arXiv subjects

Daniel Lokshtanov

Publications and source records attributed to Daniel Lokshtanov.

At least 19 recordsLinked to original sources

arXiv2026

Forbidden Subgraphs of Graphs with Low Bandwidth

A layout of a graph G is an injective function $f : V(G) \rightarrow Z$, and the bandwidth of a layout f is $bw(G,f) = max_{uv \in E(G)} |f(u) - f(v)|$. The bandwidth bw(G) of G is the minimum bandwidth of a layout of G. Computing the bandwidth of a graph is a notoriously hard problem: assuming P != NP, there is no polynomial time algorithm, even on very restricted classes of trees [Monien, SIAM Journal on Algebraic Discrete Methods, 1986], and no constant factor approximation, even on trees [Dubey et al., JCSS 2011]. Assuming the Exponential Time Hypothesis, there is no algorithm with running time $f(k)n^{o(k)}$ to determine whether an input graph has bandwidth at most k, even on very restricted classes of trees [Dregi and Lokshtanov, ICALP 2014]. In this paper we show that {\sc Bandwidth} on general graphs is FPT-approximable. In particular we give an algorithm that takes as input a graph G and an integer k, runs in time $2^{O(9^k)}n^{O(1)}$, and outputs a subtree T of G such that $bw(T) \geq k$ or a layout of G of bandwidth at most $(10^{85} k^{28})^{4^k}$. This resolves in the affirmative an open problem of Chung and Seymour [Discrete Mathematics, 1989], who asked whether the bandwidth of every graph G is upper bounded in terms of the maximum bandwidth of a subtree of G. Our theorem leads to a forbidden subgraph characterization for graphs of bounded bandwidth, and can be seen as an analog for bandwidth of the classic grid minor theorem for treewidth, the forbidden subtree theorem for pathwidth, and the forbidden subpath theorem for treedepth.

cs.DS↗

arXiv2026

Covering Points with Rectangular Boundaries

Geometric covering problems ask for a small family of geometric objects whose union covers a given point set. We study the more restrictive \emph{boundary covering} variant, where every point must lie on the boundary of a chosen object. Motivated by the framework of Langerman and Morin\,[Discret.\ Comput.\ Geom., 2005] for hyperspheres, we initiate the study of boundary covering by axis-parallel rectangles. We first consider the \emph{discrete} setting, where rectangles must be selected from a given family. We define \bcdaprfull\ (\bcdaprshort): given a point set $P\subseteq\mathbb{R}^2$, a family $\mathcal{R}$ of axis-parallel rectangles, and an integer $k$, decide whether $P$ can be covered by the boundaries of at most $k$ rectangles from $\mathcal{R}$. We prove that \bcdaprshort\ is $\mathrm{W}[1]$-hard parameterized by $k$. We then study the \emph{continuous} variant, \prbcfull\ (\prbcshort), where rectangles may be placed freely. Given $P\subseteq\mathbb{R}^2$ and $k$, the goal is to decide whether $P$ can be covered by the boundaries of at most $k$ axis-parallel rectangles. In contrast to the discrete case, we show that \prbcshort\ is fixed-parameter tractable, with running time $2^{\cO(k\log k)}\cdot n^{\cO(1)}$, where $n=|P|$. Our algorithm relies on a structural analysis of how $k$ rectangles interact with the point set, reducing \prbcshort\ to at most $2^{\cO(k\log k)}$ instances of \ddmtcsp, each solvable in polynomial time. On the hardness side, we prove NP-completeness for boundary covering by axis-aligned $L$-shapes and use this reduction to establish NP-completeness of \prbcshort.

cs.CG↗

arXiv2026

Courcelle's Theorem in Truly Linear FPT

Recently, Bumpus, Downey, Eagling-Vose, Enright, Fellows, Kutner, Larios-Jones, Martin, Rosamond, and Yates defined Truly Linear FPT (TLFPT) to be the class of parameterized problems with algorithms running in time $O(n) + f(k)$, where $n$ is the input size and $k$ the parameter [arXiv:2606.02492]. They gave several algorithmic techniques for designing TLFPT algorithms, but left parameterization by treewidth open. In this paper, we give a general method for designing TLFPT algorithms parameterized by treewidth, solving three open problems posed by Bumpus et al. In particular, we give a TLFPT algorithm for Courcelle's theorem: We show that given an $n$-vertex $m$-edge graph $G$, an integer $k$, and a $\mathsf{CMSO}_2$-formula $φ$, we can in time $O(n+m) + f(k, φ)$ either conclude that the treewidth of $G$ is more than $k$, or check whether $G$ satisfies $φ$. As a part of our algorithm, we give an approximation algorithm for treewidth that runs in time $O(n+m)$ and returns a tree decomposition whose width is at most $2^{O(k)}$ times the optimum. Our result also implies a TLFPT algorithm for computing the value of treewidth exactly.

