We study several generalizations of the Steiner Tree problem that are motivated by the design of power networks. While Steiner Tree asks for a single minimum-cost tree that connects a given set of terminal vertices, a power network typically consists of multiple trees. Each tree connects to a subset of the terminals, to avoid electrical overloads. The cost of installing a power network is therefore determined by two factors: the total length of the cables in the network and the cost of digging underground trenches into which the cables are placed. Since the digging costs can be substantial, to minimize the total cost of the network it might be necessary to place multiple cables into the same trench. These characteristics lead to variations of Steiner Tree in which the goal is to compute a minimum-cost set of Steiner trees, all with a common root, that together connect a given terminal set while balancing the power demand of the terminals in each tree. Two important variations arise depending on whether the network is intended for low-voltage or high-voltage power. In the low-voltage setting, there is substantial power loss across the cables which effectively means that the maximum depth of any tree in the solution has to be bounded. No such depth bound applies to the high-voltage setting. We investigate the parameterized complexity of several power network design problems, using the number of terminals as the parameter. While this parameterization of the standard Steiner Tree problem is fixed-parameter tractable, many of our variants are W[1]-hard. For low-voltage networks (bounded-depth trees), we present an XP-algorithm for planar inputs, which exploits a nontrivial bound on the treewidth of solution subgraphs. We provide an intricate reduction from Grid Tiling to establish that the resulting algorithm is tight under the Exponential Time Hypothesis. The XP-algorithm extends to the high-voltage setting and to general graphs, albeit at a cost in the running time. For high-voltage networks, we prove that the problem remains W[1]-hard on planar graphs. Finally, we explore a variation of the cost model for sharing digging costs in which both problems become fixed-parameter tractable.

For an optimization problem Π on graphs whose solutions are vertex sets, a vertex v is called c-essential for Π if all solutions of size at most contain v . Recent work showed that polynomial-time algorithms to detect c -essential vertices can be used to reduce the search-space of fixed-parameter tractable algorithms solving such problems parameterized by the size k of the solution. We provide several new upper- and lower bounds for detecting essential vertices. For example, we give a polynomial-time algorithm for 3 -Essential detection for Vertex Multicut , which translates into an algorithm that finds a minimum multicut of an undirected n -vertex graph G in time 2 O ( ℓ 3 ) ⋅ n O ( 1 ) , where ℓ is the number of vertices in an optimal solution that are not 3-essential. Our positive results are obtained by analyzing the integrality gaps of certain linear programs. Our lower bounds show that for sufficiently small values of c , the detection task becomes NP-hard assuming the Unique Games Conjecture . For example, we show that ( 2 − ε )-Essential detection for Directed Feedback Vertex Set is NP-hard under this conjecture, thereby proving that the existing algorithm that detects 2-essential vertices is best-possible.

The Planar Separator Theorem, which states that any planar graph 𝒢 has a separator consisting of O(√n) nodes whose removal partitions 𝒢 into components of size at most 2n/3, is a widely used tool to obtain fast algorithms on planar graphs. Intersection graphs of disks, which generalize planar graphs, do not admit such separators. It has recently been shown that disk graphs do admit so-called clique-based separators that consist of O(√n) cliques. This result has been generalized to intersection graphs of various other types of disk-like objects. Unfortunately, segment intersection graphs do not admit small clique-based separators, because they can contain arbitrarily large bicliques. This is true even in the simple case of axis-aligned segments. In this paper we therefore introduce biclique-based separators (and, in particular, star-based separators), which are separators consisting of a small number of bicliques (or stars). We prove that any c-oriented set of n segments in the plane, where c is a constant, admits a star-based separator consisting of O(√n) stars. In fact, our result is more general, as it applies to any set of n pseudo-segments that is partitioned into c subsets such that the pseudo-segments in the same subset are pairwise disjoint. We extend our result to intersection graphs of c-oriented polygons. These results immediately lead to an almost-exact distance oracle for such intersection graphs, which has O(n√n) storage and O(√n) query time, and that can report the hop-distance between any two query nodes in the intersection graph with an additive error of at most 2. This is the first distance oracle for such types of intersection graphs that has subquadratic storage and sublinear query time and that only has an additive error.

We study the parameterized complexity of a recently introduced multi-agent variant of the Kidney Exchange problem. Given a directed graph G and integers d and k, the standard problem asks whether G contains a packing of vertex-disjoint cycles, each of length ≤ d, covering at least k vertices in total. In the multi-agent setting we consider, the vertex set is partitioned over several agents who reject a cycle packing as solution if it can be modified into an alternative packing that covers more of their own vertices. A cycle packing is called rejection-proof if no agent rejects it and the problem asks whether such a packing exists that covers at least k vertices. We exploit the sunflower lemma on a set packing formulation of the problem to give a kernel for this Σ₂^P-complete problem that is polynomial in k for all constant values of d. We also provide a 2^(k log k) + n^(1) algorithm based on it and show that this FPT algorithm is asymptotically optimal under the ETH. Further, we generalize the problem by including an additional positive integer c in the input that naturally captures how much agents can modify a given cycle packing to reject it. For every constant c, the resulting problem simplifies from being Σ₂^P-complete to NP-complete. The super-exponential lower bound already holds for c = 2, though. We present an ad-hoc single-exponential algorithm for c = 1. These results reveal an interesting discrepancy between the classical and parameterized complexity of the problem and give a good view of what makes it hard.

The celebrated notion of important separators bounds the number of small ( S , T ) -separators in a graph which are ‘farthest from S ’ in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive, tailored to undirected graphs, that is phrased in terms of k-secluded vertex sets: sets with an open neighborhood of size at most k . In this terminology, the bound on important separators says that there are at most 4 k maximal k -secluded connected vertex sets C containing S but disjoint from T . We generalize this statement significantly: even when we demand that G [ C ] avoids a finite set F of forbidden induced subgraphs, the number of such maximal subgraphs is 2 O ( k ) and they can be enumerated efficiently. This enumeration algorithm allows us to give improved parameterized algorithms for Connected k -Secluded F -Free Subgraph and for deleting into scattered graph classes.

Abstract. In the [Formula: see text]-minor-free deletion problem we are given an undirected graph [Formula: see text] and the goal is to find a minimum vertex set that intersects all minor models of graphs from the family [Formula: see text]. This captures numerous important problems including Vertex cover, Feedback vertex set, Treewidth-[Formula: see text] modulator, and Vertex planarization. In the latter one, we ask for a minimum vertex set whose removal makes the graph planar. This is a special case of [Formula: see text]-minor-free deletion for the family [Formula: see text]. Whenever the family [Formula: see text] contains at least one planar graph, then [Formula: see text]-minor-free deletion is known to admit a constant-factor approximation algorithm and a polynomial kernelization [F. Fomin et al., Proceedings of the 53 rd Annual Symposium on Foundations of Computer Science, IEEE, 2012, pp. 470–479]. A polynomial kernelization is a polynomial-time algorithm that, given a graph [Formula: see text] and integer [Formula: see text], outputs a graph [Formula: see text] on [Formula: see text] vertices and integer [Formula: see text], so that [Formula: see text] if and only if [Formula: see text]. The Vertex planarization problem is arguably the simplest setting for which [Formula: see text] does not contain a planar graph and the existence of a constant-factor approximation or a polynomial kernelization remains a major open problem. In this work we show that Vertex planarization admits an algorithm which is a combination of both approaches. Namely, we present a polynomial [Formula: see text]-approximate kernelization, for some constant [Formula: see text], based on the framework of lossy kernelization [D. Lokshtanov et al., Proceedings of the 49 th Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2017, pp. 224–237]. Simply speaking, when given a graph [Formula: see text] and integer [Formula: see text], we show how to compute a graph [Formula: see text] on [Formula: see text] vertices so that any [Formula: see text]-approximate solution to [Formula: see text] can be lifted to an [Formula: see text]-approximate solution to [Formula: see text], as long as [Formula: see text]. In order to achieve this, we develop a toolkit for sparsification of planar graphs which approximately preserves all separators and near-separators between subsets of the given terminal set. Our result yields an improvement over the state-of-the-art approximation algorithms for Vertex planarization. The problem admits a polynomial-time [Formula: see text]-approximation algorithm, for any [Formula: see text], and a quasi-polynomial-time [Formula: see text]-approximation algorithm, where [Formula: see text] is the input size, both randomized [K. Kawarabayashi and A. Sidiropoulos, Proceedings of the 51 st Annual ACM SIGACT Symposium on Theory of Computing, ACM, 2019, pp. 164–175]. By pipelining these algorithms with our approximate kernelization, we improve the approximation factors to respectively [Formula: see text] and [Formula: see text].

