vertex explosions, where a vertex explosion replaces a vertex $v$ by deg$(v)$ degree-1 vertices, each incident to exactly one edge that was originally incident to $v$. For POVE, we give an FPT algorithm with running time $O(4^k \cdot m)$ and an $O(k^2)$ kernel, thereby improving over the $O(k^6)$-kernel by Ahmed et al. [GD 22] in a more general setting. Similarly, a vertex split replaces a vertex $v$ by two distinct vertices $v_1$ and $v_2$ and distributes the edges originally incident to $v$ arbitrarily to $v_1$ and $v_2$. Analogously to POVE, we define the problem variant Pathwidth-One Vertex Splitting (POVS) that uses the split operation instead of vertex explosions. Here we obtain a linear kernel and an algorithm with running time $O((6k+12)^k \cdot m)$. This answers an open question by Ahmed et al. [GD22]. Finally, we consider the problem $\Pi$ Vertex Splitting ($\Pi$-VS), which generalizes the problem POVS and asks whether a given graph can be turned into a graph of a specific graph class $\Pi$ using at most $k$ vertex splits. For graph classes $\Pi$ that can be tested in monadic second-order graph logic (MSO$_2$), we show that the problem $\Pi$-VS can be expressed as an MSO$_2$ formula, resulting in an FPT algorithm for $\Pi$-VS parameterized by $k$ if $\Pi$ additionally has bounded treewidth. We obtain the same result for the problem variant using vertex explosions

A Constant-Factor Approximation for Weighted Bond Cover

The Weighted ?-Vertex Deletion for a class ? of graphs asks, weighted graph G, for a minimum weight vertex set S such that G-S ? ?. The case when ? is minor-closed and excludes some graph as a minor has received particular attention but a constant-factor approximation remained elusive for Weighted ?-Vertex Deletion. Only three cases of minor-closed ? are known to admit constant-factor approximations, namely Vertex Cover, Feedback Vertex Set and Diamond Hitting Set. We study the problem for the class ? of ?_c-minor-free graphs, under the equivalent setting of the Weighted c-Bond Cover problem, and present a constant-factor approximation algorithm using the primal-dual method. For this, we leverage a structure theorem implicit in [Joret et al., SIDMA\u2714] which states the following: any graph G containing a ?_c-minor-model either contains a large two-terminal protrusion, or contains a constant-size ?_c-minor-model, or a collection of pairwise disjoint constant-sized connected sets that can be contracted simultaneously to yield a dense graph. In the first case, we tame the graph by replacing the protrusion with a special-purpose weighted gadget. For the second and third case, we provide a weighting scheme which guarantees a local approximation ratio. Besides making an important step in the quest of (dis)proving a constant-factor approximation for Weighted ?-Vertex Deletion, our result may be useful as a template for algorithms for other minor-closed families