arXiv subjects

William Lochet

Publications and source records attributed to William Lochet.

At least 19 recordsLinked to original sources

arXiv2025

On a tree-based variant of bandwidth and forbidding simple topological minors

We obtain structure theorems for graphs excluding a fan (a path with a universal vertex) or a dipole ($K_{2,k}$) as a topological minor. The corresponding decompositions can be computed in FPT linear time. This is motivated by the study of a graph parameter we call treebandwidth which extends the graph parameter bandwidth by replacing the linear layout by a rooted tree such that neighbours in the graph are in ancestor-descendant relation in the tree. We deduce an approximation algorithm for treebandwidth running in FPT linear time from our structure theorems. We complement this result with a precise characterisation of the parameterised complexity of computing the parameter exactly.

cs.DM↗

arXiv2024

Robust Contraction Decomposition for Minor-Free Graphs and its Applications

We prove a robust contraction decomposition theorem for $H$-minor-free graphs, which states that given an $H$-minor-free graph $G$ and an integer $p$, one can partition in polynomial time the vertices of $G$ into $p$ sets $Z_1,\dots,Z_p$ such that $\operatorname{tw}(G/(Z_i \setminus Z')) = O(p + |Z'|)$ for all $i \in [p]$ and $Z' \subseteq Z_i$. Here, $\operatorname{tw}(\cdot)$ denotes the treewidth of a graph and $G/(Z_i \setminus Z')$ denotes the graph obtained from $G$ by contracting all edges with both endpoints in $Z_i \setminus Z'$. Our result generalizes earlier results by Klein [SICOMP 2008] and Demaine et al. [STOC 2011] based on partitioning $E(G)$, and some recent theorems for planar graphs by Marx et al. [SODA 2022], for bounded-genus graphs (more generally, almost-embeddable graphs) by Bandyapadhyay et al. [SODA 2022], and for unit-disk graphs by Bandyapadhyay et al. [SoCG 2022]. The robust contraction decomposition theorem directly results in parameterized algorithms with running time $2^{\widetilde{O}(\sqrt{k})} \cdot n^{O(1)}$ or $n^{O(\sqrt{k})}$ for every vertex/edge deletion problems on $H$-minor-free graphs that can be formulated as Permutation CSP Deletion or 2-Conn Permutation CSP Deletion. Consequently, we obtain the first subexponential-time parameterized algorithms for Subset Feedback Vertex Set, Subset Odd Cycle Transversal, Subset Group Feedback Vertex Set, 2-Conn Component Order Connectivity on $H$-minor-free graphs. For other problems which already have subexponential-time parameterized algorithms on $H$-minor-free graphs (e.g., Odd Cycle Transversal, Vertex Multiway Cut, Vertex Multicut, etc.), our theorem gives much simpler algorithms of the same running time.

cs.DS↗

arXiv2024

Blow-ups and extensions of trees in tournaments

A class of acyclic digraphs $\mathscr{C}$ is linearly unavoidable if there exists a constant $c$ such that every digraph $D\in \mathscr{C}$ is contained in all tournaments of order $c\cdot |V(D)|$. The class of all acyclic digraphs is not linearly avoidable, and Fox, He, and Widgerson recently showed that this is not even the case for acyclic digraphs with bounded maximum degree. On the positive side, Thomason and H\"aggkvist proved that the class of oriented trees is linearly unavoidable. In this work, we generalize this result to acyclic digraphs obtained from an oriented tree by adding at most $k$ vertices, and $k$-blow-ups of oriented trees, for every fixed integer $k$. More precisely, we show that if $D$ is obtained from an oriented tree $F$ of order $n$ by adding $k$ universal vertices, then $D$ is contained in every tournament of order $2\cdot 3^{(k+1)(2k+1)} \cdot n$; and if $D$ is obtained from $F$ by replacing each vertex $u$ by an independent set $X_u$ of size $k$ and every arc $uv$ by all possible arcs from $X_u$ to $X_v$, then $D$ is contained in every tournament of order $2^{10+18k}k \cdot n$.