In the Steiner Tree problem we are given an undirected edge-weighted graph as input, along with a set K of vertices called terminals. The task is to output a minimum-weight connected subgraph that spans all the terminals. The famous Dreyfus-Wagner algorithm running in 3^{|K|}poly(n) time shows that the problem is fixed-parameter tractable parameterized by the number of terminals. We present fixed-parameter tractable algorithms for Steiner Tree using structurally smaller parameterizations. Our first result concerns the parameterization by a multiway cut S of the terminals, which is a vertex set S (possibly containing terminals) such that each connected component of G-S contains at most one terminal. We show that Steiner Tree can be solved in 2^{𝒪(|S|log|S|)}poly(n) time and polynomial space, where S is a minimum multiway cut for K. The algorithm is based on the insight that, after guessing how an optimal Steiner tree interacts with a multiway cut S, computing a minimum-cost solution of this type can be formulated as minimum-cost bipartite matching. Our second result concerns a new hybrid parameterization called K-free treewidth that simultaneously refines the number of terminals |K| and the treewidth of the input graph. By utilizing recent work on ℋ-Treewidth in order to find a corresponding decomposition of the graph, we give an algorithm that solves Steiner Tree in time 2^{𝒪(k)} poly(n), where k denotes the K-free treewidth of the input graph. To obtain this running time, we show how the rank-based approach for solving Steiner Tree parameterized by treewidth can be extended to work in the setting of K-free treewidth, by exploiting existing algorithms parameterized by |K| to compute the table entries of leaf bags of a tree K-free decomposition.

For a fixed graph $H$, the $H$-SUBGRAPH HITTING problem consists in deleting the minimum number of vertices from an input graph to obtain a graph without any occurrence of $H$ as a subgraph. This problem can be seen as a generalization of VERTEX COVER, which corresponds to the case $H = K_2$. We initiate a study of $H$-SUBGRAPH HITTING from the point of view of characterizing structural parameterizations that allow for polynomial kernels, within the recently active framework of taking as the parameter the number of vertex deletions to obtain a graph in a "simple" class $C$. Our main contribution is to identify graph parameters that, when $H$-SUBGRAPH HITTING is parameterized by the vertex-deletion distance to a class $C$ where any of these parameters is bounded, and assuming standard complexity assumptions and that $H$ is biconnected, allow us to prove the following sharp dichotomy: the problem admits a polynomial kernel if and only if $H$ is a clique. These new graph parameters are inspired by the notion of $C$-elimination distance introduced by Bulian and Dawar [Algorithmica 2016], and generalize it in two directions. Our results also apply to the version of the problem where one wants to hit $H$ as an induced subgraph, and imply in particular, that the problems of hitting minors and hitting (induced) subgraphs have a substantially different behavior with respect to the existence of polynomial kernels under structural parameterizations.

The NP-hard Odd Cycle Transversal problem asks for a minimum vertex set whose removal from an undirected input graph $G$ breaks all odd cycles, and thereby yields a bipartite graph. The problem is well-known to be fixed-parameter tractable when parameterized by the size $k$ of the desired solution. It also admits a randomized kernelization of polynomial size, using the celebrated matroid toolkit by Kratsch and Wahlström. The kernelization guarantees a reduction in the total $\textit{size}$ of an input graph, but does not guarantee any decrease in the size of the solution to be sought; the latter governs the size of the search space for FPT algorithms parameterized by $k$. We investigate under which conditions an efficient algorithm can detect one or more vertices that belong to an optimal solution to Odd Cycle Transversal. By drawing inspiration from the popular $\textit{crown reduction}$ rule for Vertex Cover, and the notion of $\textit{antler decompositions}$ that was recently proposed for Feedback Vertex Set, we introduce a graph decomposition called $\textit{tight odd cycle cut}$ that can be used to certify that a vertex set is part of an optimal odd cycle transversal. While it is NP-hard to compute such a graph decomposition, we develop parameterized algorithms to find a set of at least $k$ vertices that belong to an optimal odd cycle transversal when the input contains a tight odd cycle cut certifying the membership of $k$ vertices in an optimal solution. The resulting algorithm formalizes when the search space for the solution-size parameterization of Odd Cycle Transversal can be reduced by preprocessing. To obtain our results, we develop a graph reduction step that can be used to simplify the graph to the point that the odd cycle cut can be detected via color coding.

We study efficient preprocessing for the undirected Feedback Vertex Set problem, a fundamental problem in graph theory which asks for a minimum-sized vertex set whose removal yields an acyclic graph. More precisely, we aim to determine for which parameterizations this problem admits a polynomial kernel. While a characterization is known for the related Vertex Cover problem based on the recently introduced notion of bridge-depth, it remained an open problem whether this could be generalized to Feedback Vertex Set. The answer turns out to be negative; the existence of polynomial kernels for structural parameterizations for Feedback Vertex Set is governed by the elimination distance to a forest. Under the standard assumption NP⁄⊆coNP/poly, we prove that for any minor-closed graph class G, Feedback Vertex Set parameterized by the size of a modulator to G has a polynomial kernel if and only if G has bounded elimination distance to a forest. This captures and generalizes all existing kernels for structural parameterizations of the Feedback Vertex Set problem. Download : Download high-res image (47KB)Download : Download full-size image

The goal of this paper is to open up a new research direction aimed at understanding the power of preprocessing in speeding up algorithms that solve NP-hard problems exactly. We explore this direction for the classic Feedback Vertex Set problem on undirected graphs, leading to a new type of graph structure called antler decomposition, which identifies vertices that belong to an optimal solution. It is an analogue of the celebrated crown decomposition which has been used for Vertex Cover. We develop the graph structure theory around such decompositions and develop fixed-parameter tractable algorithms to find them, parameterized by the number of vertices for which they witness presence in an optimal solution. This reduces the search space of fixed-parameter tractable algorithms parameterized by the solution size that solve Feedback Vertex Set.

Abstract. We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a nonempty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a [Formula: see text]-essential vertex as one that is contained in all [Formula: see text]-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of nonessential vertices in the solution.

