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In this article we show that Maximum Partial List H-Coloring is polynomial-time solvable on P_5-free graphs for every fixed graph H. In particular, this implies that Maximum k-Colorable Subgraph is polynomial-time solvable on P_5-free graphs. This answers an open question from Agrawal, Lima, Lokshtanov, Saurabh & Sharma [SODA 2024]. This also improves the $n^{\omega(G)}$-time algorithm for Maximum Partial H-Coloring by Chudnovsky, King, Pilipczuk, Rz\k{a}\.{z}ewski & Spirkl [SIDMA 2021] to polynomial-time algorithm.

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., Comment: This is the second part following an accompanying paper arXiv:2307.08149. We split the original paper to keep paper length more manageable

Drawing a graph in the plane with as few crossings as possible is one of the central problems in graph drawing and computational geometry. Another option is to remove the smallest number of vertices or edges such that the remaining graph can be drawn without crossings. We study both problems in a book-embedding setting for ordered graphs, that is, graphs with a fixed vertex order. In this setting, the vertices lie on a straight line, called the spine, in the given order, and each edge must be drawn on one of several pages of a book such that every edge has at most a fixed number of crossings. In book embeddings, there is another way to reduce or avoid crossings; namely by using more pages. The minimum number of pages needed to draw an ordered graph without any crossings is its (fixed-vertex-order) page number. We show that the page number of an ordered graph with $n$ vertices and $m$ edges can be computed in $2^m \cdot n^{O(1)}$ time. An $O(\log n)$-approximation of this number can be computed efficiently. We can decide in $2^{O(d \sqrt{k} \log (d+k))} \cdot n^{O(1)}$ time whether it suffices to delete $k$ edges of an ordered graph to obtain a $d$-planar layout (where every edge crosses at most $d$ other edges) on one page. As an additional parameter, we consider the size $h$ of a hitting set, that is, a set of points on the spine such that every edge, seen as an open interval, contains at least one of the points. For $h=1$, we can efficiently compute the minimum number of edges whose deletion yields fixed-vertex-order page number $p$. For $h>1$, we give an XP algorithm with respect to $h+p$. Finally, we consider spine+$t$-track drawings, where some but not all vertices lie on the spine. The vertex order on the spine is given; we must map every vertex that does not lie on the spine to one of $t$ tracks, each of which is a straight line on a separate page, parallel to the spine., Comment: Appears in Proc. 19th Scandinavian Symposium on Algorithm Theory (SWAT 2024)

An edge-colored graph is said to be balanced if it has an equal number of edges of each color. Given a graph $G$ whose edges are colored using two colors and a positive integer $k$, the objective in the Edge Balanced Connected Subgraph problem is to determine if $G$ has a balanced connected subgraph containing at least $k$ edges. We first show that this problem is NP-complete and remains so even if the solution is required to be a tree or a path. Then, we focus on the parameterized complexity of Edge Balanced Connected Subgraph and its variants (where the balanced subgraph is required to be a path/tree) with respect to $k$ as the parameter. Towards this, we show that if a graph has a balanced connected subgraph/tree/path of size at least $k$, then it has one of size at least $k$ and at most $f(k)$ where $f$ is a linear function. We use this result combined with dynamic programming algorithms based on color coding and representative sets to show that Edge Balanced Connected Subgraph and its variants are FPT. Further, using polynomial-time reductions to the Multilinear Monomial Detection problem, we give faster randomized FPT algorithms for the problems. In order to describe these reductions, we define a combinatorial object called relaxed-subgraph. We define this object in such a way that balanced connected subgraphs, trees and paths are relaxed-subgraphs with certain properties. This object is defined in the spirit of branching walks known for the Steiner Tree problem and may be of independent interest., Comment: Minor changes