cs.DS↗

arXiv2026

Induced-Minor-Closed Classes have Linear, Square-Root, or Sub-Polynomial Tree-Independence

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, χ)$ where $T$ is a tree and $χ: V(T) \rightarrow 2^{V(G)}$ is a function satisfying two axioms: for every edge $uv \in E(G)$ there is an $x \in V(T)$ such that $\{u,v\} \subseteq χ(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) | u \in χ(x)\}$ induces a non-empty and connected subtree of $T$. The sets $χ(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the decomposition. A graph $H$ is an induced minor of a graph $G$ if a graph isomorphic to $H$ can be obtained from $G$ by vertex deletions and edge contractions. We prove that for every $t\in\mathbb{N}$ there exists an $ε> 0$ such that every graph $G$ either contains the complete bipartite graph $K_{t,t}$ or the wall $W_{t\times t}$ as an induced minor, or has tree-independence at most $O(2^{O((\log n)^{1-ε})})$. This leads to algorithms with running time $2^{n^{o(1)}}$, for a wide range of problems on $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs. Our result is a substantial generalization of existing bounds for the tree-independence and tree-width on various graph classes, and a partial resolution of the conjecture of Chudnovsky, E S, and Lokshtanov [Arxiv, 2025] that $\{K_{t,t}, W_{t\times t}\}$-induced minor free graphs have poly-logarithmic tree independence number. The generality comes at the cost of a sub-polynomial, rather than poly-logarithmic upper bound. Our result leads to a complete classification of induced-minor closed classes into ones that have sub-polynomial tree-independence, tree-independence equal to $\tilde{O}(\sqrt{n})$, and linear tree-independence.

math.CO↗

arXiv2026

Fine-Grained Bounds for Courcelle's Theorem

Courcelle's theorem states that there exists an algorithm that takes as input a graph $G$ of treewidth at most $t$ and a MSO formula $ϕ$, and determines whether $G$ satisfies $ϕ$ in time $f(ϕ,t) \cdot n$. It is folklore that the the function $f$ contains a tower of exponentials whose height depends as a linear function of the number of quantifier alternations of the input formula $ϕ$. A classic reduction of Frick and Grohe shows that, assuming the Exponential Time Hypothesis (ETH), the linear growth of the height of the tower is unavoidable. Nevertheless, there is still a huge gap between existing upper and lower bounds -- after all, there is quite a difference between a single exponential and a double exponential running time. In addition, this only gives us a very coarse understanding in the time complexity of Courcelle's theorem. In this paper, we prove a fine-grained version of Courcelle's theorem with nearly ETH-tight dependence on the treewidth parameter $t$ and the quantifier structure of $ϕ$ (specifically, the number of first order and second order variables in each quantifier alternation block).

cs.DS↗

arXiv2026

(Treewidth, Clique)-Boundedness and Poly-logarithmic Tree-Independence

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, χ)$ where $T$ is a tree and $χ: V(T) \rightarrow 2^{V(G)}$ is a function satisfying the following two axioms: for every edge $uv \in V(G)$ there is a $x \in V(T)$ such that $\{u,v\} \subseteq χ(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) ~:~ u \in χ(X)\}$ induces a non-empty and connected subtree of $T$. The sets $χ(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the tree decomposition. The study of graph classes with bounded tree-independence number has attracted much attention in recent years, in part due its improtant algorithmic implications. A conjecture of Dallard, Milanič and Štorgel, connecting tree-independence number to the classical notion of treewidth, was one of the motivating problems in the area. This conjecture was recently disproved, but here we prove a slight variant of it, that retains much of the algorithmic significance. As part of the proof we introduce the notion of independence-containers, which can be viewed as a generalization of the set of all maximal cliques of a graph, and is of independent interest.

