arXiv subjects

Hung P. Hoang

Publications and source records attributed to Hung P. Hoang.

18 recordsLinked to original sources

arXiv2026

Not All Degree Constraints Are Created Equal when Computing Spanning Trees

We study the computation of minimum spanning trees subject to local degree constraints. Recent work (ICALP 2026) established that three natural formalizations of this problem share the exact same parameterized complexity under standard structural graph parameters, including treewidth, pathwidth and clique-width. This applies to the cases where every vertex has a single target degree (Specified Degree MST), or a degree upper bound (Bounded Degree MST), or is equipped with a set of admissible degrees (Set of Degrees MST). In this paper, we investigate these problems under more restrictive parameterizations and reveal that their complexity landscapes fundamentally diverge on bounded-treedepth graphs. Specifically, we prove that the former two problems are fixed-parameter tractable when parameterized by the treedepth of the input graph. In sharp contrast, we show that Set of Degrees MST remains W[1]-hard parameterized by treedepth, even when combined with the feedback vertex number (i.e., deletion distance to treewidth $1$). Finally, we show that this divergence seems to be specific to treedepth: we exclude an analogous W[1]-hardness result for Set of Degrees MST w.r.t. the vertex cover number and also rule out fixed-parameter algorithms for the former two problems w.r.t. deletion distance to constant pathwidth.

cs.DS↗

arXiv2026

Parameterized Complexity of Efficient Sortation

A crucial challenge arising in the design of large-scale logistical networks is to optimize parcel sortation for routing. We study this problem under the recent graph-theoretic formalization of Van Dyk, Klause, Koenemann and Megow (IPCO 2024). The problem asks - given an input digraph D (the fulfillment network) together with a set of commodities represented as source-sink tuples - for a minimum-outdegree subgraph H of the transitive closure of D that contains a source-sink route for each of the commodities. Given the underlying motivation, we study two variants of the problem which differ in whether the routes for the commodities are assumed to be given, or can be chosen arbitrarily. We perform a thorough parameterized analysis of the complexity of both problems. Our results concentrate on three fundamental parameterizations of the problem: (1) When attempting to parameterize by the target outdegree of H, we show that the problems are paraNP-hard even in highly restricted cases; (2) When parameterizing by the number of commodities, we utilize Ramsey-type arguments and color-coding techniques to obtain parameterized algorithms for both problems; (3) When parameterizing by the structure of D, we establish fixed-parameter tractability for both problems w.r.t. treewidth, maximum degree and the maximum routing length. We combine this with lower bounds which show that omitting any of the three parameters results in paraNP-hardness.

cs.DS↗

arXiv2026

Fine-Grained Complexity of Computing Degree-Constrained Spanning Trees

We investigate the computation of minimum-cost spanning trees satisfying prescribed vertex degree constraints: Given a graph $G$ and a constraint function $D$, we ask for a (minimum-cost) spanning tree $T$ such that for each vertex $v$, $T$ achieves a degree specified by $D(v)$. Specifically, we consider three kinds of constraint functions ordered by their generality -- $D$ may either assign each vertex to a list of admissible degrees, an upper bound on the degrees, or a specific degree. Using a combination of novel techniques and state-of-the-art machinery, we obtain an almost-complete overview of the fine-grained complexity of these problems taking into account the most classical graph parameters of the input graph $G$. In particular, we present SETH-tight upper and lower bounds for these problems when parameterized by the pathwidth and cutwidth, an ETH-tight algorithm parameterized by the cliquewidth, and a nearly SETH-tight algorithm parameterized by treewidth. In order to obtain our upper bound for clique-width, we develop a novel technique of double representation through ``requirement shifting''. Using this technique, we also obtain an ETH-tight single-exponential XP algorithm for the Exact Leaf Spanning Tree problem parameterized by clique-width, which settles the final remaining open case for clique-width from the classical Cut and Count of Cygan et al. [FOCS 2011, TALG 2022]. This shows the versatility of our technique and its potential applicability to other problems as well. Additionally, in order to establish our lower and upper bounds we introduce a number of tools which may be of independent interest, including lazy coloring and ``asymptotic'' SETH-based reductions for structural parameters.