We study a general family of problems that form a common generalization of classic hitting (also referred to as covering or transversal) and packing problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of a graph G prevent us from packing $\ell$ vertex-disjoint objects of type X? This problem captures a spectrum of problems with standard hitting and packing on opposite ends. Our main motivating question is whether the combination X-HitPack can be significantly harder than these two base problems. Already for a particular choice of X, this question can be posed for many different complexity notions, leading to a large, so-far unexplored domain in the intersection of the areas of hitting and packing problems. On a high-level, we present two case studies: (1) X being all cycles, and (2) X being all copies of a fixed graph H. In each, we explore the classical complexity, as well as the parameterized complexity with the natural parameters k+l and treewidth. We observe that the combined problem can be drastically harder than the base problems: for cycles or for H being a connected graph with at least 3 vertices, the problem is \Sigma_2^P-complete and requires double-exponential dependence on the treewidth of the graph (assuming the Exponential-Time Hypothesis). In contrast, the combined problem admits qualitatively similar running times as the base problems in some cases, although significant novel ideas are required. For example, for X being all cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on graphs of treewidth tw. The key step enabling this running time relies on a combinatorial bound obtained from an algebraic (linear delta-matroid) representation of possible matchings.

An independent set in a graph G is a set of pairwise non-adjacent vertices. A graph $G$ is bipartite if its vertex set can be partitioned into two independent sets. In the Odd Cycle Transversal problem, the input is a graph $G$ along with a weight function $w$ associating a rational weight with each vertex, and the task is to find a smallest weight vertex subset $S$ in $G$ such that $G - S$ is bipartite; the weight of $S$, $w(S) = \sum_{v\in S} w(v)$. We show that Odd Cycle Transversal is polynomial-time solvable on graphs excluding $P_5$ (a path on five vertices) as an induced subgraph. The problem was previously known to be polynomial-time solvable on $P_4$-free graphs and NP-hard on $P_6$-free graphs [Dabrowski, Feghali, Johnson, Paesani, Paulusma and Rz\k{a}\.zewski, Algorithmica 2020]. Bonamy, Dabrowski, Feghali, Johnson and Paulusma [Algorithmica 2019] posed the existence of a polynomial-time algorithm on $P_5$-free graphs as an open problem, this was later re-stated by Rz\k{a}\.zewski [Dagstuhl Reports, 9(6): 2019] and by Chudnovsky, King, Pilipczuk, Rz\k{a}\.zewski, and Spirkl [SIDMA 2021], who gave an algorithm with running time $n^{O(\sqrt{n})}$.

Large networks are useful in a wide range of applications. Sometimes problem instances are composed of billions of entities. Decomposing and analyzing these structures helps us gain new insights about our surroundings. Even if the final application concerns a different problem (such as traversal, finding paths, trees, and flows), decomposing large graphs is often an important subproblem for complexity reduction or parallelization. This report is a summary of discussions that happened at Dagstuhl seminar 23331 on "Recent Trends in Graph Decomposition" and presents currently open problems and future directions in the area of (hyper)graph decomposition.

In a recent work, Esmer et al. describe a simple method - Approximate Monotone Local Search - to obtain exponential approximation algorithms from existing parameterized exact algorithms, polynomial-time approximation algorithms and, more generally, parameterized approximation algorithms. In this work, we generalize those results to the weighted setting. More formally, we consider monotone subset minimization problems over a weighted universe of size $n$ (e.g., Vertex Cover, $d$-Hitting Set and Feedback Vertex Set). We consider a model where the algorithm is only given access to a subroutine that finds a solution of weight at most $\alpha \cdot W$ (and of arbitrary cardinality) in time $c^k \cdot n^{O(1)}$ where $W$ is the minimum weight of a solution of cardinality at most $k$. In the unweighted setting, Esmer et al. determine the smallest value $d$ for which a $\beta$-approximation algorithm running in time $d^n \cdot n^{O(1)}$ can be obtained in this model. We show that the same dependencies also hold in a weighted setting in this model: for every fixed $\varepsilon>0$ we obtain a $\beta$-approximation algorithm running in time $O\left((d+\varepsilon)^{n}\right)$, for the same $d$ as in the unweighted setting. Similarly, we also extend a $\beta$-approximate brute-force search (in a model which only provides access to a membership oracle) to the weighted setting. Using existing approximation algorithms and exact parameterized algorithms for weighted problems, we obtain the first exponential-time $\beta$-approximation algorithms that are better than brute force for a variety of problems including Weighted Vertex Cover, Weighted $d$-Hitting Set, Weighted Feedback Vertex Set and Weighted Multicut.