math.CO↗

arXiv2024

Packing Short Cycles

Cycle packing is a fundamental problem in optimization, graph theory, and algorithms. Motivated by recent advancements in finding vertex-disjoint paths between a specified set of vertices that either minimize the total length of the paths [Bj\"orklund, Husfeldt, ICALP 2014; Mari, Mukherjee, Pilipczuk, and Sankowski, SODA 2024] or request the paths to be shortest [Lochet, SODA 2021], we consider the following cycle packing problems: Min-Sum Cycle Packing and Shortest Cycle Packing. In Min-Sum Cycle Packing, we try to find, in a weighted undirected graph, $k$ vertex-disjoint cycles of minimum total weight. Our first main result is an algorithm that, for any fixed $k$, solves the problem in polynomial time. We complement this result by establishing the W[1]-hardness of Min-Sum Cycle Packing parameterized by $k$. The same results hold for the version of the problem where the task is to find $k$ edge-disjoint cycles. Our second main result concerns Shortest Cycle Packing, which is a special case of Min-Sum Cycle Packing that asks to find a packing of $k$ shortest cycles in a graph. We prove this problem to be fixed-parameter tractable (FPT) when parameterized by $k$ on weighted planar graphs. We also obtain a polynomial kernel for the edge-disjoint variant of the problem on planar graphs. Deciding whether Min-Sum Cycle Packing is FPT on planar graphs and whether Shortest Cycle Packing is FPT on general graphs remain challenging open questions.

cs.DS↗

arXiv2023

Euclidean Bottleneck Steiner Tree is Fixed-Parameter Tractable

In the Euclidean Bottleneck Steiner Tree problem, the input consists of a set of $n$ points in $\mathbb{R}^2$ called terminals and a parameter $k$, and the goal is to compute a Steiner tree that spans all the terminals and contains at most $k$ points of $\mathbb{R}^2$ as Steiner points such that the maximum edge-length of the Steiner tree is minimized, where the length of a tree edge is the Euclidean distance between its two endpoints. The problem is well-studied and is known to be NP-hard. In this paper, we give a $k^{O(k)} n^{O(1)}$-time algorithm for Euclidean Bottleneck Steiner Tree, which implies that the problem is fixed-parameter tractable (FPT). This settles an open question explicitly asked by Bae et al. [Algorithmica, 2011], who showed that the $\ell_1$ and $\ell_{\infty}$ variants of the problem are FPT. Our approach can be generalized to the problem with $\ell_p$ metric for any rational $1 \le p \le \infty$, or even other metrics on $\mathbb{R}^2$.

cs.CG↗

arXiv2023

Minimum-Membership Geometric Set Cover, Revisited

We revisit a natural variant of geometric set cover, called minimum-membership geometric set cover (MMGSC). In this problem, the input consists of a set $S$ of points and a set $\mathcal{R}$ of geometric objects, and the goal is to find a subset $\mathcal{R}^*\subseteq\mathcal{R}$ to cover all points in $S$ such that the \textit{membership} of $S$ with respect to $\mathcal{R}^*$, denoted by $\mathsf{memb}(S,\mathcal{R}^*)$, is minimized, where $\mathsf{memb}(S,\mathcal{R}^*)=\max_{p\in S}|\{R\in\mathcal{R}^*: p\in R\}|$. We achieve the following two main results. * We give the first polynomial-time constant-approximation algorithm for MMGSC with unit squares. This answers a question left open since the work of Erlebach and Leeuwen [SODA'08], who gave a constant-approximation algorithm with running time $n^{O(\mathsf{opt})}$ where $\mathsf{opt}$ is the optimum of the problem (i.e., the minimum membership). * We give the first polynomial-time approximation scheme (PTAS) for MMGSC with halfplanes. Prior to this work, it was even unknown whether the problem can be approximated with a factor of $o(\log n)$ in polynomial time, while it is well-known that the minimum-size set cover problem with halfplanes can be solved in polynomial time. We also consider a problem closely related to MMGSC, called minimum-ply geometric set cover (MPGSC), in which the goal is to find $\mathcal{R}^*\subseteq\mathcal{R}$ to cover $S$ such that the ply of $\mathcal{R}^*$ is minimized, where the ply is defined as the maximum number of objects in $\mathcal{R}^*$ which have a nonempty common intersection. Very recently, Durocher et al. gave the first constant-approximation algorithm for MPGSC with unit squares which runs in $O(n^{12})$ time. We give a significantly simpler constant-approximation algorithm with near-linear running time.

cs.CG↗

arXiv2023

FPT Constant-Approximations for Capacitated Clustering to Minimize the Sum of Cluster Radii