cs.DS↗

arXiv2026

On (In)approximability of MaxMin Independent Set Reconfiguration

In the Independent Set Reconfiguration problem under the Token Addition/Removal rule, given a graph $G$ and two independent sets $I$ and $J$ of $G$, we want to transform $I$ into $J$ by adding and removing vertices, such that all the sets throughout the process are independent sets. Its approximate version called MaxMin Independent Set Reconfiguration aims to maximise the minimum size of the independent sets in the process above. We study the (in)approximability of this problem for general graphs as well as restricted graph classes. Firstly, on general graphs, we obtain a polynomial-time $(n / \log n)$-factor approximation algorithm, complementing the $\mathsf{PSPACE}$-hardness of $n^{Ω(1)}$-factor approximation due to Hirahara and Ohsaka [STOC 2024, ICALP 2024] and the $\mathsf{NP}$-hardness of $n^{1-\varepsilon}$-factor approximation due to Ito, Demaine, Harvey, Papadimitriou, Sideri, Uehara, and Uno [TCS 2011]. Secondly, we present a polynomial-time approximation algorithm for degenerate graphs as well as $\mathsf{FPT}$-approximation schemes for bounded-treewidth graphs and $H$-minor-free graphs. Lastly, we extend the above inapproximability results to bounded-degree graphs, graphs of bandwidth $n^{\frac{1}{2}+Θ(1)}$, and bipartite graphs.

cs.DS↗

arXiv2026

On the PLS-Completeness of $k$-Opt Local Search for the Traveling Salesman Problem

The $k$-Opt algorithm is a local search algorithm for the traveling salesman problem. Starting with an initial tour, it iteratively replaces at most $k$ edges in the tour with the same number of edges to obtain a better tour. Krentel (FOCS 1989) showed that the traveling salesman problem with the $k$-Opt neighborhood is complete for the class PLS (polynomial time local search). However, his proof requires $k \gg 1000$ and has a substantial gap. We provide the first rigorous proof for the PLS-completeness and at the same time drastically lower the value of $k$ to $k \geq 15$, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). Our result holds for both the general and the metric traveling salesman problem.

cs.DS↗

arXiv2026

Splitting Sandwiches Unevenly via Unique Sink Orientations and Rainbow Arrangements

The famous Ham-Sandwich theorem states that any $d$ point sets in $\mathbb{R}^d$ can be simultaneously bisected by a single hyperplane. The $α$-Ham-Sandwich theorem gives a sufficient condition for the existence of biased cuts, i.e., hyperplanes that do not cut off half but some prescribed fraction of each point set. We give two new proofs for this theorem. The first proof is completely combinatorial and highlights a strong connection between the $α$-Ham-Sandwich theorem and Unique Sink Orientations of grids. The second proof uses point-hyperplane duality and the Poincaré-Miranda theorem and allows us to generalize the result to and beyond oriented matroids. For this we introduce a new concept of rainbow arrangements, generalizing colored pseudo-hyperplane arrangements. Along the way, we also show that the realizability problem for rainbow arrangements is $\exists \mathbb{R}$-complete, which also implies that the realizability problem for grid Unique Sink Orientations is $\exists \mathbb{R}$-complete.

