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Tony Huynh

Publications and source records attributed to Tony Huynh.

At least 55 records · Page 3Linked to original sources
arXiv2019

A tight Erdős-Pósa function for planar minors

Let $H$ be a planar graph. By a classical result of Robertson and Seymour, there is a function $f:\mathbb{N} \to \mathbb{R}$ such that for all $k \in \mathbb{N}$ and all graphs $G$, either $G$ contains $k$ vertex-disjoint subgraphs each containing $H$ as a minor, or there is a subset $X$ of at most $f(k)$ vertices such that $G-X$ has no $H$-minor. We prove that this remains true with $f(k) = c k \log k$ for some constant $c=c(H)$. This bound is best possible, up to the value of $c$, and improves upon a recent result of Chekuri and Chuzhoy [STOC 2013], who established this with $f(k) = c k \log^d k$ for some universal constant $d$. The proof is constructive and yields a polynomial-time $O(\log \mathsf{OPT})$-approximation algorithm for packing subgraphs containing an $H$-minor.

math.CO↗
arXiv2019

The matroid secretary problem for minor-closed classes and random matroids

We prove that for every proper minor-closed class $M$ of matroids representable over a prime field, there exists a constant-competitive matroid secretary algorithm for the matroids in $M$. This result relies on the extremely powerful matroid minor structure theory being developed by Geelen, Gerards and Whittle. We also note that for asymptotically almost all matroids, the matroid secretary algorithm that selects a random basis, ignoring weights, is $(2+o(1))$-competitive. In fact, assuming the conjecture that almost all matroids are paving, there is a $(1+o(1))$-competitive algorithm for almost all matroids.

math.CO↗
arXiv2019

The stable set problem in graphs with bounded genus and bounded odd cycle packing number

Consider the family of graphs without $ k $ node-disjoint odd cycles, where $ k $ is a constant. Determining the complexity of the stable set problem for such graphs $ G $ is a long-standing problem. We give a polynomial-time algorithm for the case that $ G $ can be further embedded in a (possibly non-orientable) surface of bounded genus. Moreover, we obtain polynomial-size extended formulations for the respective stable set polytopes. To this end, we show that $2$-sided odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed surface. This extends the fact that odd cycles satisfy the Erdős-Pósa property in graphs embedded in a fixed orientable surface (Kawarabayashi & Nakamoto, 2007). Eventually, our findings allow us to reduce the original problem to the problem of finding a minimum-cost non-negative integer circulation of a certain homology class, which turns out to be efficiently solvable in our case.

cs.DM↗
arXiv2019

A unified Erdős-Pósa theorem for constrained cycles

A doubly group-labeled graph is an oriented graph with its edges labeled by elements of the direct sum of two groups $Γ_1,Γ_2$. A cycle in a doubly group-labeled graph is $(Γ_1,Γ_2)$-non-zero if it is non-zero in both coordinates. Our main result is a generalization of the Flat Wall Theorem of Robertson and Seymour to doubly group-labeled graphs. As an application, we determine all canonical obstructions to the Erdős-Pósa property for $(Γ_1,Γ_2)$-non-zero cycles in doubly group-labeled graphs. The obstructions imply that the half-integral Erdős-Pósa property always holds for $(Γ_1,Γ_2)$-non-zero cycles. Moreover, our approach gives a unified framework for proving packing results for constrained cycles in graphs. For example, as immediate corollaries we recover the Erdős-Pósa property for cycles and $S$-cycles and the half-integral Erdős-Pósa property for odd cycles and odd $S$-cycles. Furthermore, we recover Reed's Escher-wall Theorem. We also prove many new packing results as immediate corollaries. For example, we show that the half-integral Erdős-Pósa property holds for cycles not homologous to zero, odd cycles not homologous to zero, and $S$-cycles not homologous to zero. Moreover, the (full) Erdős-Pósa property holds for $S_1$-$S_2$-cycles and cycles not homologous to zero on an orientable surface. Finally, we also describe the canonical obstructions to the Erdős-Pósa property for cycles not homologous to zero and for odd $S$-cycles.

math.CO↗
arXiv2019

Flip distances between graph orientations

Flip graphs are a ubiquitous class of graphs, which encode relations induced on a set of combinatorial objects by elementary, local changes. Skeletons of associahedra, for instance, are the graphs induced by quadrilateral flips in triangulations of a convex polygon. For some definition of a flip graph, a natural computational problem to consider is the flip distance: Given two objects, what is the minimum number of flips needed to transform one into the other? We consider flip graphs on orientations of simple graphs, where flips consist of reversing the direction of some edges. More precisely, we consider so-called $α$-orientations of a graph $G$, in which every vertex $v$ has a specified outdegree $α(v)$, and a flip consists of reversing all edges of a directed cycle. We prove that deciding whether the flip distance between two $α$-orientations of a planar graph $G$ is at most two is \NP-complete. This also holds in the special case of perfect matchings, where flips involve alternating cycles. This problem amounts to finding geodesics on the common base polytope of two partition matroids, or, alternatively, on an alcoved polytope. It therefore provides an interesting example of a flip distance question that is computationally intractable despite having a natural interpretation as a geodesic on a nicely structured combinatorial polytope. We also consider the dual question of the flip distance betwe en graph orientations in which every cycle has a specified number of forward edges, and a flip is the reversal of all edges in a minimal directed cut. In general, the problem remains hard. However, if we restrict to flips that only change sinks into sources, or vice-versa, then the problem can be solved in polynomial time. Here we exploit the fact that the flip graph is the cover graph of a distributive lattice. This generalizes a recent result from Zhang, Qian, and Zhang.