The notion of ℋ-treewidth, where ℋ is a hereditary graph class, was recently introduced as a generalization of the treewidth of an undirected graph. Roughly speaking, a graph of ℋ-treewidth at most k can be decomposed into (arbitrarily large) ℋ-subgraphs which interact only through vertex sets of size 𝒪(k) which can be organized in a tree-like fashion. ℋ-treewidth can be used as a hybrid parameterization to develop fixed-parameter tractable algorithms for ℋ-deletion problems, which ask to find a minimum vertex set whose removal from a given graph G turns it into a member of ℋ. The bottleneck in the current parameterized algorithms lies in the computation of suitable tree ℋ-decompositions. We present FPT-approximation algorithms to compute tree ℋ-decompositions for hereditary and union-closed graph classes ℋ. Given a graph of ℋ-treewidth k, we can compute a 5-approximate tree ℋ-decomposition in time f(𝒪(k)) ⋅ n^𝒪(1) whenever ℋ-deletion parameterized by solution size can be solved in time f(k) ⋅ n^𝒪(1) for some function f(k) ≥ 2^k. The current-best algorithms either achieve an approximation factor of k^𝒪(1) or construct optimal decompositions while suffering from non-uniformity with unknown parameter dependence. Using these decompositions, we obtain algorithms solving Odd Cycle Transversal in time 2^𝒪(k) ⋅ n^𝒪(1) parameterized by bipartite-treewidth and Vertex Planarization in time 2^𝒪(k log k) ⋅ n^𝒪(1) parameterized by planar-treewidth, showing that these can be as fast as the solution-size parameterizations and giving the first ETH-tight algorithms for parameterizations by hybrid width measures.

The celebrated notion of important separators bounds the number of small $(S,T)$-separators in a graph which are 'farthest from $S$' in a technical sense. In this paper, we introduce a generalization of this powerful algorithmic primitive that is phrased in terms of $k$-secluded vertex sets: sets with an open neighborhood of size at most $k$. In this terminology, the bound on important separators says that there are at most $4^k$ maximal $k$-secluded connected vertex sets $C$ containing $S$ but disjoint from $T$. We generalize this statement significantly: even when we demand that $G[C]$ avoids a finite set $\mathcal{F}$ of forbidden induced subgraphs, the number of such maximal subgraphs is $2^{O(k)}$ and they can be enumerated efficiently. This allows us to make significant improvements for two problems from the literature. Our first application concerns the 'Connected $k$-Secluded $\mathcal{F}$-free subgraph' problem, where $\mathcal{F}$ is a finite set of forbidden induced subgraphs. Given a graph in which each vertex has a positive integer weight, the problem asks to find a maximum-weight connected $k$-secluded vertex set $C \subseteq V(G)$ such that $G[C]$ does not contain an induced subgraph isomorphic to any $F \in \mathcal{F}$. The parameterization by $k$ is known to be solvable in triple-exponential time via the technique of recursive understanding, which we improve to single-exponential. Our second application concerns the deletion problem to scattered graph classes. Here, the task is to find a vertex set of size at most $k$ whose removal yields a graph whose each connected component belongs to one of the prescribed graph classes $Π_1, \ldots, Π_d$. We obtain a single-exponential algorithm whenever each class $Π_i$ is characterized by a finite number of forbidden induced subgraphs. This generalizes and improves upon earlier results in the literature.

We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T \subseteq V(G)$, it asks whether there is a vertex set $S \subseteq V(G) \setminus T$ of size at most $k$ such that in graph $G-S$, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in $G-S$ by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of $G-S$. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time $2^{O(k \log k)} * n^{O(1)}$. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size $k$ plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph $G$ and terminal set $T \subseteq V(G)$ along with a single vertex $x \in V(G)$ that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing $x$.

A kernelization for a parameterized decision problem $\mathcal{Q}$ is a polynomial-time preprocessing algorithm that reduces any parameterized instance $(x,k)$ into an instance $(x',k')$ whose size is bounded by a function of $k$ alone and which has the same yes/no answer for $\mathcal{Q}$. Such preprocessing algorithms cannot exist in the context of counting problems, when the answer to be preserved is the number of solutions, since this number can be arbitrarily large compared to $k$. However, we show that for counting minimum feedback vertex sets of size at most $k$, and for counting minimum dominating sets of size at most $k$ in a planar graph, there is a polynomial-time algorithm that either outputs the answer or reduces to an instance $(G',k')$ of size polynomial in $k$ with the same number of minimum solutions. This shows that a meaningful theory of kernelization for counting problems is possible and opens the door for future developments. Our algorithms exploit that if the number of solutions exceeds $2^{\mathsf{poly}(k)}$, the size of the input is exponential in terms of $k$ so that the running time of a parameterized counting algorithm can be bounded by $\mathsf{poly}(n)$. Otherwise, we can use gadgets that slightly increase $k$ to represent choices among $2^{O(k)}$ options by only $\mathsf{poly}(k)$ vertices.

The q-Coloring problem asks whether the vertices of a graph can be properly colored with q colors. In this paper we perform a fine-grained analysis of the complexity of q-Coloring with respect to a hierarchy of structural parameters. We show that unless the Exponential Time Hypothesis fails, there is no constant θ such that q-Coloring parameterized by the size k of a vertex cover can be solved in O∗(θk) time for all fixed q. We prove that there are O∗((q−ɛ)k) time algorithms where k is the vertex deletion distance to several graph classes for which q-Coloring is known to be solvable in polynomial time, including all graph classes F whose (q+1)-colorable members have bounded treedepth. In contrast, we prove that if F is the class of paths – some of the simplest graphs of unbounded treedepth – then no such algorithm can exist unless the Strong Exponential Time Hypothesis fails.

We revisit the k -Secluded Tree problem. Given a vertex-weighted undirected graph G , its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k . We present a fixed-parameter tractable algorithm that solves the problem in time 2 O ( k log ⁡ k ) ⋅ n O ( 1 ) , improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k -secluded tree by branching on vertices in the open neighborhood of the current tree T . To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k -secluded supertree T ′ ⊇ T once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k -secluded trees, which allows us to count them as well.

We introduce and study two natural generalizations of the Connected Vertex Cover (VC) problem: the p-Edge-Connected and p-Vertex-Connected VC problem (where p≥2 is a fixed integer). We obtain an 2O(pk)nO(1)-time algorithm for p-Edge-Connected VC and an 2O(k2)nO(1)-time algorithm for p-Vertex-Connected VC. Thus, like Connected VC, both constrained VC problems are FPT. Furthermore, like Connected VC, neither problem admits a polynomial kernel unless NP ⊆ coNP/poly, which is highly unlikely. We prove however that both problems admit time efficient polynomial sized approximate kernelization schemes. Finally, we describe a 2(p+1)-approximation algorithm for the p-Edge-Connected VC. The proofs for the new VC problems require more sophisticated arguments than for Connected VC. In particular, for the approximation algorithm we use Gomory-Hu trees and for the approximate kernels a result on small-size spanning p-vertex/edge-connected subgraphs of a p-vertex/edge-connected graph by Nishizeki and Poljak (1994) and Nagamochi and Ibaraki (1992).

We investigate preprocessing for vertex-subset problems on graphs. While the notion of kernelization, originating in parameterized complexity theory, is a formalization of provably effective preprocessing aimed at reducing the total instance size, our focus is on finding a non-empty vertex set that belongs to an optimal solution. This decreases the size of the remaining part of the solution which still has to be found, and therefore shrinks the search space of fixed-parameter tractable algorithms for parameterizations based on the solution size. We introduce the notion of a c-essential vertex as one that is contained in all c-approximate solutions. For several classic combinatorial problems such as Odd Cycle Transversal and Directed Feedback Vertex Set, we show that under mild conditions a polynomial-time preprocessing algorithm can find a subset of an optimal solution that contains all 2-essential vertices, by exploiting packing/covering duality. This leads to FPT algorithms to solve these problems where the exponential term in the running time depends only on the number of non-essential vertices in the solution.

In the F-minor-free deletion problem we are given an undirected graph G and the goal is to find a minimum vertex set that intersects all minor models of graphs from the family F. This captures numerous important problems including Vertex cover, Feedback vertex set, Treewidth-η modulator, and Vertex planarization. In the latter one, we ask for a minimum vertex set whose removal makes the graph planar. This is a special case of F-minor-free deletion for the family F = {K5, K3,3}.