math.CO↗

arXiv2026

A Parameterized-Complexity Framework for Finding Local Optima

Local search is a fundamental optimization technique that is both widely used in practice and deeply studied in theory, yet its computational complexity remains poorly understood. The traditional frameworks, PLS and the standard algorithm problem, introduced by Johnson, Papadimitriou, and Yannakakis (1988) fail to capture the methodology of local search algorithms: PLS is concerned with finding a local optimum and not with using local search, while the standard algorithm problem restricts each improvement step to follow a fixed pivoting rule. In this work, we introduce a novel formulation of local search which provides a middle ground between these models. In particular, the task is to output not only a local optimum but also a chain of local improvements leading to it. With this framework, we aim to capture the challenge in designing a good pivoting rule. Especially, when combined with the parameterized complexity paradigm, it enables both strong lower bounds and meaningful tractability results. Unlike previous works that combined parameterized complexity with local search, our framework targets the whole task of finding a local optimum and not only a single improvement step. Focusing on two representative meta-problems -- Subset Weight Optimization Problem with the $c$-swap neighborhood and Weighted Circuit with the flip neighborhood -- we establish fixed-parameter tractability results related to the number of distinct weights, while ruling out an analogous result when parameterized by the distance to the nearest optimum via a new type of reduction.

cs.CC↗

arXiv2025

Matrix Editing Meets Fair Clustering: Parameterized Algorithms and Complexity

We study the computational problem of computing a fair means clustering of discrete vectors, which admits an equivalent formulation as editing a colored matrix into one with few distinct color-balanced rows by changing at most $k$ values. While NP-hard in both the fairness-oblivious and the fair settings, the problem is well-known to admit a fixed-parameter algorithm in the former ``vanilla'' setting. As our first contribution, we exclude an analogous algorithm even for highly restricted fair means clustering instances. We then proceed to obtain a full complexity landscape of the problem, and establish tractability results which capture three means of circumventing our obtained lower bound: placing additional constraints on the problem instances, fixed-parameter approximation, or using an alternative parameterization targeting tree-like matrices.

cs.DS↗

arXiv2025

Flipping odd matchings in geometric and combinatorial settings

We study the problem of reconfiguring odd matchings, that is, matchings that cover all but a single vertex. Our reconfiguration operation is a so-called flip where the unmatched vertex of the first matching gets matched, while consequently another vertex becomes unmatched. We consider two distinct settings: the geometric setting, in which the vertices are points embedded in the plane and all occurring odd matchings are crossing-free, and a combinatorial setting, in which we consider odd matchings in general graphs. For the latter setting, we provide a complete polynomial time checkable characterization of graphs in which any two odd matchings can be reconfigured into each another. This complements the previously known result that the flip graph is always connected in the geometric setting [Aichholzer, Brötzner, Perz, and Schnider. Flips in odd matchings]. In the combinatorial setting, we prove that the diameter of the flip graph, if connected, is linear in the number of vertices. Furthermore, we establish that deciding whether there exists a flip sequence of length $k$ transforming one given matching into another is NP-complete in both the combinatorial and the geometric settings. To prove the latter, we introduce a framework that allows us to transform partial order types into general position with only polynomial overhead. Finally, we demonstrate that when parameterized by the flip distance $k$, the problem is fixed-parameter tractable (FPT) in the geometric setting when restricted to convex point sets.

cs.CG↗

arXiv2025

A near-complete resolution of the exponential-time complexity of k-opt for the traveling salesman problem

The $k$-opt algorithm is one of the simplest and most widely used heuristics for solving the traveling salesman problem. Starting from an arbitrary tour, the $k$-opt algorithm improves the current tour in each iteration by exchanging up to $k$ edges. The algorithm continues until no further improvement of this kind is possible. For a long time, it remained an open question how many iterations the $k$-opt algorithm might require for small values of $k$, assuming the use of an optimal pivot rule. In this paper, we resolve this question for the cases $k = 3$ and $k = 4$ by proving that in both these cases an exponential number of iterations may be needed even if an optimal pivot rule is used. Combined with a recent result from Heimann, Hoang, and Hougardy (ICALP 2024), this provides a complete answer for all $k \geq 3$ regarding the number of iterations the $k$-opt algorithm may require under an optimal pivot rule. In addition we establish an analogous exponential lower bound for the 2.5-opt algorithm, a variant that generalizes 2-opt and is a restricted version of 3-opt. All our results hold for both the general and the metric traveling salesman problem.

