Mateus de Oliveira Oliveira

Mateus de Oliveira Oliveira, Farhad Vadiee

In the field of parameterized complexity theory, the study of graph width measures has been intimately connected with the development of width-based model checking algorithms for combinatorial properties on graphs. In this work, we introduce a general framework to convert a large class of width-based model-checking algorithms into algorithms that can be used to test the validity of graph-theoretic conjectures on classes of graphs of bounded width. Our framework is modular and can be applied with respect to several well-studied width measures for graphs, including treewidth and cliquewidth. As a quantitative application of our framework, we prove analytically that for several long-standing graph-theoretic conjectures, there exists an algorithm that takes a number $k$ as input and correctly determines in time double-exponential in $k^{O(1)}$ whether the conjecture is valid on all graphs of treewidth at most $k$. These upper bounds, which may be regarded as upper-bounds on the size of proofs/disproofs for these conjectures on the class of graphs of treewidth at most $k$, improve significantly on theoretical upper bounds obtained using previously available techniques.

Mateus de Oliveira Oliveira, Sam Urmian

Width-based automated theorem proving is a framework where counterexamples to graph-theoretic conjectures are searched width-wise relative to some graph width measure, such as treewidth or pathwidth. In a recent work it has been shown that dynamic programming algorithms operating on tree decompositions can be combined together with the purpose of width-based theorem proving. This approach can be used to show that several long-standing conjectures in graph theory can be tested in time \(2^{2^{k^{O(1)}}}\) on the class of graphs of treewidth at most \(k\). In this work, we give the first steps towards evaluating the viability of this framework from a practical standpoint. At the same time, we advance the framework in two directions. First, we introduce a state-canonization technique that significantly reduces the number of states evaluated during the search for a counterexample of the conjecture. Second, we introduce an early-pruning technique that can be applied in the study of conjectures of the form \(\mathcal{P}_1 \rightarrow \mathcal{P}_2\), for graph properties \(\mathcal{P}_1\) and \(\mathcal{P}_2\), where \(\mathcal{P}_1\) is a property closed under subgraphs. As a concrete application, we use our framework in the study of graph-theoretic conjectures related to coloring triangle-free graphs. In particular, our algorithm is able to show that Reed's conjecture for triangle-free graphs is valid on the class of graphs of pathwidth at most 5, and on graphs of treewidth at most 3. Perhaps more interestingly, our algorithm is able to construct in a completely automated way counterexamples to invalid strengthenings of Reed's conjecture. These are the first results showing that width-based automated theorem proving is a promising avenue in the study of graph-theoretic conjectures.

Emmanuel Arrighi, Henning Fernau, Daniel Lokshtanov et al.

In its most traditional setting, the main concern of optimization theory is the search for optimal solutions for instances of a given computational problem. A recent trend of research in artificial intelligence, called solution diversity, has focused on the development of notions of optimality that may be more appropriate in settings where subjectivity is essential. The idea is that instead of aiming at the development of algorithms that output a single optimal solution, the goal is to investigate algorithms that output a small set of sufficiently good solutions that are sufficiently diverse from one another. In this way, the user has the opportunity to choose the solution that is most appropriate to the context at hand. It also displays the richness of the solution space. When combined with techniques from parameterized complexity theory, the paradigm of diversity of solutions offers a powerful algorithmic framework to address problems of practical relevance. In this work, we investigate the impact of this combination in the field of Kemeny Rank Aggregation, a well-studied class of problems lying in the intersection of order theory and social choice theory and also in the field of order theory itself. In particular, we show that the Kemeny Rank Aggregation problem is fixed-parameter tractable with respect to natural parameters providing natural formalizations of the notions of diversity and of the notion of a sufficiently good solution. Our main results work both when considering the traditional setting of aggregation over linearly ordered votes, and in the more general setting where votes are partially ordered.