Abstract We study efficient preprocessing for the undirected Feedback Vertex Set problem, a fundamental problem in graph theory which asks for a minimum-sized vertex set whose removal yields an acyclic graph. More precisely, we aim to determine for which parameterizations this problem admits a polynomial kernel. While a characterization is known for the related Vertex Cover problem based on the recently introduced notion of bridge-depth, it remained an open problem whether this could be generalized to Feedback Vertex Set. The answer turns out to be negative; the existence of polynomial kernels for structural parameterizations for Feedback Vertex Set is governed by the elimination distance to a forest. Under the standard assumption $$\textrm{NP}\not \subseteq \textrm{coNP}/\textrm{poly}$$ , we prove that for any minor-closed graph class $$\mathcal {G}$$ , Feedback Vertex Set parameterized by the size of a modulator to $$\mathcal {G}$$ has a polynomial kernel if and only if $$\mathcal {G}$$ has bounded elimination distance to a forest. This captures and generalizes all existing kernels for structural parameterizations of the Feedback Vertex Set problem.

Abstract We revisit the k-Secluded Tree problem. Given a vertex-weighted undirected graph G, its objective is to find a maximum-weight induced subtree T whose open neighborhood has size at most k. We present a fixed-parameter tractable algorithm that solves the problem in time $$2^{\mathcal {O} (k \log k)}\cdot n^{\mathcal {O} (1)}$$ , improving on a double-exponential running time from earlier work by Golovach, Heggernes, Lima, and Montealegre. Starting from a single vertex, our algorithm grows a k-secluded tree by branching on vertices in the open neighborhood of the current tree T. To bound the branching depth, we prove a structural result that can be used to identify a vertex that belongs to the neighborhood of any k-secluded supertree $$T' \supseteq T$$ once the open neighborhood of T becomes sufficiently large. We extend the algorithm to enumerate compact descriptions of all maximum-weight k-secluded trees, which allows us to count the number of such trees containing a specified vertex in the same running time.

Abstract In the $${\varvec{\mathcal {F}}}$$ F - Minor - Free Deletion problem one is given an undirected graph $${\varvec{G}}$$ G , an integer $${\varvec{k}}$$ k , and the task is to determine whether there exists a vertex set $${\varvec{S}}$$ S of size at most $${\varvec{k}}$$ k , so that $${\varvec{G}}-{\varvec{S}}$$ G - S contains no graph from the finite family $${\varvec{\mathcal {F}}}$$ F as a minor. It is known that whenever $${\varvec{\mathcal {F}}}$$ F contains at least one planar graph, then $${\varvec{\mathcal {F}}}$$ F - Minor - Free Deletion admits a polynomial kernel, that is, there is a polynomial-time algorithm that outputs an equivalent instance of size $${\varvec{k}}^{{\varvec{\mathcal {O}}}{} {\textbf {(1)}}}$$ k O ( 1 ) [Fomin, Lokshtanov, Misra, Saurabh; FOCS 2012]. However, this result relies on non-constructive arguments based on well-quasi-ordering and does not provide a concrete bound on the kernel size. We study the Outerplanar Deletion problem, in which we want to remove at most $${\varvec{k}}$$ k vertices from a graph to make it outerplanar. This is a special case of $${\varvec{\mathcal {F}}}$$ F - Minor - Free Deletion for the family $${\varvec{\mathcal {F}}} = \{{\varvec{K}}_{{\textbf {4}}}, {\varvec{K}}_{{{\textbf {2,3}}}}\}$$ F = { K 4 , K 2 , 3 } . The class of outerplanar graphs is arguably the simplest class of graphs

We consider the Π-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties Π. Given an input graph G, this problem asks whether there is a subset of at most k vertices whose removal ensures the resulting graph does not contain a graph from Π as an induced subgraph. We introduce the concept of characterizing a graph property Π by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for Π-Free Deletion parameterized by the size of a vertex cover. The resulting framework captures problems such as AT-Free Deletion, Wheel-free Deletion, and Interval Deletion. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. (2014) [18].

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a predetermined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$ admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

We investigate computational problems involving large weights through the lens of kernelization, which is a framework of polynomial-time preprocessing aimed at compressing the instance size. Our main focus is the weighted Clique problem, where we are given an edge-weighted graph and the goal is to detect a clique of total weight equal to a prescribed value. We show that the weighted variant, parameterized by the number of vertices $n$, is significantly harder than the unweighted problem by presenting an $O(n^{3 - \varepsilon})$ lower bound on the size of the kernel, under the assumption that NP $\not \subseteq$ coNP/poly. This lower bound is essentially tight: we show that we can reduce the problem to the case with weights bounded by $2^{O(n)}$, which yields a randomized kernel of $O(n^3)$ bits. We generalize these results to the weighted $d$-Uniform Hyperclique problem, Subset Sum, and weighted variants of Boolean Constraint Satisfaction Problems (CSPs). We also study weighted minimization problems and show that weight compression is easier when we only want to preserve the collection of optimal solutions. Namely, we show that for node-weighted Vertex Cover on bipartite graphs it is possible to maintain the set of optimal solutions using integer weights from the range $[1, n]$, but if we want to maintain the ordering of the weights of all inclusion-minimal solutions, then weights as large as $2^{Ω(n)}$ are necessary.

We study the parameterized complexity of various classic vertex-deletion problems such as Odd cycle transversal, Vertex planarization, and Chordal vertex deletion under hybrid parameterizations. Existing FPT algorithms for these problems either focus on the parameterization by solution size, detecting solutions of size k in time f(k) · nO(1), or width parameterizations, finding arbitrarily large optimal solutions in time f(w) · nO(1) for some width measure w like treewidth. We unify these lines of research by presenting FPT algorithms for parameterizations that can simultaneously be arbitrarily much smaller than the solution size and the treewidth.

For a fixed finite family of graphs F, the F-Minor-Free Deletion problem takes as input a graph G and integer ℓ and asks whether a size-ℓ vertex set X exists such that G−X is F-minor-free. {K2}-Minor-Free Deletion and {K3}-Minor-Free Deletion encode Vertex Cover and Feedback Vertex Set respectively. When parameterized by the feedback vertex number of G these two problems are known to admit a polynomial kernelization. We show {P3}-Minor-Free Deletion parameterized by the feedback vertex number is MK[2]-hard. This rules out the existence of a polynomial kernel assuming NP⊈coNP/poly. Our hardness result generalizes to any F containing only graphs with a connected component of at least 3 vertices, using as parameter the vertex-deletion distance to treewidth min⁡tw(F), where min⁡tw(F) denotes the minimum treewidth of the graphs in F. For all other families F we present a polynomial Turing kernelization. Our results extend to F-Subgraph-Free Deletion.

We show that Odd Cycle Transversal and Vertex Multiway Cut admit deterministic polynomial kernels when restricted to planar graphs and parameterized by the solution size. This answers a question of Saurabh. On the way to these results, we provide an efficient sparsification routine in the flavor of the sparsification routine used for the Steiner Tree problem in planar graphs [Pilipczuk et al., ACM Trans. Algorithms, 14 (2018), 53]. It differs from the previous work because it preserves the existence of low-cost subgraphs that are not necessarily Steiner trees in the original plane graph, but structures that turn into (supergraphs of) Steiner trees after adding all edges between pairs of vertices that lie on a common face. We also show connections between Vertex Multiway Cut and the Vertex Planarization problem, where the existence of a polynomial kernel remains an important open problem.