cs.DS↗

arXiv2024

Generating all invertible matrices by row operations

We show that all invertible $n \times n$ matrices over any finite field $\mathbb{F}_q$ can be generated in a Gray code fashion. More specifically, there exists a listing such that (1) each matrix appears exactly once, and (2) two consecutive matrices differ by adding or subtracting one row from a previous or subsequent row, or by multiplying or diving a row by the generator of the multiplicative group of $\mathbb{F}_q$. This even holds if the addition and subtraction of each row is allowed to some specific rows satisfying a certain mild condition. Moreover, we can prescribe the first and the last matrix if $n\ge 3$, or $n=2$ and $q>2$. In other words, the corresponding flip graph on all invertible $n \times n$ matrices over $\mathbb{F}_q$ is Hamilton connected if it is not a cycle. This solves yet another special case of Lovász conjecture on Hamiltonicity of vertex-transitive graphs.

math.CO↗

arXiv2024

The $k$-Opt algorithm for the Traveling Salesman Problem has exponential running time for $k \ge 5$

The $k$-Opt algorithm is a local search algorithm for the Traveling Salesman Problem. Starting with an initial tour, it iteratively replaces at most $k$ edges in the tour with the same number of edges to obtain a better tour. Krentel (FOCS 1989) showed that the Traveling Salesman Problem with the $k$-Opt neighborhood is complete for the class PLS (polynomial time local search) and that the $k$-Opt algorithm can have exponential running time for any pivot rule. However, his proof requires $k \gg 1000$ and has a substantial gap. We show the two properties above for a much smaller value of $k$, addressing an open question by Monien, Dumrauf, and Tscheuschner (ICALP 2010). In particular, we prove the PLS-completeness for $k \geq 17$ and the exponential running time for $k \geq 5$.

cs.DS↗

arXiv2024

On approximating the rank of graph divisors

Baker and Norine initiated the study of graph divisors as a graph-theoretic analogue of the Riemann-Roch theory for Riemann surfaces. One of the key concepts of graph divisor theory is the {\it rank} of a divisor on a graph. The importance of the rank is well illustrated by Baker's {\it Specialization lemma}, stating that the dimension of a linear system can only go up under specialization from curves to graphs, leading to a fruitful interaction between divisors on graphs and curves. Due to its decisive role, determining the rank is a central problem in graph divisor theory. Kiss and Tóthméresz reformulated the problem using chip-firing games, and showed that computing the rank of a divisor on a graph is NP-hard via reduction from the Minimum Feedback Arc Set problem. In this paper, we strengthen their result by establishing a connection between chip-firing games and the Minimum Target Set Selection problem. As a corollary, we show that the rank is difficult to approximate to within a factor of $O(2^{\log^{1-\varepsilon}n})$ for any $\varepsilon > 0$ unless $P=NP$. Furthermore, assuming the Planted Dense Subgraph Conjecture, the rank is difficult to approximate to within a factor of $O(n^{1/4-\varepsilon})$ for any $\varepsilon>0$.

math.CO↗

arXiv2024

Conflict-Free Coloring: Graphs of Bounded Clique Width and Intersection Graphs

A conflict-free coloring of a graph $G$ is a (partial) coloring of its vertices such that every vertex $u$ has a neighbor whose assigned color is unique in the neighborhood of $u$. There are two variants of this coloring, one defined using the open neighborhood and one using the closed neighborhood. For both variants, we study the problem of deciding whether the conflict-free coloring of a given graph $G$ is at most a given number $k$. In this work, we investigate the relation of clique-width and minimum number of colors needed (for both variants) and show that these parameters do not bound one another. Moreover, we consider specific graph classes, particularly graphs of bounded clique-width and types of intersection graphs, such as distance hereditary graphs, interval graphs and unit square and disk graphs. We also consider Kneser graphs and split graphs. We give (often tight) upper and lower bounds and determine the complexity of the decision problem on these graph classes, which improve some of the results from the literature. Particularly, we settle the number of colors needed for an interval graph to be conflict-free colored under the open neighborhood model, which was posed as an open problem.