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.

Treewidth (tw) is an important parameter that, when bounded, yields tractability for many problems. For example, graph problems expressible in Monadic Second Order (MSO) logic and QUANTIFIED SAT or, more generally, QUANTIFIED CSP, are FPT parameterized by the tw of the input's (primal) graph plus the length of the MSO-formula [Courcelle, Information & Computation 1990] and the quantifier rank [Chen, ECAI 2004], resp. The algorithms from these (meta-)results have running times whose dependence on tw is a tower of exponents. A conditional lower bound by Fichte et al. [LICS 2020] shows that, for QUANTIFIED SAT, the height of this tower is equal to the number of quantifier alternations. Lower bounds showing that at least double-exponential factors in the running time are necessary are rare: there are very few (for tw and vertex cover vc parameterizations) and they are for problems that are complete for #NP, $\Sigma_2^p$, $\Pi_2^p$, or higher levels of the polynomial hierarchy. We show, for the first time, that it is not necessary to go higher up in the polynomial hierarchy to obtain such lower bounds. We design a novel, yet simple versatile technique based on Sperner families to obtain such lower bounds and apply it to 3 problems: METRIC DIMENSION, STRONG METRIC DIMENSION, and GEODETIC SET. We prove that they do not admit $2^{2^{o(tw)}} \cdot n^{O(1)}$-time algorithms, even on bounded diameter graphs, unless the ETH fails. For STRONG METRIC DIMENSION, the lower bound holds even for vc. We complement our lower bounds with matching upper bounds., Comment: Shortened abstract to meet arxiv requirements. We split the original paper to keep paper length more manageable. This is the first part following an accompanying paper arXiv:2405.01344; and will be presented at ICALP 2024

The goal of this paper is to understand how exponential-time approximation algorithms can be obtained from existing polynomial-time approximation algorithms, existing parameterized exact algorithms, and existing parameterized approximation algorithms. More formally, we consider a monotone subset minimization problem over a universe of size $n$ (e.g., Vertex Cover or Feedback Vertex Set). We have access to an algorithm that finds an $\alpha$-approximate solution in time $c^k \cdot n^{O(1)}$ if a solution of size $k$ exists (and more generally, an extension algorithm that can approximate in a similar way if a set can be extended to a solution with $k$ further elements). Our goal is to obtain a $d^n \cdot n^{O(1)}$ time $\beta$-approximation algorithm for the problem with $d$ as small as possible. That is, for every fixed $\alpha,c,\beta \geq 1$, we would like to determine the smallest possible $d$ that can be achieved in a model where our problem-specific knowledge is limited to checking the feasibility of a solution and invoking the $\alpha$-approximate extension algorithm. Our results completely resolve this question: (1) For every fixed $\alpha,c,\beta \geq 1$, a simple algorithm (``approximate monotone local search'') achieves the optimum value of $d$. (2) Given $\alpha,c,\beta \geq 1$, we can efficiently compute the optimum $d$ up to any precision $\varepsilon > 0$. Earlier work presented algorithms (but no lower bounds) for the special case $\alpha = \beta = 1$ [Fomin et al., J. ACM 2019] and for the special case $\alpha = \beta > 1$ [Esmer et al., ESA 2022]. Our work generalizes these results and in particular confirms that the earlier algorithms are optimal in these special cases., Comment: 80 pages, 5 figures