math.CO↗
arXiv2018

Extension Complexity of the Correlation Polytope

We prove that for every $n$-vertex graph $G$, the extension complexity of the correlation polytope of $G$ is $2^{O(\mathrm{tw}(G) + \log n)}$, where $\mathrm{tw}(G)$ is the treewidth of $G$. Our main result is that this bound is tight for graphs contained in minor-closed classes.

cs.DM↗
arXiv2018

A tight Erdős-Pósa function for wheel minors

Let $W_t$ denote the wheel on $t+1$ vertices. We prove that for every integer $t \geq 3$ there is a constant $c=c(t)$ such that for every integer $k\geq 1$ and every graph $G$, either $G$ has $k$ vertex-disjoint subgraphs each containing $W_t$ as minor, or there is a subset $X$ of at most $c k \log k$ vertices such that $G-X$ has no $W_t$ minor. This is best possible, up to the value of $c$. We conjecture that the result remains true more generally if we replace $W_t$ with any fixed planar graph $H$.

cs.DM↗
arXiv2018

Strengthening Convex Relaxations of 0/1-Sets Using Boolean Formulas

In convex integer programming, various procedures have been developed to strengthen convex relaxations of sets of integer points. On the one hand, there exist several general-purpose methods that strengthen relaxations without specific knowledge of the set $ S $, such as popular linear programming or semi-definite programming hierarchies. On the other hand, various methods have been designed for obtaining strengthened relaxations for very specific sets that arise in combinatorial optimization. We propose a new efficient method that interpolates between these two approaches. Our procedure strengthens any convex set $ Q \subseteq \mathbb{R}^n $ containing a set $ S \subseteq \{0,1\}^n $ by exploiting certain additional information about $ S $. Namely, the required extra information will be in the form of a Boolean formula $ ϕ$ defining the target set $ S $. The aim of this work is to analyze various aspects regarding the strength of our procedure. As one result, interpreting an iterated application of our procedure as a hierarchy, our findings simplify, improve, and extend previous results by Bienstock and Zuckerberg on covering problems.

math.CO↗
arXiv2018

Intertwining connectivities in representable matroids

Let $M$ be a representable matroid, and $Q, R, S, T$ subsets of the ground set. We prove that, if $M$ is sufficiently large, then there is an element $e$ such that deleting or contracting $e$ preserves both the $Q$-$R$ and the $S$-$T$ connectivities. For matroids representable over a finite field we prove a stronger result: we show that we can remove $e$ such that both a connectivity and a minor of $M$ are preserved.

math.CO↗
arXiv2017

Characterizing Polytopes Contained in the $0/1$-Cube with Bounded Chvátal-Gomory Rank

Let $S \subseteq \{0,1\}^n$ and $R$ be any polytope contained in $[0,1]^n$ with $R \cap \{0,1\}^n = S$. We prove that $R$ has bounded Chvátal-Gomory rank (CG-rank) provided that $S$ has bounded notch and bounded gap, where the notch is the minimum integer $p$ such that all $p$-dimensional faces of the $0/1$-cube have a nonempty intersection with $S$, and the gap is a measure of the size of the facet coefficients of $\mathsf{conv}(S)$. Let $H[\bar{S}]$ denote the subgraph of the $n$-cube induced by the vertices not in $S$. We prove that if $H[\bar{S}]$ does not contain a subdivision of a large complete graph, then both the notch and the gap are bounded. By our main result, this implies that the CG-rank of $R$ is bounded as a function of the treewidth of $H[\bar{S}]$. We also prove that if $S$ has notch $3$, then the CG-rank of $R$ is always bounded. Both results generalize a recent theorem of Cornuéjols and Lee, who proved that the CG-rank is bounded by a constant if the treewidth of $H[\bar{S}]$ is at most $2$.