math.CO↗

arXiv2026

Induced Minors and Coarse Tree Decompositions

Let $G$ be a graph, $S \subseteq V(G)$ be a vertex set in $G$ and $r$ be a positive integer. The distance $r$-independence number of $S$ is the size of the largest subset $I \subseteq S$ such that no pair $u$, $v$ of vertices in $I$ have a path on at most $r$ edges between them in $G$. It has been conjectured [Chudnovsky et al., arXiv, 2025] that for every positive integer $t$ there exist positive integers $c$, $d$ such that every graph $G$ that excludes both the complete bipartite graph $K_{t,t}$ and the grid $\boxplus_t$ as an induced minor has a tree decomposition in which every bag has (distance $1$) independence number at most $c(\log n)^d$. We prove a weaker version of this conjecture where every bag of the tree decomposition has distance $16(\log n + 1)$-independence number at most $c(\log n)^d$. On the way we also prove a version of the conjecture where every bag of the decomposition has distance $8$-independence number at most $2^{c (\log n)^{1-(1/d)}}$.

math.CO↗

arXiv2026

Parameterized Approximation of Rectangle Stabbing

In the Rectangle Stabbing problem, input is a set ${\cal R}$ of axis-parallel rectangles and a set ${\cal L}$ of axis parallel lines in the plane. The task is to find a minimum size set ${\cal L}^* \subseteq {\cal L}$ such that for every rectangle $R \in {\cal R}$ there is a line $\ell \in {\cal L}^*$ such that $\ell$ intersects $R$. Gaur et al. [Journal of Algorithms, 2002] gave a polynomial time $2$-approximation algorithm, while Dom et al. [WALCOM 2009] and Giannopolous et al. [EuroCG 2009] independently showed that, assuming FPT $\neq$ W[1], there is no algorithm with running time $f(k)(|{\cal L}||{\cal R}|)^{O(1)}$ that determines whether there exists an optimal solution with at most $k$ lines. We give the first parameterized approximation algorithm for the problem with a ratio better than $2$. In particular we give an algorithm that given ${\cal R}$, ${\cal L}$, and an integer $k$ runs in time $k^{O(k)}(|{\cal L}||{\cal R}|)^{O(1)}$ and either correctly concludes that there does not exist a solution with at most $k$ lines, or produces a solution with at most $\frac{7k}{4}$ lines. We complement our algorithm by showing that unless FPT $=$ W[1], the Rectangle Stabbing problem does not admit a $(\frac{5}{4}-ε)$-approximation algorithm running in $f(k)(|{\cal L}||{\cal R}|)^{O(1)}$ time for any function $f$ and $ε> 0$.

cs.CG↗

arXiv2026

Induced minors and subpolynomial treewidth

Given a family $\mathcal{H}$ of graphs, we say that a graph $G$ is $\mathcal{H}$-induced-minor-free if no induced minor of $G$ is isomorphic to a member of $\mathcal{H}$, We denote by $W_{t\times t}$ the $t$-by-$t$ hexagonal grid, and by $K_{t,t}$ the complete bipartite graph with both sides of the bipartition of size $t$. We show that the class of $\{K_{t,t},W_{t\times t}\}$-induced minor-free graphs with bounded clique number has subpolynomial treewidth. Specifically, we prove that for every integer $t$ there exist $ε\in (0,1]$ and $c \in \mathbb{N}$ such that every $n$-vertex $\{K_{t,t},W_{t\times t}\}$-induced minor-free graph with no clique of size $t$ has treewidth at most $2^{c\log^{1-ε}n}$.

math.CO↗

arXiv2026

Maximum Partial List H-Coloring on P_5-free graphs in polynomial time

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 $n^{O(ω(G))}$-time algorithm for Maximum Partial H-Coloring by Chudnovsky, King, Pilipczuk, Rzążewski & Spirkl [SIDMA 2021] to polynomial-time algorithm.

cs.DS↗

arXiv2026

Faster Parameterized Vertex Multicut

In the {\sc Vertex Multicut} problem the input consists of a graph $G$, integer $k$, and a set $\mathbf{T} = \{(s_1, t_1), \ldots, (s_p, t_p)\}$ of pairs of vertices of $G$. The task is to find a set $X$ of at most $k$ vertices such that, for every $(s_i, t_i) \in \mathbf{T}$, there is no path from $s_i$ to $t_i$ in $G - X$. Marx and Razgon [STOC 2011 and SICOMP 2014] and Bousquet, Daligault, and Thomassé [STOC 2011 and SICOMP 2018] independently and simultaneously gave the first algorithms for {\sc Vertex Multicut} with running time $f(k)n^{O(1)}$. The running time of their algorithms is $2^{O(k^3)}n^{O(1)}$ and $2^{O(k^{O(1)})}n^{O(1)}$, respectively. As part of their result, Marx and Razgon introduce the {\em shadow removal} technique, which was subsequently applied in algorithms for several parameterized cut and separation problems. The shadow removal step is the only step of the algorithm of Marx and Razgon which requires $2^{O(k^3)}n^{O(1)}$ time. Chitnis et al. [TALG 2015] gave an improved version of the shadow removal step, which, among other results, led to a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Vertex Multicut}. We give a faster algorithm for the {\sc Vertex Multicut} problem with running time $k^{O(k)}n^{O(1)}$. Our main technical contribution is a refined shadow removal step for vertex separation problems that only introduces an overhead of $k^{O(k)}\log n$ time. The new shadow removal step implies a $k^{O(k^2)}n^{O(1)}$ time algorithm for {\sc Directed Subset Feedback Vertex Set} and a $k^{O(k)}n^{O(1)}$ time algorithm for {\sc Directed Multiway Cut}, improving over the previously best known algorithms of Chitnis et al. [TALG 2015].