Constraint satisfaction problems form a nicely behaved class of problems that lends itself to complexity classification results. From the point of view of parameterized complexity, a natural task is to classify the parameterized complexity of MinCSP problems parameterized by the number of unsatisfied constraints. In other words, we ask whether we can delete at most $k$ constraints, where $k$ is the parameter, to get a satisfiable instance. In this work, we take a step towards classifying the parameterized complexity for an important infinite-domain CSP: Allen's interval algebra (IA). This CSP has closed intervals with rational endpoints as domain values and employs a set $A$ of 13 basic comparison relations such as ``precedes'' or ``during'' for relating intervals. IA is a highly influential and well-studied formalism within AI and qualitative reasoning that has numerous applications in, for instance, planning, natural language processing and molecular biology. We provide an FPT vs. W[1]-hard dichotomy for MinCSP$(\Gamma)$ for all $\Gamma \subseteq A$. IA is sometimes extended with unions of the relations in $A$ or first-order definable relations over $A$, but extending our results to these cases would require first solving the parameterized complexity of Directed Symmetric Multicut, which is a notorious open problem. Already in this limited setting, we uncover connections to new variants of graph cut and separation problems. This includes hardness proofs for simultaneous cuts or feedback arc set problems in directed graphs, as well as new tractable cases with algorithms based on the recently introduced flow augmentation technique. Given the intractability of MinCSP$(A)$ in general, we then consider (parameterized) approximation algorithms and present a factor-$2$ fpt-approximation algorithm.

We consider the well-studied Robust $(k, z)$-Clustering problem, which generalizes the classic $k$-Median, $k$-Means, and $k$-Center problems. Given a constant $z\ge 1$, the input to Robust $(k, z)$-Clustering is a set $P$ of $n$ weighted points in a metric space $(M,\delta)$ and a positive integer $k$. Further, each point belongs to one (or more) of the $m$ many different groups $S_1,S_2,\ldots,S_m$. Our goal is to find a set $X$ of $k$ centers such that $\max_{i \in [m]} \sum_{p \in S_i} w(p) \delta(p,X)^z$ is minimized. This problem arises in the domains of robust optimization [Anthony, Goyal, Gupta, Nagarajan, Math. Oper. Res. 2010] and in algorithmic fairness. For polynomial time computation, an approximation factor of $O(\log m/\log\log m)$ is known [Makarychev, Vakilian, COLT $2021$], which is tight under a plausible complexity assumption even in the line metrics. For FPT time, there is a $(3^z+\epsilon)$-approximation algorithm, which is tight under GAP-ETH [Goyal, Jaiswal, Inf. Proc. Letters, 2023]. Motivated by the tight lower bounds for general discrete metrics, we focus on \emph{geometric} spaces such as the (discrete) high-dimensional Euclidean setting and metrics of low doubling dimension, which play an important role in data analysis applications. First, for a universal constant $\eta_0 >0.0006$, we devise a $3^z(1-\eta_{0})$-factor FPT approximation algorithm for discrete high-dimensional Euclidean spaces thereby bypassing the lower bound for general metrics. We complement this result by showing that even the special case of $k$-Center in dimension $\Theta(\log n)$ is $(\sqrt{3/2}- o(1))$-hard to approximate for FPT algorithms. Finally, we complete the FPT approximation landscape by designing an FPT $(1+\epsilon)$-approximation scheme (EPAS) for the metric of sub-logarithmic doubling dimension., Comment: 26 pages, 5 figures

This paper considers the well-studied algorithmic regime of designing a $(1+\epsilon)$-approximation algorithm for a $k$-clustering problem that runs in time $f(k,\epsilon)poly(n)$ (sometimes called an efficient parameterized approximation scheme or EPAS for short). Notable results of this kind include EPASes in the high-dimensional Euclidean setting for $k$-center [Bad\u{o}iu, Har-Peled, Indyk; STOC'02] as well as $k$-median, and $k$-means [Kumar, Sabharwal, Sen; J. ACM 2010]. However, existing EPASes handle only basic objectives (such as $k$-center, $k$-median, and $k$-means) and are tailored to the specific objective and metric space. Our main contribution is a clean and simple EPAS that settles more than ten clustering problems (across multiple well-studied objectives as well as metric spaces) and unifies well-known EPASes. Our algorithm gives EPASes for a large variety of clustering objectives (for example, $k$-means, $k$-center, $k$-median, priority $k$-center, $\ell$-centrum, ordered $k$-median, socially fair $k$-median aka robust $k$-median, or more generally monotone norm $k$-clustering) and metric spaces (for example, continuous high-dimensional Euclidean spaces, metrics of bounded doubling dimension, bounded treewidth metrics, and planar metrics). Key to our approach is a new concept that we call bounded $\epsilon$-scatter dimension--an intrinsic complexity measure of a metric space that is a relaxation of the standard notion of bounded doubling dimension. Our main technical result shows that two conditions are essentially sufficient for our algorithm to yield an EPAS on the input metric $M$ for any clustering objective: (i) The objective is described by a monotone (not necessarily symmetric!) norm, and (ii) the $\epsilon$-scatter dimension of $M$ is upper bounded by a function of $\epsilon$., Comment: 40 pages, 6 figures