math.OC↗
arXiv2017

Extension complexity of stable set polytopes of bipartite graphs

The extension complexity $\mathsf{xc}(P)$ of a polytope $P$ is the minimum number of facets of a polytope that affinely projects to $P$. Let $G$ be a bipartite graph with $n$ vertices, $m$ edges, and no isolated vertices. Let $\mathsf{STAB}(G)$ be the convex hull of the stable sets of $G$. It is easy to see that $n \leqslant \mathsf{xc} (\mathsf{STAB}(G)) \leqslant n+m$. We improve both of these bounds. For the upper bound, we show that $\mathsf{xc} (\mathsf{STAB}(G))$ is $O(\frac{n^2}{\log n})$, which is an improvement when $G$ has quadratically many edges. For the lower bound, we prove that $\mathsf{xc} (\mathsf{STAB}(G))$ is $Ω(n \log n)$ when $G$ is the incidence graph of a finite projective plane. We also provide examples of $3$-regular bipartite graphs $G$ such that the edge vs stable set matrix of $G$ has a fooling set of size $|E(G)|$.

cs.DM↗
arXiv2017

Even-cycle decompositions of graphs with no odd-$K_4$-minor

An even-cycle decomposition of a graph G is a partition of E(G) into cycles of even length. Evidently, every Eulerian bipartite graph has an even-cycle decomposition. Seymour (1981) proved that every 2-connected loopless Eulerian planar graph with an even number of edges also admits an even-cycle decomposition. Later, Zhang (1994) generalized this to graphs with no $K_5$-minor. Our main theorem gives sufficient conditions for the existence of even-cycle decompositions of graphs in the absence of odd minors. Namely, we prove that every 2-connected loopless Eulerian odd-$K_4$-minor-free graph with an even number of edges has an even-cycle decomposition. This is best possible in the sense that `odd-$K_4$-minor-free' cannot be replaced with `odd-$K_5$-minor-free.' The main technical ingredient is a structural characterization of the class of odd-$K_4$-minor-free graphs, which is due to Lovász, Seymour, Schrijver, and Truemper.

math.CO↗
arXiv2016

Tree-chromatic number is not equal to path-chromatic number

For a graph $G$ and a tree-decomposition $(T, \mathcal{B})$ of $G$, the chromatic number of $(T, \mathcal{B})$ is the maximum of $χ(G[B])$, taken over all bags $B \in \mathcal{B}$. The tree-chromatic number of $G$ is the minimum chromatic number of all tree-decompositions $(T, \mathcal{B})$ of $G$. The path-chromatic number of $G$ is defined analogously. In this paper, we introduce an operation that always increases the path-chromatic number of a graph. As an easy corollary of our construction, we obtain an infinite family of graphs whose path-chromatic number and tree-chromatic number are different. This settles a question of Seymour. Our results also imply that the path-chromatic numbers of the Mycielski graphs are unbounded.

math.CO↗
arXiv2016

The excluded minors for isometric realizability in the plane

Let $G$ be a graph and $p \in [1, \infty]$. The parameter $f_p(G)$ is the least integer $k$ such that for all $m$ and all vectors $(r_v)_{v \in V(G)} \subseteq \mathbb{R}^m$, there exist vectors $(q_v)_{v \in V(G)} \subseteq \mathbb{R}^k$ satisfying $$\|r_v-r_w\|_p=\|q_v-q_w\|_p, \ \text{ for all }\ vw\in E(G).$$ It is easy to check that $f_p(G)$ is always finite and that it is minor monotone. By the graph minor theorem of Robertson and Seymour, there are a finite number of excluded minors for the property $f_p(G) \leq k$. In this paper, we determine the complete set of excluded minors for $f_\infty(G) \leq 2$. The two excluded minors are the wheel on $5$ vertices and the graph obtained by gluing two copies of $K_4$ along an edge and then deleting that edge. We also show that the same two graphs are the complete set of excluded minors for $f_1(G) \leq 2$. In addition, we give a family of examples that show that $f_\infty$ is unbounded on the class of planar graphs and $f_\infty$ is not bounded as a function of tree-width.

math.MG↗
arXiv2016

Explicit bounds for graph minors

Let $Σ$ be a surface with boundary $b(Σ)$, $\mathcal{L}$ be a collection of $k$ disjoint $b(Σ)$-paths in $Σ$, and $P$ be a non-separating $b(Σ)$-path in $Σ$. We prove that there is a homeomorphism $ϕ: Σ\to Σ$ that fixes each point of $b(Σ)$ and such that $ϕ(\mathcal{L})$ meets $P$ at most $2k$ times. With this theorem, we derive explicit constants in the graph minor algorithms of Robertson and Seymour. We reprove a result concerning redundant vertices for graphs on surfaces, but with explicit bounds. That is, we prove that there exists a computable integer $t:=t(Σ,k)$ such that if $v$ is a '$t$-protected' vertex in a surface $Σ$, then $v$ is redundant with respect to any $k$-linkage.

math.CO↗
arXiv2015

Strongly even-cycle decomposable graphs

A graph is strongly even-cycle decomposable if the edge set of every subdivision with an even number of edges can be partitioned into cycles of even length. We prove that several fundamental composition operations that preserve the property of being Eulerian also yield strongly even-cycle decomposable graphs. As an easy application of our theorems, we give an exact characterization of the set of strongly even-cycle decomposable cographs.

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