Clustering with capacity constraints is a fundamental problem that attracted significant attention throughout the years. In this paper, we give the first FPT constant-factor approximation algorithm for the problem of clustering points in a general metric into $k$ clusters to minimize the sum of cluster radii, subject to non-uniform hard capacity constraints. In particular, we give a $(15+\epsilon)$-approximation algorithm that runs in $2^{0(k^2\log k)}\cdot n^3$ time. When capacities are uniform, we obtain the following improved approximation bounds: A (4 + $\epsilon$)-approximation with running time $2^{O(k\log(k/\epsilon))}n^3$, which significantly improves over the FPT 28-approximation of Inamdar and Varadarajan [ESA 2020]; a (2 + $\epsilon$)-approximation with running time $2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$ and a $(1+\epsilon)$-approximation with running time $2^{O(kd\log ((k/\epsilon)))}n^{3}$ in the Euclidean space; and a (1 + $\epsilon$)-approximation in the Euclidean space with running time $2^{O(k/\epsilon^2 \cdot\log(k/\epsilon))}dn^3$ if we are allowed to violate the capacities by (1 + $\epsilon$)-factor. We complement this result by showing that there is no (1 + $\epsilon$)-approximation algorithm running in time $f(k)\cdot n^{O(1)}$, if any capacity violation is not allowed.

cs.DS↗

arXiv2022

Edge separators for graphs excluding a minor

We prove that every $n$-vertex $K_t$-minor-free graph $G$ of maximum degree $\Delta$ has a set $F$ of $O(t^2(\log t)^{1/4}\sqrt{\Delta n})$ edges such that every component of $G - F$ has at most $n/2$ vertices. This is best possible up to the dependency on $t$ and extends earlier results of Diks, Djidjev, Sykora, and Vr\v{t}o (1993) for planar graphs, and of Sykora and Vr\v{t}o (1993) for bounded-genus graphs. Our result is a consequence of the following more general result: The line graph of $G$ is isomorphic to a subgraph of the strong product $H \boxtimes K_{\lfloor p \rfloor}$ for some graph $H$ with treewidth at most $t-2$ and $p = \sqrt{(t-3)\Delta |E(G)|} + \Delta$.

math.CO↗

arXiv2022

Detours in Directed Graphs

We study two "above guarantee" versions of the classical Longest Path problem on undirected and directed graphs and obtain the following results. In the first variant of Longest Path that we study, called Longest Detour, the task is to decide whether a graph has an (s,t)-path of length at least dist_G(s,t)+k (where dist_G(s,t) denotes the length of a shortest path from s to t). Bez\'akov\'a et al. proved that on undirected graphs the problem is fixed-parameter tractable (FPT) by providing an algorithm of running time 2^{O (k)} n. Further, they left the parameterized complexity of the problem on directed graphs open. Our first main result establishes a connection between Longest Detour on directed graphs and 3-Disjoint Paths on directed graphs. Using these new insights, we design a 2^{O(k)} n^{O(1)} time algorithm for the problem on directed planar graphs. Further, the new approach yields a significantly faster FPT algorithm on undirected graphs. In the second variant of Longest Path, namely Longest Path Above Diameter, the task is to decide whether the graph has a path of length at least diam(G)+k (diam(G) denotes the length of a longest shortest path in a graph G). We obtain dichotomy results about Longest Path Above Diameter on undirected and directed graphs. For (un)directed graphs, Longest Path Above Diameter is NP-complete even for k=1. However, if the input undirected graph is 2-connected, then the problem is FPT. On the other hand, for 2-connected directed graphs, we show that Longest Path Above Diameter is solvable in polynomial time for each k\in{1,\dots, 4} and is NP-complete for every k\geq 5. The parameterized complexity of Longest Path Above Diameter on general directed graphs remains an interesting open problem.

cs.DS↗

arXiv2021

How to Find a Good Explanation for Clustering?

$k$-means and $k$-median clustering are powerful unsupervised machine learning techniques. However, due to complicated dependences on all the features, it is challenging to interpret the resulting cluster assignments. Moshkovitz, Dasgupta, Rashtchian, and Frost [ICML 2020] proposed an elegant model of explainable $k$-means and $k$-median clustering. In this model, a decision tree with $k$ leaves provides a straightforward characterization of the data set into clusters. We study two natural algorithmic questions about explainable clustering. (1) For a given clustering, how to find the "best explanation" by using a decision tree with $k$ leaves? (2) For a given set of points, how to find a decision tree with $k$ leaves minimizing the $k$-means/median objective of the resulting explainable clustering? To address the first question, we introduce a new model of explainable clustering. Our model, inspired by the notion of outliers in robust statistics, is the following. We are seeking a small number of points (outliers) whose removal makes the existing clustering well-explainable. For addressing the second question, we initiate the study of the model of Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous algorithmic analysis sheds some light on the influence of parameters like the input size, dimension of the data, the number of outliers, the number of clusters, and the approximation ratio, on the computational complexity of explainable clustering.

