Thibault Lechien, Jorik Jooken, Patrick De Causmaecker
Predicting and comparing algorithm performance on graph instances is challenging for multiple reasons. First, there is usually no standard set of instances to benchmark performance. Second, using existing graph generators results in a restricted spectrum of difficulty and the resulting graphs are usually not diverse enough to draw sound conclusions. That is why recent work proposes a new methodology to generate a diverse set of instances by using an evolutionary algorithm. We can then analyze the resulting graphs and get key insights into which attributes are most related to algorithm performance. We can also fill observed gaps in the instance space in order to generate graphs with previously unseen combinations of features. This methodology is applied to the instance space of the Hamiltonian completion problem using two different solvers, namely the Concorde TSP Solver and a multi-start local search algorithm.
Jorik Jooken, Pieter Leyman, Patrick De Causmaecker
This paper proposes a local search algorithm for a specific combinatorial optimisation problem in graph theory: the Hamiltonian Completion Problem (HCP) on undirected graphs. In this problem, the objective is to add as few edges as possible to a given undirected graph in order to obtain a Hamiltonian graph. This problem has mainly been studied in the context of various specific kinds of undirected graphs (e.g. trees, unicyclic graphs and series-parallel graphs). The proposed algorithm, however, concentrates on solving HCP for general undirected graphs. It can be considered to belong to the category of matheuristics, because it integrates an exact linear time solution for trees into a local search algorithm for general graphs. This integration makes use of the close relation between HCP and the minimum path partition problem, which makes the algorithm equally useful for solving the latter problem. Furthermore, a benchmark set of problem instances is constructed for demonstrating the quality of the proposed algorithm. A comparison with state-of-the-art solvers indicates that the proposed algorithm is able to achieve high-quality results.
Marien Abreu, Jan Goedgebeur, Jorik Jooken, Federico Romaniello, Tibo Van den Eede
A graph is pseudo 2-factor isomorphic if all of its 2-factors have the same parity of number of cycles. Abreu et al. [J. Comb. Theory, Ser. B. 98 (2008) 432--442] conjectured that $K_{3,3}$, the Heawood graph and the Pappus graph are the only essentially 4-edge-connected pseudo 2-factor isomorphic cubic bipartite graphs. This conjecture was disproved by Goedgebeur [Discr. Appl. Math. 193 (2015) 57--60] who constructed a counterexample $\mathcal{G}$ (of girth 6) on 30 vertices. Using a computer search, he also showed that this is the only counterexample up to at least 40 vertices and that there are no counterexamples of girth greater than 6 up to at least 48 vertices. In this manuscript, we show that the Gray graph -- which has 54 vertices and girth 8 -- is also a counterexample to the pseudo 2-factor isomorphic graph conjecture. Next to the graph $\mathcal{G}$, this is the only other known counterexample. Using a computer search, we show that there are no smaller counterexamples of girth 8 and show that there are no other counterexamples up to at least 42 vertices of any girth. Moreover, we also verified that there are no further counterexamples among the known censuses of symmetrical graphs. Recall that a graph is 2-factor Hamiltonian if all of its 2-factors are Hamiltonian cycles. As a by-product of the computer searches performed for this paper, we have verified that the $2$-factor Hamiltonian conjecture of Funk et al. [J. Comb. Theory, Ser. B. 87(1) (2003) 138--144], which is still open, holds for cubic bipartite graphs of girth at least 8 up to 52 vertices, and up to 42 vertices for any girth.
Yiao Ju, Jorik Jooken, Jan Goedgebeur, Shenwei Huang
In this paper, we are interested in $4$-colouring algorithms for graphs that do not contain an induced path on $6$ vertices nor an induced bull, i.e., the graph with vertex set $\{v_1,v_2,v_3,v_4,v_5\}$ and edge set $\{v_1v_2,v_2v_3,v_3v_4,v_2v_5,v_3v_5\}$. Such graphs are referred to as $(P_6,\text{bull})$-free graphs. A graph $G$ is \emph{$k$-vertex-critical} if $χ(G)=k$, and every proper induced subgraph $H$ of $G$ has $χ(H)<k$. In the current paper, we investigate the structure of $5$-vertex-critical $(P_6,\text{bull})$-free graphs and show that there are only finitely many such graphs, thereby answering a question of Maffray and Pastor. A direct corollary of this is that there exists a polynomial-time algorithm to decide if a $(P_6,\text{bull})$-free graph is $4$-colourable such that this algorithm can also provide a certificate that can be verified in polynomial time and serves as a proof of 4-colourability or non-4-colourability.