The circumference of a graph $G$ is the length of a longest cycle in $G$, or $+\infty$ if $G$ has no cycle. Birmel\'e (2003) showed that the treewidth of a graph $G$ is at most its circumference minus $1$. We strengthen this result for $2$-connected graphs as follows: If $G$ is $2$-connected, then its treedepth is at most its circumference. The bound is best possible and improves on an earlier quadratic upper bound due to Marshall and Wood (2015)., Comment: v3: Revised following the referees's comments

We show that the $b$-Coloring problem is complete for the class XNLP when parameterized by the pathwidth of the input graph. Besides determining the precise parameterized complexity of this problem, this implies that b-Coloring parameterized by pathwidth is $W[t]$-hard for all $t$, and resolves the parameterized complexity of $b$-Coloring parameterized by treewidth.

One of the first application of the recently introduced technique of \emph{flow-augmentation} [Kim et al., STOC 2022] is a fixed-parameter algorithm for the weighted version of \textsc{Directed Feedback Vertex Set}, a landmark problem in parameterized complexity. In this note we explore applicability of flow-augmentation to other weighted graph separation problems parameterized by the size of the cutset. We show the following. -- In weighted undirected graphs \textsc{Multicut} is FPT, both in the edge- and vertex-deletion version. -- The weighted version of \textsc{Group Feedback Vertex Set} is FPT, even with an oracle access to group operations. -- The weighted version of \textsc{Directed Subset Feedback Vertex Set} is FPT. Our study reveals \textsc{Directed Symmetric Multicut} as the next important graph separation problem whose parameterized complexity remains unknown, even in the unweighted setting., Comment: 17 pages, 1 figure

In the Directed Steiner Network problem, the input is a directed graph G, a subset T of k vertices of G called the terminals, and a demand graph D on T. The task is to find a subgraph H of G with the minimum number of edges such that for every edge (s,t) in D, the solution H contains a directed s to t path. In this paper we investigate how the complexity of the problem depends on the demand pattern when G is planar. Formally, if \mathcal{D} is a class of directed graphs closed under identification of vertices, then the \mathcal{D}-Steiner Network (\mathcal{D}-SN) problem is the special case where the demand graph D is restricted to be from \mathcal{D}. For general graphs, Feldmann and Marx [ICALP 2016] characterized those families of demand graphs where the problem is fixed-parameter tractable (FPT) parameterized by the number k of terminals. They showed that if \mathcal{D} is a superset of one of the five hard families, then \mathcal{D}-SN is W[1]-hard parameterized by k, otherwise it can be solved in time f(k)n^{O(1)}. For planar graphs an interesting question is whether the W[1]-hard cases can be solved by subexponential parameterized algorithms. Chitnis et al. [SICOMP 2020] showed that, assuming the ETH, there is no f(k)n^{o(k)} time algorithm for the general \mathcal{D}-SN problem on planar graphs, but the special case called Strongly Connected Steiner Subgraph can be solved in time f(k) n^{O(\sqrt{k})} on planar graphs. We present a far-reaching generalization and unification of these two results: we give a complete characterization of the behavior of every $\mathcal{D}$-SN problem on planar graphs. We show that assuming ETH, either the problem is (1) solvable in time 2^{O(k)}n^{O(1)}, and not in time 2^{o(k)}n^{O(1)}, or (2) solvable in time f(k)n^{O(\sqrt{k})}, but not in time f(k)n^{o(\sqrt{k})}, or (3) solvable in time f(k)n^{O(k)}, but not in time f(k)n^{o({k})}., Comment: 87 pages, 18 figures