cs.DS↗

arXiv2021

Subexponential Parameterized Algorithms for Cut and Cycle Hitting Problems on H-Minor-Free Graphs

We design the first subexponential-time (parameterized) algorithms for several cut and cycle-hitting problems on $H$-minor free graphs. In particular, we obtain the following results (where $k$ is the solution-size parameter). 1. $2^{O(\sqrt{k}\log k)} \cdot n^{O(1)}$ time algorithms for Edge Bipartization and Odd Cycle Transversal; 2. a $2^{O(\sqrt{k}\log^4 k)} \cdot n^{O(1)}$ time algorithm for Edge Multiway Cut and a $2^{O(r \sqrt{k} \log k)} \cdot n^{O(1)}$ time algorithm for Vertex Multiway Cut, where $r$ is the number of terminals to be separated; 3. a $2^{O((r+\sqrt{k})\log^4 (rk))} \cdot n^{O(1)}$ time algorithm for Edge Multicut and a $2^{O((\sqrt{rk}+r) \log (rk))} \cdot n^{O(1)}$ time algorithm for Vertex Multicut, where $r$ is the number of terminal pairs to be separated; 4. a $2^{O(\sqrt{k} \log g \log^4 k)} \cdot n^{O(1)}$ time algorithm for Group Feedback Edge Set and a $2^{O(g \sqrt{k}\log(gk))} \cdot n^{O(1)}$ time algorithm for Group Feedback Vertex Set, where $g$ is the size of the group. 5. In addition, our approach also gives $n^{O(\sqrt{k})}$ time algorithms for all above problems with the exception of $n^{O(r+\sqrt{k})}$ time for Edge/Vertex Multicut and $(ng)^{O(\sqrt{k})}$ time for Group Feedback Edge/Vertex Set. We obtain our results by giving a new decomposition theorem on graphs of bounded genus, or more generally, an $h$-almost-embeddable graph for any fixed constant $h$. In particular we show the following. Let $G$ be an $h$-almost-embeddable graph for a constant $h$. Then for every $p\in\mathbb{N}$, there exist disjoint sets $Z_1,\dots,Z_p \subseteq V(G)$ such that for every $i \in \{1,\dots,p\}$ and every $Z'\subseteq Z_i$, the treewidth of $G/(Z_i\backslash Z')$ is $O(p+|Z'|)$. Here $G/(Z_i\backslash Z')$ is the graph obtained from $G$ by contracting edges with both endpoints in $Z_i \backslash Z'$.

cs.DS↗

arXiv2020

EPTAS for $k$-means Clustering of Affine Subspaces

We consider a generalization of the fundamental $k$-means clustering for data with incomplete or corrupted entries. When data objects are represented by points in $\mathbb{R}^d$, a data point is said to be incomplete when some of its entries are missing or unspecified. An incomplete data point with at most $\Delta$ unspecified entries corresponds to an axis-parallel affine subspace of dimension at most $\Delta$, called a $\Delta$-point. Thus we seek a partition of $n$ input $\Delta$-points into $k$ clusters minimizing the $k$-means objective. For $\Delta=0$, when all coordinates of each point are specified, this is the usual $k$-means clustering. We give an algorithm that finds an $(1+ \epsilon)$-approximate solution in time $f(k,\epsilon, \Delta) \cdot n^2 \cdot d$ for some function $f$ of $k,\epsilon$, and $\Delta$ only.

cs.DS↗

arXiv2020

Powers of paths in tournaments

In this short note we prove that every tournament contains the $k$-th power of a directed path of linear length. This improves upon recent results of Yuster and of Gir\~ao. We also give a complete solution for this problem when $k=2$, showing that there is always a square of a directed path of length $\lceil 2n/3 \rceil-1$, which is best possible.

math.CO↗

arXiv2020

A Polynomial Kernel for Line Graph Deletion

The line graph of a graph $G$ is the graph $L(G)$ whose vertex set is the edge set of $G$ and there is an edge between $e,f\in E(G)$ if $e$ and $f$ share an endpoint in $G$. A graph is called line graph if it is a line graph of some graph. We study the Line-Graph-Edge Deletion problem, which asks whether we can delete at most $k$ edges from the input graph $G$ such that the resulting graph is a line graph. More precisely, we give a polynomial kernel for Line-Graph-Edge Deletion with $\mathcal{O}(k^{5})$ vertices. This answers an open question posed by Falk H\"{u}ffner at Workshop on Kernels (WorKer) in 2013.