We analyze two classic variants of the T RAVELING S ALESMAN P ROBLEM ( TSP ) using the toolkit of fine-grained complexity. Our first set of results is motivated by the B ITONIC TSP problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in O ( n 2 ) time. While the near-quadratic dependency of similar dynamic programs for L ONGEST C OMMON S UBSEQUENCE and D ISCRETE F réchet D istance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in O ( n log 2 n ) time and its bottleneck version in O ( n log 3 n ) time. In the more general pyramidal TSP problem, the points to be visited are labeled 1,… , n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) TSP problem also work for the pyramidal TSP problem in the plane. Our second set of results concerns the popular k - OPT heuristic for TSP in the graph setting. More precisely, we study the k - OPT decision problem, which asks whether a given tour can be improved by a k - OPT move that replaces k edges in the tour by k new edges. A simple algorithm solves k - OPT in O ( n k ) time for fixed k . For 2- OPT , this is easily seen to be optimal. For k =3, we prove that an algorithm with a runtime of the form Õ( n 3−ɛ ) exists if and only if A LL -P AIRS S HORTEST P ATHS in weighted digraphs has such an algorithm. For general k - OPT , it is known that a runtime of f ( k ) · n o ( k / log k ) would contradict the Exponential Time Hypothesis. The results for k =2,3 may suggest that the actual time complexity of k - OPT is Θ ( n k ). We show that this is not the case, by presenting an algorithm that finds the best k -move in O ( n ⌊ 2 k /3 ⌋+1 ) time for fixed k ≥ 3. This implies that 4- OPT can be solved in O ( n 3 ) time, matching the best-known algorithm for 3- OPT . Finally, we show how to beat the quadratic barrier for k =2 in two important settings, namely, for points in the plane and when we want to solve 2- OPT repeatedly.

In the Boolean maximum constraint satisfaction problem - Max CSP(Γ) - one is given a collection of weighted applications of constraints from a finite constraint language Γ, over a common set of variables, and the goal is to assign Boolean values to the variables so that the total weight of satisfied constraints is maximized. There exists a concise dichotomy theorem providing a criterion on Γ for the problem to be polynomial-time solvable and stating that otherwise it becomes NP-hard. We study the NP-hard cases through the lens of kernelization and provide a complete characterization of Max CSP(Γ) with respect to the optimal compression size. Namely, we prove that Max CSP(Γ) parameterized by the number of variables n is either polynomial-time solvable, or there exists an integer d ≥ 2 depending on Γ, such that: 1) An instance of Max CSP(Γ) can be compressed into an equivalent instance with 𝒪(n^d log n) bits in polynomial time, 2) Max CSP(Γ) does not admit such a compression to 𝒪(n^{d-ε}) bits unless NP ⊆ co-NP / poly. Our reductions are based on interpreting constraints as multilinear polynomials combined with the framework of constraint implementations. As another application of our reductions, we reveal tight connections between optimal running times for solving Max CSP(Γ). More precisely, we show that obtaining a running time of the form 𝒪(2^{(1-ε)n}) for particular classes of Max CSPs is as hard as breaching this barrier for Max d-SAT for some d.

We study the kernelization complexity of structural parameterizations of the Vertex Cover problem. Here, the goal is to find a polynomial-time preprocessing algorithm that can reduce any instance $(G,k)$ of the Vertex Cover problem to an equivalent one, whose size is polynomial in the size of a pre-determined complexity parameter of $G$. A long line of previous research deals with parameterizations based on the number of vertex deletions needed to reduce $G$ to a member of a simple graph class $\mathcal{F}$, such as forests, graphs of bounded tree-depth, and graphs of maximum degree two. We set out to find the most general graph classes $\mathcal{F}$ for which Vertex Cover parameterized by the vertex-deletion distance of the input graph to $\mathcal{F}$, admits a polynomial kernelization. We give a complete characterization of the minor-closed graph families $\mathcal{F}$ for which such a kernelization exists. We introduce a new graph parameter called bridge-depth, and prove that a polynomial kernelization exists if and only if $\mathcal{F}$ has bounded bridge-depth. The proof is based on an interesting connection between bridge-depth and the size of minimal blocking sets in graphs, which are vertex sets whose removal decreases the independence number.

We investigate the List H-Coloring problem, the generalization of graph coloring that asks whether an input graph G admits a homomorphism to the undirected graph H (possibly with loops), such that each vertex v ∈ V(G) is mapped to a vertex on its list L(v) ⊆ V(H). An important result by Feder, Hell, and Huang [JGT 2003] states that List H-Coloring is polynomial-time solvable if H is a so-called bi-arc graph, and NP-complete otherwise. We investigate the NP-complete cases of the problem from the perspective of polynomial-time sparsification: can an n-vertex instance be efficiently reduced to an equivalent instance of bitsize 𝒪(n^(2-ε)) for some ε > 0? We prove that if H is not a bi-arc graph, then List H-Coloring does not admit such a sparsification algorithm unless NP ⊆ coNP/poly. Our proofs combine techniques from kernelization lower bounds with a study of the structure of graphs H which are not bi-arc graphs.

We continue the investigation of polynomial-time sparsification for NP-complete Boolean Constraint Satisfaction Problems (CSPs). The goal in sparsification is to reduce the number of constraints in a problem instance without changing the answer, such that a bound on the number of resulting constraints can be given in terms of the number of variables n. We investigate how the worst-case sparsification size depends on the types of constraints allowed in the problem formulation—the constraint language—and identify constraint languages giving the best-possible and worst-possible behavior for worst-case sparsifiability. Two algorithmic results are presented. The first result essentially shows that for any arity k, the only constraint type for which no nontrivial sparsification is possible has exactly one falsifying assignment, and corresponds to logical OR (up to negations). Our second result concerns linear sparsification, that is, a reduction to an equivalent instance with $$O(n)$$ constraints. Using linear algebra over rings of integers modulo prime powers, we give an elegant necessary and sufficient condition for a constraint type to be captured by a degree-1 polynomial over such a ring, which yields linear sparsifications. The combination of these algorithmic results allows us to prove two characterizations that capture the optimal sparsification sizes for a range of Boolean CSPs. For NP-complete Boolean CSPs whose constraints are symmetric (the satisfaction depends only on the number of 1 values in the assignment, not on their positions), we give a complete characterization of which constraint languages allow for a linear sparsification. For Boolean CSPs in which every constraint has arity at most three, we characterize the optimal size of sparsifications in terms of the largest OR that can be expressed by the constraint language.

Garnero et al. (2015) recently introduced a framework based on dynamic programming to make applications of the protrusion replacement technique constructive and to obtain explicit upper bounds on the involved constants. They show that for several graph problems, for every boundary size t one can find an explicit set R t of representatives . Any subgraph H with a boundary of size t can be replaced with a representative H ′ ∈ R t such that the effect of this replacement on the optimum can be deduced from H and H ′ alone. Their upper bounds on the size of the graphs in R t grow triple-exponentially with t . In this paper we complement their results by lower bounds on the sizes of representatives, in terms of the boundary size t . For example, we show that each set of planar representatives R t for Independent Set or Dominating Set contains a graph with Ω ( 2 t ∕ 4 t ) vertices. This lower bound even holds for sets that only represent the planar subgraphs of bounded pathwidth. To obtain our results we provide a lower bound on the number of equivalence classes of the canonical equivalence relation for Independent Set on t -boundaried graphs. We also find an elegant characterization of the number of equivalence classes in general graphs, in terms of the number of monotone functions of a certain kind. Our results show that the number of equivalence classes is at most 2 2 t , improving on earlier bounds of the form ( t + 1 ) 2 t .