Jorik Jooken, Pieter Leyman, Tony Wauters, Patrick De Causmaecker
In this article we propose a heuristic algorithm to explore search space trees associated with instances of combinatorial optimization problems. The algorithm is based on Monte Carlo tree search, a popular algorithm in game playing that is used to explore game trees and represents the state-of-the-art algorithm for a number of games. Several enhancements to Monte Carlo tree search are proposed that make the algorithm more suitable in a combinatorial optimization context. These enhancements exploit the combinatorial structure of the problem and aim to efficiently explore the search space tree by pruning subtrees, using a heuristic simulation policy, reducing the domains of variables by eliminating dominated value assignments and using a beam width. The algorithm was implemented with its components specifically tailored to two combinatorial optimization problems: the quay crane scheduling problem with non-crossing constraints and the 0-1 knapsack problem. For the first problem our algorithm surpasses the state-of-the-art results and several new best solutions are found for a benchmark set of instances. For the second problem our algorithm typically produces near-optimal solutions that are slightly worse than the state-of-the-art results, but it needs only a small fraction of the time to do so. These results indicate that the algorithm is competitive with the state-of-the-art for two entirely different combinatorial optimization problems.
Wen Xia, Jorik Jooken, Jan Goedgebeur, Shenwei Huang
Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. A graph $G$ is $k$-vertex-critical if every proper induced subgraph of $G$ has chromatic number less than $k$, but $G$ has chromatic number $k$. The study of $k$-vertex-critical graphs for specific graph classes is an important topic in algorithmic graph theory because if the number of such graphs that are in a given hereditary graph class is finite, then there exists a polynomial-time certifying algorithm to decide the $k$-colorability of a graph in the class. In this paper, we show that: (1) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,K_{1,4}+P_1)$-free graphs; (2) for $s \ge 1$, there are finitely many 5-vertex-critical $(P_5,K_{1,s}+P_1)$-free graphs; (3) for $k \ge 1$, there are finitely many $k$-vertex-critical $(P_5,\overline{K_3+2P_1})$-free graphs. Moreover, we characterize all $5$-vertex-critical $(P_5,H)$-free graphs where $H \in \{K_{1,3}+P_1,K_{1,4}+P_1,\overline{K_3+2P_1}\}$ using an exhaustive graph generation algorithm.
Geoffrey Exoo, Jan Goedgebeur, Jorik Jooken, Louis Stubbe, Tibo Van den Eede
The cage problem concerns finding $(k,g)$-graphs, which are $k$-regular graphs with girth $g$, of the smallest possible number of vertices. The central goal is to determine $n(k,g)$, the minimum order of such a graph, and to identify corresponding extremal graphs. In this paper, we study the cage problem and several of its variants from a computational perspective. Four complementary graph generation algorithms are developed based on exhaustive generation of lifts, a tabu search heuristic, a hill climbing heuristic and excision techniques. Using these methods, we establish new upper bounds for eleven cases of the classical cage problem: $n(3,16) \leq 936$, $n(3,17) \leq 2048$, $n(4,9) \leq 270$, $n(4,10) \leq 320$, $n(4,11) \leq 713$, $n(5,9) \leq 1116$, $n(6,11) \leq 7783$, $n(8,7) \leq 774$, $n(10,7) \leq 1608$, $n(12,7) \leq 2890$ and $n(14,7) \leq 4716$. Notably, our results improve upon several of the best-known bounds, some of which have stood unchanged for 22 years. Moreover, the improvement for $n(4,10)$, from the longstanding upper bound of 384 down to 320, is surprising and constitutes a substantial improvement. While the main focus is on the cage problem, we also adapted our algorithms for variants of the cage problem that received attention in the literature. For these variants, additional improvements are obtained, further narrowing the gaps between known lower and upper bounds.