cs.DS↗

arXiv2020

Packing and covering balls in graphs excluding a minor

We prove that for every integer $t\ge 1$ there exists a constant $c_t$ such that for every $K_t$-minor-free graph $G$, and every set $S$ of balls in $G$, the minimum size of a set of vertices of $G$ intersecting all the balls of $S$ is at most $c_t$ times the maximum number of vertex-disjoint balls in $S$. This was conjectured by Chepoi, Estellon, and Vax\`es in 2007 in the special case of planar graphs and of balls having the same radius.

math.CO↗

arXiv2019

A Polynomial Time Algorithm for the $k$-Disjoint Shortest Paths Problem

The disjoint paths problem is a fundamental problem in algorithmic graph theory and combinatorial optimization. For a given graph $G$ and a set of $k$ pairs of terminals in $G$, it asks for the existence of $k$ vertex-disjoint paths connecting each pair of terminals. The proof of Robertson and Seymour [JCTB 1995] of the existence of an $n^3$ algorithm for any fixed $k$ is one of the highlights of their Graph Minors project. In this paper, we focus on the version of the problem where all the paths are required to be shortest paths. This problem, called the disjoint shortest paths problem, was introduced by Eilam-Tzoreff [DAM 1998] where she proved that the case $k = 2$ admits a polynomial time algorithm. This problem has received some attention lately, especially since the proof of the existence of a polynomial time algorithm in the directed case when $k = 2$ by B\'erczi and Kobayashi [ESA 2017]. However, the existence of a polynomial algorithm when $k = 3$ in the undirected version remained open since 1998. In this paper we show that for any fixed $k$, the disjoint shortest paths problem admits a polynomial time algorithm. In fact for any fixed $C$, the algorithm can be extended to treat the case where each path connecting the pair $(s,t)$ has length at most $d(s,t) + C$.

math.CO↗

arXiv2019

A Polynomial Kernel for Paw-Free Editing

For a fixed graph $H$, the $H$-free-editing problem asks whether we can modify a given graph $G$ by adding or deleting at most $k$ edges such that the resulting graph does not contain $H$ as an induced subgraph. The problem is known to be NP-complete for all fixed $H$ with at least $3$ vertices and it admits a $2^{O(k)}n^{O(1)}$ algorithm. Cai and Cai showed that the $H$-free-editing problem does not admit a polynomial kernel whenever $H$ or its complement is a path or a cycle with at least $4$ edges or a $3$-connected graph with at least $1$ edge missing. Their results suggest that if $H$ is not independent set or a clique, then $H$-free-editing admits polynomial kernels only for few small graphs $H$, unless $\textsf{coNP} \in \textsf{NP/poly}$. Therefore, resolving the kernelization of $H$-free-editing for small graphs $H$ plays a crucial role in obtaining a complete dichotomy for this problem. In this paper, we positively answer the question of compressibility for one of the last two unresolved graphs $H$ on $4$ vertices. Namely, we give the first polynomial kernel for paw-free editing with $O(k^{6})$vertices.

math.CO↗

arXiv2019

The directed 2-linkage problem with length constraints

The {\sc weak 2-linkage} problem for digraphs asks for a given digraph and vertices $s_1,s_2,t_1,t_2$ whether $D$ contains a pair of arc-disjoint paths $P_1,P_2$ such that $P_i$ is an $(s_i,t_i)$-path. This problem is NP-complete for general digraphs but polynomially solvable for acyclic digraphs \cite{fortuneTCS10}. Recently it was shown \cite{bercziESA17} that if $D$ is equipped with a weight function $w$ on the arcs which satisfies that all edges have positive weight, then there is a polynomial algorithm for the variant of the weak-2-linkage problem when both paths have to be shortest paths in $D$. In this paper we consider the unit weight case and prove that for every pair constants $k_1,k_2$, there is a polynomial algorithm which decides whether the input digraph $D$ has a pair of arc-disjoint paths $P_1,P_2$ such that $P_i$ is an $(s_i,t_i)$-path and the length of $P_i$ is no more than $d(s_i,t_i)+k_i$, for $i=1,2$, where $d(s_i,t_i)$ denotes the length of the shortest $(s_i,t_i)$-path. We prove that, unless the exponential time hypothesis (ETH) fails, there is no polynomial algorithm for deciding the existence of a solution $P_1,P_2$ to the {\sc weak 2-linkage} problem where each path $P_i$ has length at most $d(s_i,t_i)+ c\log^{1+\epsilon}{}n$ for some constant $c$. We also prove that the {\sc weak 2-linkage} problem remains NP-complete if we require one of the two paths to be a shortest path while the other path has no restriction on the length.

cs.CC↗