We investigate polynomial-time preprocessing for the problem of hitting forbidden minors in a graph, using the framework of kernelization. For a fixed finite set of connected graphs F, the F-Deletion problem is the following: given a graph G and integer k, is it possible to delete k vertices from G to ensure the resulting graph does not contain any graph from F as a minor? Earlier work by Fomin, Lokshtanov, Misra, and Saurabh [FOCS'12] showed that when F contains a planar graph, an instance (G,k) can be reduced in polynomial time to an equivalent one of size kO(1). In this work we focus on structural measures of the complexity of an instance, with the aim of giving nontrivial preprocessing guarantees for instances whose solutions are large. Motivated by several impossibility results, we parameterize the F-Deletion problem by the size of a vertex modulator whose removal results in a graph of constant treedepth η. We prove that for each set F of connected graphs and constant η, the F-Deletion problem parameterized by the size of a treedepth-η modulator has a polynomial kernel. Our kernelization is fully explicit and does not depend on protrusion reduction or well-quasi-ordering, which are sources of algorithmic non-constructivity in earlier works on F-Deletion. Our main technical contribution is to analyze how models of a forbidden minor in a graph G with modulator X, interact with the various connected components of G−X. Using the language of labeled minors, we analyze the fragments of potential forbidden minor models that can remain after removing an optimal F-Deletion solution from a single connected component of G−X. By bounding the number of different types of behavior that can occur by a polynomial in |X|, we obtain a polynomial kernel using a recursive preprocessing strategy. Our results extend earlier work for specific instances of F-Deletion such as Vertex Cover and Feedback Vertex Set. It also generalizes earlier preprocessing results for F-Deletion parameterized by a vertex cover, which is a treedepth-one modulator.

The theory of kernelization can be used to rigorously analyze data reduction for graph coloring problems. Here, the aim is to reduce a q-Coloring input to an equivalent but smaller input whose size is provably bounded in terms of structural properties, such as the size of a minimum vertex cover. In this paper we settle two open problems about data reduction for q-Coloring. First, we obtain a kernel of bitsize $${\mathcal {O}}(k^{q-1}\log {k})$$ for q-Coloring parameterized by Vertex Cover for any $$q\ge 3$$ . This size bound is optimal up to $$k^{o(1)}$$ factors assuming $$\mathsf {NP} \not \subseteq \mathsf {coNP/poly}$$ , and improves on the previous-best kernel of size $${\mathcal {O}}(k^q)$$ . We generalize this result for deciding q-colorability of a graph G, to deciding the existence of a homomorphism from G to an arbitrary fixed graph H. Furthermore, we can replace the parameter vertex cover by the less restrictive parameter twin-cover. We prove that H-Coloring parameterized by Twin-Cover has a kernel of size $${\mathcal {O}}(k^{\varDelta (H)}\log k)$$ . Our second result shows that 3-Coloring does not admit non-trivial sparsification: assuming $$\mathsf {NP} \not \subseteq \mathsf {coNP/poly}$$ , the parameterization by the number of vertices n admits no (generalized) kernel of size $${\mathcal {O}}(n^{2-\varepsilon })$$ for any $$\varepsilon > 0$$ . Previously, such a lower bound was only known for coloring with $$q \ge 4$$ colors.

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether $$k$$ -Path admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size $$k^{\mathscr {O} (1)}$$ ? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and $$K_{3,t}$$ -minor-free graphs. Moreover, we show that $$k$$ -Path even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path, has a separation that can safely be reduced after communication with the oracle.

Computing the smallest number q such that the vertices of a given graph can be properly q-colored, known as the chromatic number, is one of the oldest and most fundamental problems in combinatorial optimization. The q-Coloring problem has been studied intensively using the framework of parameterized algorithmics, resulting in a very good understanding of the best-possible algorithms for several parameterizations based on the structure of the graph. For example, algorithms are known to solve the problem on graphs of treewidth tw in time O⁎(qtw), while a running time of O⁎((q−ε)tw) is impossible assuming the Strong Exponential Time Hypothesis (SETH). While there is an abundance of work for parameterizations based on decompositions of the graph by vertex separators, almost nothing is known about parameterizations based on edge separators. We fill this gap by studying q-Coloring parameterized by cutwidth, and parameterized by pathwidth in bounded-degree graphs. Our research uncovers interesting new ways to exploit small edge separators. We present two algorithms for q-Coloring parameterized by cutwidth ctw: a deterministic one that runs in time O⁎(2ω⋅ctw), where ω is the square matrix multiplication exponent, and a randomized one with runtime O⁎(2ctw). In sharp contrast to earlier work, the running time is independent of q. The dependence on cutwidth is optimal: we prove that even 3-Coloring cannot be solved in O⁎((2−ε)ctw) time assuming SETH. Our algorithms rely on a new rank bound for a matrix that describes compatible colorings. Combined with a simple communication protocol for evaluating a product of two polynomials, this also yields an O⁎((⌊d/2⌋+1)pw) time randomized algorithm for q-Coloring on graphs of pathwidth pw and maximum degree d. Such a runtime was first obtained by Björklund, but only for graphs with few proper colorings. We also prove that this result is optimal in the sense that no O⁎((⌊d/2⌋+1−ε)pw)-time algorithm exists assuming SETH.

Traditionally, reconfiguration problems ask the question whether a given solution of an optimization problem can be transformed to a target solution in a sequence of small steps that preserve feasibility of the intermediate solutions. In this paper, rather than asking this question from an algorithmic perspective, we analyze the combinatorial structure behind it. We consider the problem of reconfiguring one independent set into another, using two different processes: (1) exchanging exactly k vertices in each step, or (2) removing or adding one vertex in each step while ensuring the intermediate sets contain at most k fewer vertices than the initial solution. We are interested in determining the minimum value of k for which this reconfiguration is possible, and bound these threshold values in terms of several structural graph parameters. For hereditary graph classes we identify structures that cause the reconfiguration threshold to be large.

The Chordal Vertex Deletion (ChVD) problem asks to delete a minimum number of vertices from an input graph to obtain a chordal graph. In this paper we develop a polynomial kernel for ChVD under the parameterization by the solution size. Using a new Erdös--Pósa-type packing/covering duality for holes in nearly chordal graphs, we present a polynomial-time algorithm that reduces any instance $(G,k)$ of ChVD to an equivalent instance with ${poly}(k)$ vertices. The existence of a polynomial kernel answers an open problem posed by Marx in 2006 [D. Marx, “Chordal Deletion Is Fixed-Parameter Tractable,” in Graph-Theoretic Concepts in Computer Science, Lecture Notes in Comput. Sci. 4271, Springer, 2006, pp. 37--48]. To obtain the kernelization, we develop the first ${poly}({\sc opt})$-approximation algorithm for ChVD, which is of independent interest. In polynomial time, it either decides that $G$ has no chordal deletion set of size $k$, or outputs a solution of size $\mathcal{O}(k^4\log^2k)$.

The notion of Turing kernelization investigates whether a polynomial-time algorithm can solve an NP-hard problem, when it is aided by an oracle that can be queried for the answers to bounded-size subproblems. One of the main open problems in this direction is whether k-PATH admits a polynomial Turing kernel: can a polynomial-time algorithm determine whether an undirected graph has a simple path of length k, using an oracle that answers queries of size k^{O(1)}? We show this can be done when the input graph avoids a fixed graph H as a topological minor, thereby significantly generalizing an earlier result for bounded-degree and K_{3,t}-minor-free graphs. Moreover, we show that k-PATH even admits a polynomial Turing kernel when the input graph is not H-topological-minor-free itself, but contains a known vertex modulator of size bounded polynomially in the parameter, whose deletion makes it so. To obtain our results, we build on the graph minors decomposition to show that any H-topological-minor-free graph that does not contain a k-path has a separation that can safely be reduced after communication with the oracle.