cs.DS↗

arXiv2026

On the complexity of Multipacking

A multipacking in an undirected graph $G=(V,E)$ is a set $M\subseteq V$ such that for every vertex $v\in V$ and for every integer $r\geq 1$, the ball of radius $r$ around $v$ contains at most $r$ vertices of $M$, that is, there are at most $r$ vertices in $M$ at a distance at most $r$ from $v$ in $G$. The Multipacking problem asks whether a graph contains a multipacking of size at least $k$. For more than a decade, it remained an open question whether the Multipacking problem is NP-complete or solvable in polynomial time. Whereas the problem is known to be polynomial-time solvable for certain graph classes (e.g., strongly chordal graphs, grids, etc). Foucaud, Gras, Perez, and Sikora [Algorithmica 2021] made a step towards solving the open question by showing that the Multipacking problem is NP-complete for directed graphs and it is W[1]-hard when parameterized by the solution size. In this paper, we prove that the Multipacking problem is NP-complete for undirected graphs, which answers the open question. Moreover, the problem is W[2]-hard for undirected graphs when parameterized by the solution size. Furthermore, we have shown that the problem is NP-complete and W[2]-hard (when parameterized by the solution size) even for various subclasses: chordal, bipartite, and claw-free graphs. Whereas, it is NP-complete for regular, and CONV graphs (intersection graphs of convex sets in the plane). Additionally, the problem is NP-complete and W[2]-hard (when parameterized by the solution size) for chordal $\cap$ $\frac{1}{2}$-hyperbolic graphs, which is a superclass of strongly chordal graphs where the problem is polynomial-time solvable. On the positive side, we present an exact exponential-time algorithm for the Multipacking problem on $n$-vertex general graphs, which breaks the $2^n$ barrier by achieving a running time of $O^*(1.58^n)$.

cs.CC↗

arXiv2026

Finding large sparse induced subgraphs in graphs of small (but not very small) tree-independence number

The independence number of a tree decomposition is the size of a largest independent set contained in a single bag. The tree-independence number of a graph $G$ is the minimum independence number of a tree decomposition of $G$. As shown recently by Lima et al. [ESA~2024], a large family of optimization problems asking for a maximum-weight induced subgraph of bounded treewidth, satisfying a given \textsf{CMSO}$_2$ property, can be solved in polynomial time in graphs whose tree-independence number is bounded by some constant~$k$. However, the complexity of the algorithm of Lima et al. grows rapidly with $k$, making it useless if the tree-independence number is superconstant. In this paper we present a refined version of the algorithm. We show that the same family of problems can be solved in time~$n^{\mathcal{O}(k)}$, where $n$ is the number of vertices of the instance, $k$ is the tree-independence number, and the $\mathcal{O}(\cdot)$-notation hides factors depending on the treewidth bound of the solution and the considered \textsf{CMSO}$_2$ property. This running time is quasipolynomial for classes of graphs with polylogarithmic tree-independence number; several such classes were recently discovered. Furthermore, the running time is subexponential for many natural classes of geometric intersection graphs -- namely, ones that admit balanced clique-based separators of sublinear size.

cs.DS↗

arXiv2026

FPT Approximations for Connected Maximum Coverage

We revisit connectivity-constrained coverage through a unifying model, Partial Connected Red-Blue Dominating Set. Given a red-blue bipartite graph $G$ and an auxiliary connectivity graph $G_{conn}$ on red vertices, and integers $k, t$, the task is to find a $k$-sized subset of red vertices that dominates at least $t$ blue vertices, and that induces a connected subgraph in $G_{conn}$. This formulation captures connected variants of Max Coverage, Partial Dominating Set, and Partial Vertex Cover studied in prior literature. After identifying (parameterized) inapproximability results inherited from known problems, we first show that the problem is fixed-parameter tractable by $t$. Furthermore, when the bipartite graph excludes $K_{d,d}$ as a subgraph, we design (resp. efficient) parameterized approximation schemes for approximating $t$ (resp. $k$). Notably, these FPT approximations do not impose any restrictions on $G_{conn}$. Together, these results chart the boundary between hardness and FPT-approximability for connectivity-constrained coverage.