The leafage of a chordal graph G is the minimum integer l such that G can be realized as an intersection graph of subtrees of a tree with l leaves. We consider structural parameterization by the leafage of classical domination and cut problems on chordal graphs. Fomin, Golovach, and Raymond [ESA 2018, Algorithmica 2020] proved, among other things, that Dominating Set on chordal graphs admits an algorithm running in time $2^{O(l^2)} n^{O(1)}$. We present a conceptually much simpler algorithm that runs in time $2^{O(l)} n^{O(1)}$. We extend our approach to obtain similar results for Connected Dominating Set and Steiner Tree. We then consider the two classical cut problems MultiCut with Undeletable Terminals and Multiway Cut with Undeletable Terminals. We prove that the former is W[1]-hard when parameterized by the leafage and complement this result by presenting a simple $n^{O(l)}$-time algorithm. To our surprise, we find that Multiway Cut with Undeletable Terminals on chordal graphs can be solved, in contrast, in $n^{O(1)}$-time., Comment: 48 pages, 3 figures, 1 table

We show fixed-parameter tractability of the Directed Multicut problem with three terminal pairs (with a randomized algorithm). This problem, given a directed graph $G$, pairs of vertices (called terminals) $(s_1,t_1)$, $(s_2,t_2)$, and $(s_3,t_3)$, and an integer $k$, asks to find a set of at most $k$ non-terminal vertices in $G$ that intersect all $s_1t_1$-paths, all $s_2t_2$-paths, and all $s_3t_3$-paths. The parameterized complexity of this case has been open since Chitnis, Cygan, Hajiaghayi, and Marx proved fixed-parameter tractability of the 2-terminal-pairs case at SODA 2012, and Pilipczuk and Wahlstr\"{o}m proved the W[1]-hardness of the 4-terminal-pairs case at SODA 2016. On the technical side, we use two recent developments in parameterized algorithms. Using the technique of directed flow-augmentation [Kim, Kratsch, Pilipczuk, Wahlstr\"{o}m, STOC 2022] we cast the problem as a CSP problem with few variables and constraints over a large ordered domain.We observe that this problem can be in turn encoded as an FO model-checking task over a structure consisting of a few 0-1 matrices. We look at this problem through the lenses of twin-width, a recently introduced structural parameter [Bonnet, Kim, Thomass\'{e}, Watrigant, FOCS 2020]: By a recent characterization [Bonnet, Giocanti, Ossona de Mendes, Simon, Thomass\'{e}, Toru\'{n}czyk, STOC 2022] the said FO model-checking task can be done in FPT time if the said matrices have bounded grid rank. To complete the proof, we show an irrelevant vertex rule: If any of the matrices in the said encoding has a large grid minor, a vertex corresponding to the ``middle'' box in the grid minor can be proclaimed irrelevant -- not contained in the sought solution -- and thus reduced.

For a graph $G$, a subset $S \subseteq V(G)$ is called a \emph{resolving set} 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 {\sc Metric Dimension} problem takes as input a graph $G$ and a positive integer $k$, and asks whether there exists a resolving set of size at most $k$. This problem was introduced in the 1970s and is known to be \NP-hard~[GT~61 in Garey and Johnson's book]. In the realm of parameterized complexity, Hartung and Nichterlein~[CCC~2013] proved that the problem is \W[2]-hard when parameterized by the natural parameter $k$. They also observed that it is \FPT\ when parameterized by the vertex cover number and asked about its complexity under \emph{smaller} parameters, in particular the feedback vertex set number. We answer this question by proving that {\sc Metric Dimension} is \W[1]-hard when parameterized by the combined parameter feedback vertex set number plus pathwidth. This also improves the result of Bonnet and Purohit~[IPEC 2019] which states that the problem is \W[1]-hard parameterized by the pathwidth. On the positive side, we show that {\sc Metric Dimension} is \FPT\ when parameterized by either the distance to cluster or the distance to co-cluster, both of which are smaller parameters than the vertex cover number.

