arXiv subjects

Peter Gartland

Publications and source records attributed to Peter Gartland.

8 recordsLinked to original sources

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.

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arXiv2024

Tree Independence Number IV. Even-hole-free Graphs

We prove that the tree independence number of every even-hole-free graph is at most polylogarithmic in its number of vertices. More explicitly, we prove that there exists a constant c>0 such that for every integer n>1 every n-vertex even-hole-free graph has a tree decomposition where each bag has stability (independence) number at most c log^10 n. 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 even-hole-free.

math.CO↗

arXiv2024

Induced subgraphs and tree decompositions XV. Even-hole-free graphs with bounded clique number have logarithmic treewidth

We prove that for every integer $t\geq 1$ there exists an integer $c_t\geq 1$ such that every $n$-vertex even-hole-free graph with no clique of size $t$ has treewidth at most $c_t\log{n}$. This resolves a conjecture of Sintiari and Trotignon, who also proved that the logarithmic bound is asymptotically best possible. It follows that several \textsf{NP}-hard problems such as \textsc{Stable Set}, \textsc{Vertex Cover}, \textsc{Dominating Set} and \textsc{Coloring} admit polynomial-time algorithms on this class of graphs. As a consequence, for every positive integer $r$, $r$-{\sc Coloring} can be solved in polynomial time on even-hole-free graphs without any assumptions on clique size. As part of the proof, we show that there is an integer $d$ such that every even-hole-free graph has a balanced separator which is contained in the (closed) neighborhood of at most $d$ vertices. This is of independent interest; for instance, it implies the existence of efficient approximation algorithms for certain \textsf{NP}-hard problems while restricted to the class of all even-hole-free graphs.

math.CO↗

arXiv2023

On Induced Versions of Menger's Theorem on Sparse Graphs

Let $A$ and $B$ be sets of vertices in a graph $G$. Menger's theorem states that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k-1$ vertices. Let $Δ$ be the maximum degree of $G$. We show that there exists a function $f(Δ) = (Δ+1)^{Δ^2+1}$, so that for every positive integer $k$, either there exists a collection of $k$ vertex-disjoint and pairwise anticomplete paths between $A$ and $B$, or $A$ can be separated from $B$ by a set of at most $k \cdot f(Δ)$ vertices. We also show that the result can be generalized from bounded-degree graphs to graphs excluding a topological minor. On the negative side, we show that no such relation holds on graphs that have degeneracy 2 and arbitrarily large girth, even when $k = 2$. Similar results were obtained independently and concurrently by Hendrey, Norin, Steiner, and Turcotte [arXiv:2309.07905].

math.CO↗

arXiv2021

Finding large induced sparse subgraphs in $C_{>t}$-free graphs in quasipolynomial time

For an integer $t$, a graph $G$ is called {\em{$C_{>t}$-free}} if $G$ does not contain any induced cycle on more than~$t$ vertices. We prove the following statement: for every pair of integers $d$ and $t$ and a CMSO$_2$ statement~$ϕ$, there exists an algorithm that, given an $n$-vertex $C_{>t}$-free graph $G$ with weights on vertices, finds in time $n^{O(\log^4 n)}$ a maximum-weight vertex subset $S$ such that $G[S]$ has degeneracy at most $d$ and satisfies $ϕ$. The running time can be improved to $n^{O(\log^2 n)}$ assuming $G$ is $P_t$-free, that is, $G$ does not contain an induced path on $t$ vertices. This expands the recent results of the authors [to appear at FOCS 2020 and SOSA 2021] on the {\sc{Maximum Weight Independent Set}} problem on $P_t$-free graphs in two directions: by encompassing the more general setting of $C_{>t}$-free graphs, and by being applicable to a much wider variety of problems, such as {\sc{Maximum Weight Induced Forest}} or {\sc{Maximum Weight Induced Planar Graph}}.

cs.DS↗

arXiv2020

Dominated Minimal Separators are Tame (Nearly All Others are Feral)