Yidong Zhou, Jorik Jooken, Baoyuan Shan, Jan Goedgebeur, Shenwei Huang
A graph $G$ is $k$-vertex-critical if $χ(G)=k$, but $χ(G')<k$ for every proper induced subgraph $G'$ of $G$. For a family of graphs $\mathcal{F}$, $G$ is $\mathcal{F}$-free if no graph $F \in \mathcal{F}$ is an induced subgraph of $G$. We show that there are exactly three 4-vertex-critical $\{P_7,C_3\}$-free graphs containing an induced $C_7$, thereby settling the first of the two cases of a conjecture by Goedgebeur and Schaudt [J.~Graph Theory, 87:188--207, 2018]. Moreover, we show that all $\{P_5+P_1,C_3\}$-free graphs are $3$-colorable and by combining our result with known results from the literature, we completely characterize the maximum chromatic number of $\{F,C_3\}$-free graphs if $F$ is a six-vertex induced subgraph of $P_7$. Finally, we construct an infinite family of $4$-vertex-critical $\{4K_2,C_3\}$-free graphs. These graphs are also $\{P_{11},C_3\}$-free and this is the first value of $t$ for which an infinite family of $4$-vertex-critical $\{P_{t},C_3\}$-free graphs is known.
Stijn Cambie, Jan Goedgebeur, Jorik Jooken, Tibo Van den Eede
For integers $k,g,d$, a $(k;g,d)$-cage (or simply girth-diameter cage) is a smallest $k$-regular graph of girth $g$ and diameter $d$ (if it exists). The order of a $(k;g,d)$-cage is denoted by $n(k;g,d)$. We determine asymptotic lower and upper bounds for the ratio between the order and the diameter of girth-diameter cages as the diameter goes to infinity. We also prove that this ratio can be computed in constant time for fixed $k$ and $g$. We theoretically determine the exact values $n(3;g,d)$, and count the number of corresponding girth-diameter cages, for $g \in \{4,5\}$. Moreover, we design and implement an exhaustive graph generation algorithm and use it to determine the exact order of several open cases and obtain -- often exhaustive -- sets of the corresponding girth-diameter cages. The largest case we generated and settled with our algorithm is a $(3;7,35)$-cage of order 136.
Jorik Jooken
Computers and algorithms play an ever-increasing role in obtaining new results in graph theory. In this survey, we present a broad range of techniques used in computer-assisted graph theory, including the exhaustive generation of all pairwise non-isomorphic graphs within a given class, the use of searchable databases containing graphs and invariants as well as other established and emerging algorithmic paradigms. We cover approaches based on mixed integer linear programming, semidefinite programming, dynamic programming, SAT solving, metaheuristics and machine learning. The techniques are illustrated with numerous detailed results covering several important subareas of graph theory such as extremal graph theory, graph coloring, structural graph theory, spectral graph theory, regular graphs, topological graph theory, special sets in graphs, algebraic graph theory and chemical graph theory. We also present some smaller new results that demonstrate how readily a computer-assisted graph theory approach can be applied once the appropriate tools have been developed.
Jan Goedgebeur, Jorik Jooken, Gwenaël Joret, Tibo Van den Eede
We present a new algorithm for improving lower bounds on $ex(n;\{C_3,C_4\})$, the maximum size (number of edges) of an $n$-vertex graph of girth at least 5. The core of our algorithm is a variant of a hill-climbing heuristic introduced by Exoo, McKay, Myrvold and Nadon (2011) to find small cages. Our algorithm considers a range of values of $n$ in multiple passes. In each pass, the hill-climbing heuristic for a specific value of $n$ is initialized with a few graphs obtained by modifying near-extremal graphs previously found for neighboring values of $n$, allowing to `propagate' good patterns that were found. Focusing on the range $n\in \{74,75, \dots, 198\}$, which is currently beyond the scope of exact methods, our approach yields improvements on existing lower bounds for $ex(n;\{C_3,C_4\})$ for all $n$ in the range, except for two values of $n$ ($n=96,97$).
Tala Eagling-Vose, Jorik Jooken, Felicia Lucke, Barnaby Martin, Daniël Paulusma
We consider Colouring on graphs that are $H$-subgraph-free for some fixed graph $H$, which are graphs that do not contain $H$ as a subgraph. To classify the complexity of Colouring on $H$-subgraph-free graphs for connected $H$, it remains to consider when $H$ is a tree of maximum degree $4$ with exactly one vertex of degree $4$, or a tree of maximum degree $3$ with at least two vertices of degree $3$. We let $H$ be a so-called subdivided ``H''-graph, which is either a subdivided $\mathbb{H}_0$: a tree of maximum degree $4$ that is a star, or a subdivided $\mathbb{H}_1$: a tree of maximum degree $3$ with exactly two vertices of degree $3$. We develop new decomposition theorems resulting in polynomial-time algorithms, and in combination with known results, fully classify all cases $\mathbb{H}_0$ and $\mathbb{H}_1$. To illustrate the wider applicability of our techniques, we also employ them to obtain similar new polynomial-time results for two other classic graph problems: Stable Cut and, in part, Feedback Vertex Set.