We generalize the monotone local search approach of Fomin, Gaspers, Lokshtanov and Saurabh [J.ACM 2019], by establishing a connection between parameterized approximation and exponential-time approximation algorithms for monotone subset minimization problems. In a monotone subset minimization problem the input implicitly describes a non-empty set family over a universe of size $n$ which is closed under taking supersets. The task is to find a minimum cardinality set in this family. Broadly speaking, we use approximate monotone local search to show that a parameterized $\alpha$-approximation algorithm that runs in $c^k \cdot n^{O(1)}$ time, where $k$ is the solution size, can be used to derive an $\alpha$-approximation randomized algorithm that runs in $d^n \cdot n^{O(1)}$ time, where $d$ is the unique value in $d \in (1,1+\frac{c-1}{\alpha})$ such that $\mathcal{D}(\frac{1}{\alpha}\|\frac{d-1}{c-1})=\frac{\ln c}{\alpha}$ and $\mathcal{D}(a \|b)$ is the Kullback-Leibler divergence. This running time matches that of Fomin et al. for $\alpha=1$, and is strictly better when $\alpha >1$, for any $c > 1$. Furthermore, we also show that this result can be derandomized at the expense of a sub-exponential multiplicative factor in the running time. We demonstrate the potential of approximate monotone local search by deriving new and faster exponential approximation algorithms for Vertex Cover, $3$-Hitting Set, Directed Feedback Vertex Set, Directed Subset Feedback Vertex Set, Directed Odd Cycle Transversal and Undirected Multicut. For instance, we get a $1.1$-approximation algorithm for Vertex Cover with running time $1.114^n \cdot n^{O(1)}$, improving upon the previously best known $1.1$-approximation running in time $1.127^n \cdot n^{O(1)}$ by Bourgeois et al. [DAM 2011]., Comment: 27 pages, full version of a paper accepted at ESA 2022

For a positive integer $\ell \geq 3$, the $C_\ell$-Contractibility problem takes as input an undirected simple graph $G$ and determines whether $G$ can be transformed into a graph isomorphic to $C_\ell$ (the induced cycle on $\ell$ vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that $C_4$-Contractibility is NP-complete in general graphs. It is easy to verify that $C_3$-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that $C_{\ell}$-Contractibility is \NP-complete\ on bipartite graphs for $\ell = 6$ and posed as open problems the status of the problem when $\ell$ is 4 or 5. In this paper, we show that both $C_5$-Contractibility and $C_4$-Contractibility are NP-complete on bipartite graphs.

The Edge Multicut problem is a classical cut problem where given an undirected graph $G$, a set of pairs of vertices $\mathcal{P}$, and a budget $k$, the goal is to determine if there is a set $S$ of at most $k$ edges such that for each $(s,t) \in \mathcal{P}$, $G-S$ has no path from $s$ to $t$. Edge Multicut has been relatively recently shown to be fixed-parameter tractable (FPT), parameterized by $k$, by Marx and Razgon [SICOMP 2014], and independently by Bousquet et al. [SICOMP 2018]. In the weighted version of the problem, called Weighted Edge Multicut one is additionally given a weight function $\mathtt{wt} : E(G) \to \mathbb{N}$ and a weight bound $w$, and the goal is to determine if there is a solution of size at most $k$ and weight at most $w$. Both the FPT algorithms for Edge Multicut by Marx et al. and Bousquet et al. fail to generalize to the weighted setting. In fact, the weighted problem is non-trivial even on trees and determining whether Weighted Edge Multicut on trees is FPT was explicitly posed as an open problem by Bousquet et al. [STACS 2009]. In this article, we answer this question positively by designing an algorithm which uses a very recent result by Kim et al. [STOC 2022] about directed flow augmentation as subroutine. We also study a variant of this problem where there is no bound on the size of the solution, but the parameter is a structural property of the input, for example, the number of leaves of the tree. We strengthen our results by stating them for the more general vertex deletion version., Comment: Full version of the paper accepted for WG 2022