cs.DM↗

arXiv2023

Drawings of Complete Multipartite Graphs Up to Triangle Flips

For a drawing of a labeled graph, the rotation of a vertex or crossing is the cyclic order of its incident edges, represented by the labels of their other endpoints. The extended rotation system (ERS) of the drawing is the collection of the rotations of all vertices and crossings. A drawing is simple if each pair of edges has at most one common point. Gioan's Theorem states that for any two simple drawings of the complete graph $K_n$ with the same crossing edge pairs, one drawing can be transformed into the other by a sequence of triangle flips (a.k.a. Reidemeister moves of Type 3). This operation refers to the act of moving one edge of a triangular cell formed by three pairwise crossing edges over the opposite crossing of the cell, via a local transformation. We investigate to what extent Gioan-type theorems can be obtained for wider classes of graphs. A necessary (but in general not sufficient) condition for two drawings of a graph to be transformable into each other by a sequence of triangle flips is that they have the same ERS. As our main result, we show that for the large class of complete multipartite graphs, this necessary condition is in fact also sufficient. We present two different proofs of this result, one of which is shorter, while the other one yields a polynomial time algorithm for which the number of needed triangle flips for graphs on $n$ vertices is bounded by $O(n^{16})$. The latter proof uses a Carathéodory-type theorem for simple drawings of complete multipartite graphs, which we believe to be of independent interest. Moreover, we show that our Gioan-type theorem for complete multipartite graphs is essentially tight in the sense that having the same ERS does not remain sufficient when removing or adding very few edges.

cs.CG↗

arXiv2022

Combinatorial generation via permutation languages. V. Acyclic orientations

In 1993, Savage, Squire, and West described an inductive construction for generating every acyclic orientation of a chordal graph exactly once, flipping one arc at a time. We provide two generalizations of this result. Firstly, we describe Gray codes for acyclic orientations of hypergraphs that satisfy a simple ordering condition, which generalizes the notion of perfect elimination order of graphs. This unifies the Savage-Squire-West construction with a recent algorithm for generating elimination trees of chordal graphs. Secondly, we consider quotients of lattices of acyclic orientations of chordal graphs, and we provide a Gray code for them, addressing a question raised by Pilaud. This also generalizes a recent algorithm for generating lattice congruences of the weak order on the symmetric group. Our algorithms are derived from the Hartung-Hoang-Mütze-Williams combinatorial generation framework, and they yield simple algorithms for computing Hamilton paths and cycles on large classes of polytopes, including chordal nestohedra and quotientopes. In particular, we derive an efficient implementation of the Savage-Squire-West construction. Along the way, we give an overview of old and recent results about the polyhedral and order-theoretic aspects of acyclic orientations of graphs and hypergraphs.

math.CO↗

arXiv2021

Assistance and Interdiction Problems on Interval Graphs

We introduce a novel framework of graph modifications specific to interval graphs. We study interdiction problems with respect to these graph modifications. Given a list of original intervals, each interval has a replacement interval such that either the replacement contains the original, or the original contains the replacement. The interdictor is allowed to replace up to $k$ original intervals with their replacements. Using this framework we also study the contrary of interdiction problems which we call assistance problems. We study these problems for the independence number, the clique number, shortest paths, and the scattering number. We obtain polynomial time algorithms for most of the studied problems. Via easy reductions, it follows that on interval graphs, the most vital nodes problem with respect to shortest path, independence number and Hamiltonicity can be solved in polynomial time.

cs.DS↗

arXiv2021

A Subexponential Algorithm for ARRIVAL

The ARRIVAL problem is to decide the fate of a train moving along the edges of a directed graph, according to a simple (deterministic) pseudorandom walk. The problem is in $NP \cap coNP$ but not known to be in $P$. The currently best algorithms have runtime $2^{Θ(n)}$ where $n$ is the number of vertices. This is not much better than just performing the pseudorandom walk. We develop a subexponential algorithm with runtime $2^{O(\sqrt{n}\log n)}$. We also give a polynomial-time algorithm if the graph is almost acyclic. Both results are derived from a new general approach to solve ARRIVAL instances.

cs.DS↗