Polynomial Kernelization for Removing Induced Claws and Diamonds

A graph is called (claw,diamond)-free if it contains neither a claw (a $K_{1,3}$) nor a diamond (a $K_4$ with an edge removed) as an induced subgraph. Equivalently, (claw,diamond)-free graphs can be characterized as line graphs of triangle-free graphs, or as linear dominoes, i.e., graphs in which every vertex is in at most two maximal cliques and every edge is in exactly one maximal clique. In this paper we consider the parameterized complexity of the (claw,diamond)-free Edge Deletion problem, where given a graph $G$ and a parameter $k$, the question is whether one can remove at most $k$ edges from $G$ to obtain a (claw,diamond)-free graph. Our main result is that this problem admits a polynomial kernel. We complement this finding by proving that, even on instances with maximum degree $6$, the problem is NP-complete and cannot be solved in time $2^{o(k)}\cdot |V(G)|^{O(1)}$ unless the Exponential Time Hypothesis fai

Subexponential parameterized algorithms for graphs of polynomial growth

We show that for a number of parameterized problems for which only $2^{O(k)} n^{O(1)}$ time algorithms are known on general graphs, subexponential parameterized algorithms with running time $2^{O(k^{1-\frac{1}{1+\delta}} \log^2 k)} n^{O(1)}$ are possible for graphs of polynomial growth with growth rate (degree) $\delta$, that is, if we assume that every ball of radius $r$ contains only $O(r^\delta)$ vertices. The algorithms use the technique of low-treewidth pattern covering, introduced by Fomin et al. [FOCS 2016] for planar graphs; here we show how this strategy can be made to work for graphs with polynomial growth. Formally, we prove that, given a graph $G$ of polynomial growth with growth rate $\delta$ and an integer $k$, one can in randomized polynomial time find a subset $A \subseteq V(G)$ such that on one hand the treewidth of $G[A]$ is $O(k^{1-\frac{1}{1+\delta}} \log k)$, and on the other hand for every set $X \subseteq V(G)$ of size at most $k$, the probability that $X \subseteq A$ is $2^{-O(k^{1-\frac{1}{1+\delta}} \log^2 k)}$. Together with standard dynamic programming techniques on graphs of bounded treewidth, this statement gives subexponential parameterized algorithms for a number of subgraph search problems, such as Long Path or Steiner Tree, in graphs of polynomial growth. We complement the algorithm with an almost tight lower bound for Long Path: unless the Exponential Time Hypothesis fails, no parameterized algorithm with running time $2^{k^{1-\frac{1}{\delta}-\varepsilon}}n^{O(1)}$ is possible for any $\varepsilon > 0$ and an integer $\delta \geq 3$

Cutwidth: obstructions and algorithmic aspects

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most $k$ are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most $k$. We prove that every minimal immersion obstruction for cutwidth at most $k$ has size at most $2^{O(k^3\log k)}$. As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $2^{O(k^2\log k)}\cdot n$, where $k$ is the optimum width and $n$ is the number of vertices. While being slower by a $\log k$-factor in the exponent than the fastest known algorithm, given by Thilikos, Bodlaender, and Serna in [Cutwidth I: A linear time fixed parameter algorithm, J. Algorithms, 56(1):1--24, 2005] and [Cutwidth II: Algorithms for partial $w$-trees of bounded degree, J. Algorithms, 56(1):25--49, 2005], our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts

Parameterized Approximation for Maximum Weight Independent Set of Rectangles and Segments

In the Maximum Weight Independent Set of Rectangles problem (MWISR) we aregiven a weighted set of $n$ axis-parallel rectangles in the plane. The task isto find a subset of pairwise non-overlapping rectangles with the maximumpossible total weight. This problem is NP-hard and the best-knownpolynomial-time approximation algorithm, due to by Chalermsook and Walczak(SODA 2021), achieves approximation factor $O(\log\log n )$. While in theunweighted setting, constant factor approximation algorithms are known, due toMitchell (FOCS 2021) and to G\'alvez et al. (SODA 2022), it remains open toextend these techniques to the weighted setting. In this paper, we consider MWISR through the lens of parameterizedapproximation. Grandoni et al. (ESA 2019) gave a $(1-\epsilon)$-approximationalgorithm with running time $k^{O(k/\epsilon^8)} n^{O(1/\epsilon^8)}$ time,where $k$ is the number of rectangles in an optimum solution. Unfortunately,their algorithm works only in the unweighted setting and they left it as anopen problem to give a parameterized approximation scheme in the weightedsetting. Our contribution is a partial answer to the open question of Grandoni et al.(ESA 2019). We give a parameterized approximation algorithm for MWISR thatgiven a parameter $k$, finds a set of non-overlapping rectangles of weight atleast $(1-\epsilon) \text{opt}_k$ in $2^{O(k \log(k/\epsilon))}n^{O(1/\epsilon)}$ time, where $\text{opt}_k$ is the maximum weight of asolution of cardinality at most $k$. Note that thus, our algorithm may return asolution consisting of more than $k$ rectangles. To complement this apparentweakness, we also propose a parameterized approximation scheme with runningtime $2^{O(k^2 \log(k/\epsilon))} n^{O(1)}$ that finds a solution withcardinality at most $k$ and total weight at least $(1-\epsilon)\text{opt}_k$for the special case of axis-parallel segments.<br

Bounding Generalized Coloring Numbers of Planar Graphs Using Coin Models

We study Koebe orderings of planar graphs: vertex orderings obtained bymodelling the graph as the intersection graph of pairwise internally-disjointdiscs in the plane, and ordering the vertices by non-increasing radii of theassociated discs. We prove that for every $d\in \mathbb{N}$, any such orderinghas $d$-admissibility bounded by $O(d/\ln d)$ and weak $d$-coloring numberbounded by $O(d^4 \ln d)$. This in particular shows that the $d$-admissibilityof planar graphs is bounded by $O(d/\ln d)$, which asymptotically matches aknown lower bound due to Dvo\v{r}\'ak and Siebertz.<br

Fixed-Parameter Tractability of Multicut in Directed Acyclic Graphs

The Multicut problem, given a graph G, a set of terminal pairs $\mathcal{T}=\{(s_i,t_i)\ |\ 1\leq i\leq r\}$, and an integer $p$, asks whether one can find a cutset consisting of at most $p$ nonterminal vertices that separates all the terminal pairs, i.e., after removing the cutset, $t_i$ is not reachable from $s_i$ for each $1\leq i\leq r$. The fixed-parameter tractability of Multicut in undirected graphs, parameterized by the size of the cutset only, has been recently proved by Marx and Razgon [SIAM J. Comput., 43 (2014), pp. 355--388] and, independently, by Bousquet, Daligault, and Thomassé [Proceedings of STOC, ACM, 2011, pp. 459--468], after resisting attacks as a long-standing open problem. In this paper we prove that Multicut is fixed-parameter tractable on directed acyclic graphs when parameterized both by the size of the cutset and the number of terminal pairs. We complement this result by showing that this is implausible for parameterization by the size of the cutset only, as this version of the problem remains $W[1]$-hard

Turing Kernelization for Finding Long Paths in Graphs Excluding a Topological Minor

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

Independence Number of Intersection Graphs of Axis-Parallel Segments

We prove that for any triangle-free intersection graph of $n$ axis-parallelsegments in the plane, the independence number $\alpha$ of this graph is atleast $\alpha \ge n/4 + \Omega(\sqrt{n})$. We complement this with aconstruction of a graph in this class satisfying $\alpha \le n/4 + c \sqrt{n}$for an absolute constant $c$, which demonstrates the optimality of our result.<br