cs.DS↗

arXiv2025

Beyond Exact Fairness: Envy-Free Incomplete Connected Fair Division

We study the problem of Envy-Free Incomplete Connected Fair Division, where exactly p vertices of an undirected graph must be allocated to agents such that each agent receives a connected share and does not envy another agent's share. Focusing on agents with additive valuations, we show that the problem remains computationally hard when parameterized by p and the number of agents. This result holds even for star graphs and with the input numbers given in unary representation, thereby resolving an open problem posed by Gahlawat and Zehavi (FSTTCS 2023). In stark contrast, we show that if one is willing to tolerate even the slightest amount of envy, then the problem becomes efficient with respect to the natural parameters. Specifically, we design an Efficient Parameterized Approximation Scheme parameterized by p and the number of agent types. Our algorithm works on general graphs and remains efficient even when the input numbers are provided in binary representation.

cs.DS↗

arXiv2025

A Quasi-Polynomial Time Algorithm for 3-Coloring Circle Graphs

A graph $G$ is a circle graph if it is an intersection graph of chords of a unit circle. We give an algorithm that takes as input an $n$ vertex circle graph $G$, runs in time at most $n^{O(\log n)}$ and finds a proper $3$-coloring of $G$, if one exists. As a consequence we obtain an algorithm with the same running time to determine whether a given ordered graph $(G, \prec)$ has a $3$-page book embedding. This gives a partial resolution to the well known open problem of Dujmović and Wood [Discret. Math. Theor. Comput. Sci. 2004], Eppstein [2014], and Bachmann, Rutter and Stumpf [J. Graph Algorithms Appl. 2024] of whether 3-Coloring on circle graphs admits a polynomial time algorithm.

cs.DS↗

arXiv2025

Maximum Weight Independent Set in Graphs with no Long Claws in Quasi-Polynomial Time

We show that the Maximum Weight Independent Set problem (MWIS) can be solved in quasi-polynomial time on $H$-free graphs (graphs excluding a fixed graph $H$ as an induced subgraph) for every $H$ whose every connected component is a path or a subdivided claw (i.e., a tree with at most three leaves). This completes the dichotomy of the complexity of MWIS in $\mathcal{F}$-free graphs for any finite set $\mathcal{F}$ of graphs into NP-hard cases and cases solvable in quasi-polynomial time, and corroborates the conjecture that the cases not known to be NP-hard are actually polynomial-time solvable. The key graph-theoretic ingredient in our result is as follows. Fix an integer $t \geq 1$. Let $S_{t,t,t}$ be the graph created from three paths on $t$ edges by identifying one endpoint of each path into a single vertex. We show that, given a graph $G$, one can in polynomial time find either an induced $S_{t,t,t}$ in $G$, or a balanced separator consisting of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods in $G$, or an extended strip decomposition of $G$ (a decomposition almost as useful for recursion for MWIS as a partition into connected components) with each particle of weight multiplicatively smaller than the weight of $G$. This is a strengthening of a result of Majewski, Masařík, Novotná, Okrasa, Pilipczuk, Rzążewski, and Sokołowski [ICALP 2022] which provided such an extended strip decomposition only after the deletion of $\mathcal{O}(\log |V(G)|)$ vertex neighborhoods. To reach the final result, we employ an involved branching strategy that relies on the structural lemma presented above.

cs.DS↗

arXiv2025

Tree independence number II. Three-path-configurations

A three-path-configuration is a graph consisting of three pairwise internally-disjoint paths the union of every two of which is an induced cycle of length at least four. A graph is 3PC-free if no induced subgraph of it is a three-path-configuration. We prove that 3PC-free graphs have poly-logarithmic tree-independence number. More explicitly, we show that there exists a constant $c$ such that every $n$-vertex 3PC-free graph graph has a tree decomposition in which every bag has stability number at most $c (\log n)^2$. This implies that the Maximum Weight Independent Set problem, as well as several other natural algorithmic problems, that are known to be NP-hard in general, can be solved in quasi-polynomial time if the input graph is 3PC-free.

math.CO↗