{\sc Directed Feedback Vertex Set (DFVS)} is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the {\sc ${\cal H}$-free SCC Deletion} problem. Here, one is given a digraph $D$, an integer $k$ and the objective is to decide whether there is a vertex set of size at most $k$ whose deletion leaves a digraph where every strong component excludes graphs in the fixed finite family ${\cal H}$ as (not necessarily induced) subgraphs. When ${\cal H}$ comprises only the digraph with a single arc, then this problem is precisely DFVS. Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if ${\cal H}$ only contains rooted graphs or if ${\cal H}$ contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the recent results of G\"{o}ke et al. [CIAC 2019] for the {\sc 1-Out-Regular Vertex Deletion} and {\sc Bounded Size Strong Component Vertex Deletion} problems. Moreover, we design algorithms for the two above mentioned problems, whose running times are better and match with the best bounds for {\sc DFVS}, without using the heavy machinery of shadow removal as is done by G\"{o}ke et al. [CIAC 2019].

The family of judicious partitioning problems, introduced by Bollob\'as and Scott to the field of extremal combinatorics, has been extensively studied from a structural point of view for over two decades. This rich realm of problems aims to counterbalance the objectives of classical partitioning problems such as Min Cut, Min Bisection and Max Cut. While these classical problems focus solely on the minimization/maximization of the number of edges crossing the cut, judicious (bi)partitioning problems ask the natural question of the minimization/maximization of the number of edges lying in the (two) sides of the cut. In particular, Judicious Bipartition (JB) seeks a bipartition that is "judicious" in the sense that neither side is burdened by too many edges, and Balanced JB also requires that the sizes of the sides themselves are "balanced" in the sense that neither of them is too large. Both of these problems were defined in the work by Bollob\'as and Scott, and have received notable scientific attention since then. In this paper, we shed light on the study of judicious partitioning problems from the viewpoint of algorithm design. Specifically, we prove that BJB is FPT (which also proves that JB is FPT).

We present two new combinatorial tools for the design of parameterized algorithms. The first is a simple linear time randomized algorithm that given as input a $d$-degenerate graph $G$ and an integer $k$, outputs an independent set $Y$, such that for every independent set $X$ in $G$ of size at most $k$, the probability that $X$ is a subset of $Y$ is at least $\left({(d+1)k \choose k} \cdot k(d+1)\right)^{-1}$.The second is a new (deterministic) polynomial time graph sparsification procedure that given a graph $G$, a set $T = \{\{s_1, t_1\}, \{s_2, t_2\}, \ldots, \{s_\ell, t_\ell\}\}$ of terminal pairs and an integer $k$, returns an induced subgraph $G^\star$ of $G$ that maintains all the inclusion minimal multicuts of $G$ of size at most $k$, and does not contain any $(k+2)$-vertex connected set of size $2^{{\cal O}(k)}$. In particular, $G^\star$ excludes a clique of size $2^{{\cal O}(k)}$ as a topological minor. Put together, our new tools yield new randomized fixed parameter tractable (FPT) algorithms for Stable $s$-$t$ Separator, Stable Odd Cycle Transversal and Stable Multicut on general graphs, and for Stable Directed Feedback Vertex Set on $d$-degenerate graphs, resolving two problems left open by Marx et al. [ACM Transactions on Algorithms, 2013]. All of our algorithms can be derandomized at the cost of a small overhead in the running time., Comment: 35 pages

The product dimension of a graph G is defined as the minimum natural number l such that G is an induced subgraph of a direct product of l complete graphs. In this paper we study the product dimension of forests, bounded treewidth graphs and k-degenerate graphs. We show that every forest on n vertices has a product dimension at most 1.441logn+3. This improves the best known upper bound of 3logn for the same due to Poljak and Pultr. The technique used in arriving at the above bound is extended and combined with a result on existence of orthogonal Latin squares to show that every graph on n vertices with a treewidth at most t has a product dimension at most (t+2)(logn+1). We also show that every k-degenerate graph on n vertices has a product dimension at most \ceil{8.317klogn}+1. This improves the upper bound of 32klogn for the same by Eaton and Rodl., Comment: 12 pages, 3 figures