The Chordal Vertex Deletion (ChVD) problem asks to delete a minimum number of vertices from an input graph to obtain a chordal graph. In this paper we develop a polynomial kernel for ChVD under the parameterization by the solution size. Using a new Erdos-Posa type packing/covering duality for holes in nearly-chordal graphs, we present a polynomial-time algorithm that reduces any instance (G, k) of ChVD to an equivalent instance with poly(k) vertices. The existence of a polynomial kernel answers an open problem of Marx from 2006 [WG 2006, LNCS 4271, 37–48]. To obtain the kernelization, we develop the first poly(oPT)- approximation algorithm for ChVD, which is of independent interest. In polynomial time, it either decides that G has no chordal deletion set of size k, or outputs a solution of size O(k4 log2 k).

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize $$O(n^{2-\varepsilon })$$ for $$\varepsilon > 0$$ , unless $$\mathsf {NP \subseteq coNP/poly}$$ and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have $$O(k^{2-\varepsilon })$$ edges, unless $$\mathsf {NP \subseteq coNP/poly}$$ . We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with $$O(n^{d-1})$$ clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by $$\left( {\begin{array}{c}n\\ d-1\end{array}}\right) $$ . We show that our kernel is tight under the assumption that $$\mathsf {NP} \nsubseteq \mathsf {coNP}/\mathsf {poly}$$ .

The F -M inor -F ree D eletion problem asks, for a fixed set F and an input consisting of a graph G and integer k , whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. At FOCS 2012, Fomin et al. showed that the special case when F contains at least one planar graph has a kernel of size f ( F ) ċ k g ( F ) for some functions f and g . They left open whether this P lanar F -M inor -F ree D eletion problem has kernels whose size is uniformly polynomial, of the form f ( F ) ċ k c for some universal constant c . We prove that some P lanar F -M inor -F ree D eletion problems do not have uniformly polynomial kernels (unless NP ⊆ coNP/poly), not even when parameterized by the vertex cover number. On the positive side, we consider the problem of determining whether k vertices can be removed to obtain a graph of treedepth at most η. We prove that this problem admits uniformly polynomial kernels with O ( k 6 ) vertices for every fixed η.

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic TSP problem: given a set of $n$ points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in $O(n^2)$ time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Frechet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in $O(n \log^2 n)$ time and its bottleneck version in $O(n \log^3 n)$ time. Our second set of results concerns the popular $k$-OPT heuristic for TSP in the graph setting. More precisely, we study the $k$-OPT decision problem, which asks whether a given tour can be improved by a $k$-OPT move that replaces $k$ edges in the tour by $k$ new edges. A simple algorithm solves $k$-OPT in $O(n^k)$ time for fixed $k$. For 2-OPT, this is easily seen to be optimal. For $k=3$ we prove that an algorithm with a runtime of the form $\tilde{O}(n^{3-ε})$ exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. The results for $k=2,3$ may suggest that the actual time complexity of $k$-OPT is $Θ(n^k)$. We show that this is not the case, by presenting an algorithm that finds the best $k$-move in $O(n^{\lfloor 2k/3 \rfloor + 1})$ time for fixed $k \geq 3$. This implies that 4-OPT can be solved in $O(n^3)$ time, matching the best-known algorithm for 3-OPT. Finally, we show how to beat the quadratic barrier for $k=2$ in two important settings, namely for points in the plane and when we want to solve 2-OPT repeatedly.

In this article, the steering committee of the Parameterized Algorithms and Computational Experiments challenge (PACE) reports on the first iteration of the challenge. Where did PACE come from, how did it go, who won, and what's next?

The CONSTRAINED BIPARTITE VERTEX COVER problem asks, for a bipartite graph G with partite sets A and B, and integers k_A and k_B, whether there is a vertex cover for G containing at most k_A vertices from A and k_B vertices from B. The problem has an easy kernel with 2 * k_A * k_B edges and 4 k_A * k_B vertices, based on the fact that every vertex in A of degree more than k_B has to be included in the solution, together with every vertex in B of degree more than k_A. We show that the number of vertices and edges in this kernel are asymptotically essentially optimal in terms of the product k_A * k_B. We prove that if there is a polynomial-time algorithm that reduces any instance (G,A,B,k_A,k_B) of CONSTRAINED BIPARTITE VERTEX COVER to an equivalent instance (G',A',B',k'_A,k'_B) such that k'_A in (k_A)^{O(1)}, k'_B in (k_B)^{O(1)}, and |V(G')| in O((k_A * k_B)^{1 - epsilon}), for some epsilon > 0, then NP subseteq coNP/poly and the polynomial-time hierarchy collapses. Using a different construction, we prove that if there is a polynomial-time algorithm that reduces any n-vertex instance into an equivalent instance (of a possibly different problem) that can be encoded in O(n^{2- epsilon}) bits, then NP subseteq coNP/poly.

We present several sparsification lower and upper bounds for classic problems in graph theory and logic. For the problems 4-Coloring, (Directed) Hamiltonian Cycle, and (Connected) Dominating Set, we prove that there is no polynomial-time algorithm that reduces any n-vertex input to an equivalent instance, of an arbitrary problem, with bitsize O(n^{2-epsilon}) for epsilon > 0, unless NP is a subset of coNP/poly and the polynomial-time hierarchy collapses. These results imply that existing linear-vertex kernels for k-Nonblocker and k-Max Leaf Spanning Tree (the parametric duals of (Connected) Dominating Set) cannot be improved to have O(k^{2-epsilon}) edges, unless NP is a subset of NP/poly. We also present a positive result and exhibit a non-trivial sparsification algorithm for d-Not-All-Equal-SAT. We give an algorithm that reduces an n-variable input with clauses of size at most d to an equivalent input with O(n^{d-1}) clauses, for any fixed d. Our algorithm is based on a linear-algebraic proof of Lovász that bounds the number of hyperedges in critically 3-chromatic d-uniform n-vertex hypergraphs by binom{n}{d-1}. We show that our kernel is tight under the assumption that NP is not a subset of NP/poly.

We study two fundamental problems related to finding subgraphs: (1) given graphs G and H, Subgraph Test asks if H is isomorphic to a subgraph of G, (2) given graphs G, H, and an integer t, PACKING asks if G contains t vertex-disjoint subgraphs isomorphic to H. For every graph class ℱ, let ℱ-Subgraph Test and ℱ-Packing be the special cases of the two problems where H is restricted to be in F. Our goal is to study which classes ℱ make the two problems tractable in one of the following senses: (randomized) polynomial-time solvable, admits a polynomial (many-one) kernel (that is, has a polynomial-time preprocessing procedure that creates an equivalent instance whose size is polynomially bounded by the size of the solution), or admits a polynomial Turing kernel (that is, has an adaptive polynomial-time procedure that reduces the problem to a polynomial number of instances, each of which has size bounded polynomially by the size of the solution). To obtain a more robust setting, we restrict our attention to hereditary classes F. It is known that if every component of every graph in ℱ has at most two vertices, then ℱ-Packing is polynomial-time solvable, and NP-hard otherwise. We identify a simple combinatorial property (every component of every graph in ℱ either has bounded size or is a bipartite graph with one of the sides having bounded size) such that if a hereditary class ℱ has this property, then ℱ-Packing admits a polynomial kernel, and has no polynomial (many-one) kernel otherwise, unless the polynomial hierarchy collapses. Furthermore, if ℱ does not have this property, then ℱ-Packing is either WK[1]-hard, W[1]-hard, or Long Path-hard, giving evidence that it does not admit polynomial Turing kernels either. For ℱ-Subgraph Test, we show that if every graph of a hereditary class ℱ satisfies the property that it is possible to delete a bounded number of vertices such that every remaining component has size at most two, then F-Subgraph Test is solvable in randomized polynomial time and it is NP-hard otherwise. We introduce a combinatorial property called (a, b, c, d)-splittability and show that if every graph in a hereditary class ℱ has this property, then F-Subgraph Test admits a polynomial Turing kernel and it is WK[1]-hard, W[1]-hard, or Long Path-hard otherwise. We do not give a complete characterization of the cases when F-Subgraph Test admits polynomial many-one kernels, but show examples that this question is much more fragile than the characterization for Turing kernels.