Jorik Jooken, Pieter Leyman, Patrick De Causmaecker
Decades of research on the 0-1 knapsack problem led to very efficient algorithms that are able to quickly solve large problem instances to optimality. This prompted researchers to also investigate whether relatively small problem instances exist that are hard for existing solvers and investigate which features characterize their hardness. Previously the authors proposed a new class of hard 0-1 knapsack problem instances and demonstrated that the properties of so-called inclusionwise maximal solutions (IMSs) can be important hardness indicators for this class. In the current paper, we formulate several new computationally challenging problems related to the IMSs of arbitrary 0-1 knapsack problem instances. Based on generalizations of previous work and new structural results about IMSs, we formulate polynomial and pseudopolynomial time algorithms for solving these problems. From this we derive a set of 14 computationally expensive features, which we calculate for two large datasets on a supercomputer in approximately 540 CPU-hours. We show that the proposed features contain important information related to the empirical hardness of a problem instance that was missing in earlier features from the literature by training machine learning models that can accurately predict the empirical hardness of a wide variety of 0-1 knapsack problem instances. Using the instance space analysis methodology, we also show that hard 0-1 knapsack problem instances are clustered together around a relatively dense region of the instance space and several features behave differently in the easy and hard parts of the instance space.
Jan Goedgebeur, Jorik Jooken, On-Hei Solomon Lo, Ben Seamone, Carol T. Zamfirescu
We fully disprove a conjecture of Haythorpe on the minimum number of hamiltonian cycles in regular hamiltonian graphs, thereby extending a result of Zamfirescu, as well as correct and complement Haythorpe's computational enumerative results from [Experim. Math. 27 (2018) 426-430]. Thereafter, we use the Lovász Local Lemma to extend Thomassen's independent dominating set method. Regarding the limitations of this method, we answer a question of Haxell, Seamone, and Verstraete, and settle the first open case of a problem of Thomassen. Motivated by an observation of Aldred and Thomassen, we prove that for every $κ\in \{ 2, 3 \}$ and any positive integer $k$, there are infinitely many non-regular graphs of connectivity $κ$ containing exactly one hamiltonian cycle and in which every vertex has degree $3$ or $2k$.
Louis Carpentier, Jorik Jooken, Jan Goedgebeur
We propose a new methodology to develop heuristic algorithms using tree decompositions. Traditionally, such algorithms construct an optimal solution of the given problem instance through a dynamic programming approach. We modify this procedure by introducing a parameter $W$ that dictates the number of dynamic programming states to consider. We drop the exactness guarantee in favour of a shorter running time. However, if $W$ is large enough such that all valid states are considered, our heuristic algorithm proves optimality of the constructed solution. In particular, we implement a heuristic algorithm for the Maximum Happy Vertices problem using this approach. Our algorithm more efficiently constructs optimal solutions compared to the exact algorithm for graphs of bounded treewidth. Furthermore, our algorithm constructs higher quality solutions than state-of-the-art heuristic algorithms Greedy-MHV and Growth-MHV for instances of which at least 40\% of the vertices are initially coloured, at the cost of a larger running time.
Jorik Jooken
Let $\partial_H(u)$ be the set of edges incident with a vertex $u$ in the graph $H$. We say that a graph $G$ is $H$-colorable if there exist total functions $f : E(G) \rightarrow E(H)$ and $g : V(G) \rightarrow V(H)$ such that $f$ is a proper edge-coloring of $G$ and for each vertex $u \in V(G)$ we have $f(\partial_G(u))=\partial_H(g(u))$. Let $\bar{X}$ be the graph obtained by adding three parallel edges between two degree one vertices of the graph $K_{1,4}$. Let $\hat{A}$ be the graph obtained by adding two pendant edges to two different vertices of a triangle and then adding two edges between the degree two vertex and the two adjacent degree three vertices. Malnegro and Ozeki [Discrete Math. 347(3):113844 (2024)] asked whether every 4-regular graph with an even number of vertices and an even cycle decomposition of size 3 admits an $\bar{X}$-coloring or an $\hat{A}$-coloring and whether every 2-connected planar 4-regular graph with an even number of vertices admits such a coloring. Additionally, they conjectured that for every 2-edge-connected simple cubic graph $G$ with an even number of edges, the line graph $L(G)$ is $\bar{X}$-colorable. In this short note, we discuss two algorithms for deciding whether a graph $G$ is $H$-colorable. We give a negative answer to the two questions and disprove the conjecture by finding suitable graphs, as verified by two independent algorithms.