A class ${\cal F}$ of graphs is called {\em tame} if there exists a constant $k$ so that every graph in ${\cal F}$ on $n$ vertices contains at most $O(n^k)$ minimal separators, {\em strongly-quasi-tame} if every graph in ${\cal F}$ on $n$ vertices contains at most $O(n^{k \log n})$ minimal separators, and {\em feral} if there exists a constant $c > 1$ so that ${\cal F}$ contains $n$-vertex graphs with at least $c^n$ minimal separators for arbitrarily large $n$. The classification of graph classes into tame or feral has numerous algorithmic consequences, and has recently received considerable attention. A key graph-theoretic object in the quest for such a classification is the notion of a $k$-{\em creature}. In a recent manuscript [Abrishami et al., Arxiv 2020] conjecture that every hereditary class ${\cal F}$ that excludes $k$-creatures for some fixed constant $k$ is tame. We give a counterexample to this conjecture and prove the weaker result that a hereditary class ${\cal F}$ is strongly quasi-tame if it excludes $k$-creatures for some fixed constant $k$ and additionally every minimal separator can be dominated by another fixed constant $k'$ number of vertices. The tools developed also lead to a number of additional results of independent interest. {\bf (i) We obtain a complete classification of all hereditary graph classes defined by a finite set of forbidden induced subgraphs into strongly quasi-tame or feral. This generalizes Milanič and Pivač [WG'19]. {\bf (ii)} We show that hereditary class that excludes $k$-creatures and additionally excludes all cycles of length at least $c$, for some constant $c$, are tame. This generalizes the result of [Chudnovsky et al., Arxiv 2019]. {\bf (iii)} We show that every hereditary class that excludes $k$-creatures and additionally excludes a complete graph on $c$ vertices for some fixed constant $c$ is tame.

cs.DM↗

arXiv2020

Independent Set on P$_k$-Free Graphs in Quasi-Polynomial Time

We present an algorithm that takes as input a graph $G$ with weights on the vertices, and computes a maximum weight independent set $S$ of $G$. If the input graph $G$ excludes a path $P_k$ on $k$ vertices as an induced subgraph, the algorithm runs in time $n^{O(k^2 \log^3 n)}$. Hence, for every fixed $k$ our algorithm runs in quasi-polynomial time. This resolves in the affirmative an open problem of [Thomassé, SODA'20 invited presentation]. Previous to this work, polynomial time algorithms were only known for $P_4$-free graphs [Corneil et al., DAM'81], $P_5$-free graphs [Lokshtanov et al., SODA'14], and $P_6$-free graphs [Grzesik et al., SODA'19]. For larger values of $t$, only $2^{O(\sqrt{kn\log n})}$ time algorithms [Bascó et al., Algorithmica'19] and quasi-polynomial time approximation schemes [Chudnovsky et al., SODA'20] were known. Thus, our work is the first to offer conclusive evidence that Independent Set on $P_k$-free graphs is not NP-complete for any integer $k$. Additionally we show that for every graph $H$, if there exists a quasi-polynomial time algorithm for Independent Set on $C$-free graphs for every connected component $C$ of $H$, then there also exists a quasi-polynomial time algorithm for {\sc Independent Set} on $H$-free graphs. This lifts our quasi-polynomial time algorithm to $T_k$-free graphs, where $T_k$ has one component that is a $P_k$, and $k-1$ components isomorphic to a fork (the unique $5$-vertex tree with a degree $3$ vertex).

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arXiv2018

A New Characterization of $\mathcal{V}$-Posets

In 2016, Hasebe and Tsujie gave a recursive characterization of the set of induced $N$-free and bowtie-free posets; Misanantenaina and Wagner studied these orders further, naming them "$\mathcal{V}$-posets". Here we offer a new characterization of $\mathcal{V}$-posets by introducing a property we refer to as autonomy. A poset $\cP$ is said to be autonomous if there exists a directed acyclic graph $D$ (with adjacency matrix $U$) whose transitive closure is $\cP$, with the property that any total ordering of the vertices of $D$ so that Gaussian elimination of $U^TU$ proceeds without row swaps is a linear extension of $\cP$. Autonomous posets arise from the theory of pressing sequences in graphs, a problem with origins in phylogenetics. The pressing sequences of a graph can be partitioned into families corresponding to posets; because of the interest in enumerating pressing sequences, we investigate when this partition has only one block, that is, when the pressing sequences are all linear extensions of a single autonomous poset. We also provide an efficient algorithm for recognition of autonomy using structural information and the forbidden subposet characterization, and we discuss a few open questions that arise in connection with these posets.

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