Jorik Jooken

Marien Abreu, Jan Goedgebeur, Jorik Jooken et al.

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.

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.

Jorik Jooken, Pieter Leyman, Tony Wauters et al.

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.

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.