Jorik Jooken, Denys Lohvynov
A $k$-regular graph of girth $g$ is called vertex-girth-regular if every vertex is contained in the same number of cycles of length $g$. For integers $n, k, g$ and $λ$, we denote such a graph on $n$ vertices in which every vertex lies on exactly $λ$ cycles of length $g$ by a $\text{vgr}(n,k,g,λ)$-graph. It is well-known that any vertex-girth-regular graph satisfies $λ\le \frac{k(k-1)^{\left\lfloor \frac{g}{2} \right\rfloor}}{2}$. Graphs for which $λ$ is close to this bound are of particular interest in connection with the cage problem, since requiring many girth cycles through every vertex is a natural way to isolate highly structured candidates for small regular graphs of prescribed girth. In this paper, we prove that for every $k\ge 3$ and every integer $0< \varepsilon \leq \frac{k-1}{2}$, there does not exist a $\text{vgr}(n,k,5,\frac{k(k-1)^2}{2}-\varepsilon)$-graph. Previous non-existence results had already settled all odd girths at least $7$ and very recently also girth $3$, leaving girth $5$ as the only girth for which no non-trivial non-existence result was known. Thus, our result resolves the final remaining case and completes the picture for odd girths.
Wen Xia, Jorik Jooken, Jan Goedgebeur, Shenwei Huang
Given two graphs $H_1$ and $H_2$, a graph is $(H_1,H_2)$-free if it contains no induced subgraph isomorphic to $H_1$ nor $H_2$. Let $P_t$ be the path on $t$ vertices. A dart is the graph obtained from a diamond by adding a new vertex and making it adjacent to exactly one vertex with degree 3 in the diamond. In this paper, we show that there are finitely many $k$-vertex-critical $(P_5,dart)$-free graphs for $k \ge 1$ To prove these results, we use induction on $k$ and perform a careful structural analysis via Strong Perfect Graph Theorem combined with the pigeonhole principle based on the properties of vertex-critical graphs. Moreover, for $k \in \{5, 6, 7\}$ we characterize all $k$-vertex-critical $(P_5,dart)$-free graphs using a computer generation algorithm. Our results imply the existence of a polynomial-time certifying algorithm to decide the $k$-colorability of $(P_5,dart)$-free graphs for $k \ge 1$ where the certificate is either a $k$-coloring or a $(k+1)$-vertex-critical induced subgraph.
Stijn Cambie, Jorik Jooken
The occupancy fraction of a graph is a (normalized) measure on the size of independent sets under the hard-core model, depending on a variable (fugacity) $λ.$ We present a criterion for finding the graph with minimum occupancy fraction among graphs with a fixed order, and disprove five conjectures on the extremes of the occupancy fraction and (normalized) independence polynomial for certain graph classes of regular graphs with a given girth.
Robert Jajcay, Jorik Jooken, István Porupsánszki
A vertex-girth-regular $vgr(v,k,g,λ)$-graph is a $k$-regular graph of girth $g$ and order $v$ in which every vertex belongs to exactly $λ$ cycles of length $g$. While all vertex-transitive graphs are necessarily vertex-girth-regular, the majority of vertex-girth-regular graphs are not vertex-transitive. Similarly, while many of the smallest $k$-regular graphs of girth $g$, the so-called $(k,g)$-cages, are vertex-girth-regular, infinitely many vertex-girth-regular graphs of degree $k$ and girth $g$ exist for many pairs $k,g$. Due to these connections, the study of vertex-girth-regular graphs promises insights into the relations between the classes of extremal, highly symmetric, and locally regular graphs of given degree and girth. This paper lays the foundation to such study by investigating the fundamental properties of $vgr(v,k,g,λ)$-graphs, specifically the relations necessarily satisfied by the parameters $v,k,g$ and $λ$ to admit the existence of a corresponding vertex-girth-regular graph, by presenting constructions of infinite families of $vgr(v,k,g,λ)$-graphs, and by establishing lower bounds on the number $v$ of vertices in a $vgr(v,k,g,λ)$-graph. It also includes computational results determining the orders of smallest cubic and quartic graphs of small girths.