The problem of testing whether a graph is planar has been studied for over half a century, and is known to be solvable in (n) time using a myriad of different approaches and techniques. Robertson and Seymour established the existence of a cubic algorithm for the more general problem of deciding whether an n-vertex graph can be made planar by at most k vertex deletions, for every fixed k. Of the known algorithms for k-Vertex Planarization, the algorithm of Marx and Schlotter (WG 2007, Algorithmica 2012) running in time achieves the best running time dependence on k. The algorithm of Kawarabayashi (FOCS 2009), running in time f(k)n for some f(k) ∊ that is not stated explicitly, achieves the best dependence on n. In this paper we present an algorithm for k-Vertex Planarization with running time 2 (k log k)·n, significantly improving the running time dependence on k without compromising the linear dependence on n. Our main technical contribution is a novel scheme to reduce the treewidth of the input graph to (k) in time 2O(klogk) · n. It combines new insights into the structure of graphs that become planar after contracting a matching, with a Baker-type subroutine that reduces the number of disjoint paths through planar parts of the graph that are not affected by the sought solution. To solve the reduced instances we formulate a dynamic programming algorithm for Weighted Vertex Planarization on graphs of treewidth w with running time 2 (w log w) · n, thereby improving over previous double-exponential algorithms. While Kawarabayashi's planarization algorithm relies heavily on deep results from the graph minors project, our techniques are elementary and practically self-contained. We expect them to be applicable to related edge-deletion and contraction variants of planarization problems.

We introduce the framework of cross-composition for proving kernelization lower bounds. A classical problem $L$ \and/or-cross-composes into a parameterized problem $\mathcal{Q}$ if it is possible to efficiently construct an instance of $\mathcal{Q}$ with polynomially bounded parameter value that expresses the logical and or or of a sequence of instances of $L$. Building on work by Bodlaender et al. and using results of Fortnow and Santhanam, Dell and van Melkebeek, and Drucker, we show that if an NP-hard problem and/or-cross-composes into a parameterized problem $\mathcal{Q}$, then $\mathcal{Q}$ does not admit a polynomial kernel unless $\mbox{NP}\subseteq \mbox{coNP/poly}$ and the polynomial hierarchy collapses. Our technique generalizes and strengthens the techniques of using composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (nonstandard) parameterizations, e.g., Clique, Chromatic Number, Weighted Feedback Vertex Set, and Weighted Odd Cycle Transversal do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set and Odd Cycle Transversal under structural parameterizations. After learning of our results, several teams of authors have successfully applied the cross-composition framework to different parameterized problems. For completeness, our presentation of the framework includes several extensions based on this follow-up work. For example, we show how a relaxed version of or-cross-compositions may be used to give lower bounds on the degree of the polynomial in the kernel size.

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 $\mathcal{O}((k')^{2})$ 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 Θ(k 2) edges are optimal for the kernel size. In this paper we consider the Vertex Cover problem with a different parameter, the size $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ of a minimum feedback vertex set for G. This refined parameter is structurally smaller than the parameter k associated to the vertex covering number $\mathop{\mathrm{\mbox {\textsc{vc}}}}(G)$ since $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)\leq\mathop{\mathrm{\mbox{\textsc{vc}}}}(G)$ and the difference can be arbitrarily large. We give a kernel for Vertex Cover with a number of vertices that is cubic in $\mathop{\mathrm{\mbox{\textsc{fvs}}}}(G)$ : 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 $|V(G')| \in\mathcal{O}(|X'|^{3})$ . 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 ⊆ coNP/poly and the polynomial hierarchy collapses to the third level.

The notion of treewidth plays an important role in theoretical and practical studies of graph problems. It has been recognized that, especially in practical environments, when computing the treewidth of a graph it is invaluable to first apply an array of preprocessing rules that simplify and shrink it. This work seeks to prove rigorous performance guarantees for such preprocessing rules---known rules as well as more recent ones---by studying them in the framework of kernelization from parameterized complexity. It is known that the NP-complete problem of determining whether a given graph $G$ has treewidth at most $k$ admits no polynomial-time preprocessing algorithm that reduces any input instance to size polynomial in $k$, unless NP $\subseteq$ coNP/poly and the polynomial hierarchy collapses to its third level. In this paper we therefore consider structural graph measures larger than treewidth, and determine whether efficient preprocessing can shrink the instance size to a polynomial in such a parameter value. We prove that, given an instance $(G,k)$ of treewidth, we can efficiently reduce its size to $\mathcal{O}(\mathrm{\textsc{fvs}}(G)^4)$ vertices, where $\mathrm{\textsc{fvs}}(G)$ is the size of a minimum feedback vertex set in $G$. We can also prove a size reduction to $\mathcal{O}(\mathrm{\textsc{vc}}(G)^3)$ vertices, where $\mathrm{\textsc{vc}}(G)$ is the size of a minimum vertex cover. Phrased in the language of parameterized complexity, we show that Treewidth has a polynomial kernel when parameterized by the size of a given feedback vertex set, and also by the size of a vertex cover. In contrast we show that Treewidth parameterized by the vertex-deletion distance to a single clique and Weighted Treewidth parameterized by the size of a vertex cover do not admit polynomial kernelizations unless NP $\subseteq$ coNP/poly.

We introduce a new technique for proving kernelization lower bounds, called cross-composition. A classical problem L cross-composes into a parameterized problem $Q$ if an instance of Q with polynomially bounded parameter value can express the logical OR of a sequence of instances of L. Building on work by Bodlaender et al. (ICALP 2008) and using a result by Fortnow and Santhanam (STOC 2008) we show that if an NP-hard problem cross-composes into a parameterized problem Q then Q does not admit a polynomial kernel unless the polynomial hierarchy collapses. Our technique generalizes and strengthens the recent techniques of using OR-composition algorithms and of transferring the lower bounds via polynomial parameter transformations. We show its applicability by proving kernelization lower bounds for a number of important graphs problems with structural (non-standard) parameterizations, e.g., Chromatic Number, Clique, and Weighted Feedback Vertex Set do not admit polynomial kernels with respect to the vertex cover number of the input graphs unless the polynomial hierarchy collapses, contrasting the fact that these problems are trivially fixed-parameter tractable for this parameter. We have similar lower bounds for Feedback Vertex Set.

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.

Computing the Dodgson Score of a candidate in an election is a hard computational problem, which has been analyzed using classical and parameterized analysis. In this paper we resolve two open problems regarding the parameterized complexity of DODGSON SCORE. We show that DODGSON SCORE parameterized by the target score value $k$ does not have a polynomial kernel unless the polynomial hierarchy collapses to the third level; this complements a result of Fellows, Rosamond and Slinko who obtain a non-trivial kernel of exponential size for a generalization of this problem. We also prove that DODGSON SCORE parameterized by the number $n$ of votes is hard for $W[1]$.

Road network data is often incomplete, making it hard to perform network analysis. This paper discusses the problem of extending partial road networks with reasonable links, using the concept of dilation (also known as crow flight conversion coefficient). To this end, we study how to connect a point (relevant location) inside a polygon (face of the known part of the road network) to the boundary so that the dilation from that point to any point on the boundary is not too large. We provide algorithms and heuristics, and give a computational and experimental analysis.