CORDIS - Resultados de investigaciones de la UE
CORDIS

# Cuts and decompositions: algorithms and combinatorial properties

## Periodic Reporting for period 4 - CUTACOMBS (Cuts and decompositions: algorithms and combinatorial properties)

From the point of view of algorithm design and complexity theory, it is of primary importance to understand the computational complexity of a given task. For example, computing shortest path between two given points can be done efficiently (as witnessed by navigation apps such as Google Maps), while finding a shortest sightseeing tour through a few dozen of points of interest (in unspecified order) can be computationally challenging as the problem is NP-hard.

The project focuses on the interplay of graph theory and algorithm design: which graph-theoretic developments can help in designing efficient algorithms and in which scenarios this is possible.

One of the main directions in the grant is a systematic study of complexity of basic computational problems, such as Maximum Independent Set, in various graph classes, that is, when the input of the problem is restricted to have some particular properties. Surprisingly, there is a wide range of graph classes for which the complexity status of these basic problems is open. The graph classes in question largely coincide with the graph classes on the front of many questions in structural and extremal graph theory, such as the Erdos-Hajnal conjecture or the theory of chi-boundedness. This suggested that these two areas may benefit from an exchange of ideas and techniques, and this is exactly what was witnessed in the project.

In other main direction of the grant, we predicted applicability of the recent toolbox for decomposing graphs into smaller, simpler pieces, for the Graph Isomorphism problem in sparse graphs. This classic problem, asking whether two given graphs are isomorphic, is not known to be efficiently (i.e. polynomial-time) solvable in theory, but is efficiently solvable in practice by heuristic methods.

A third main direction of the grant considered algorithms for graph separation problems in directed graphs. As an example, consider the st-Cut problem: given a directed graph with a designated source and sink vertices, delete as few arcs as possible so that no path from the source to the sink remains in the graph. This classic problem is known to be efficiently solvable even in the weighted setting (i.e. arcs have weights and we want to delete arcs of minimum total weight), but becomes hard if one restricts both the cardinality and total weight of the cut (so-called bicriteria setting). This problem, and many other related problems, lacked fixed-pameter algorithm: an algorithm that is efficient if the cardinality of the cut in question is small. We conjectured that a algorithmic or graph theory technique is missing that will provide a uniform reason for tractability.

In all three aforementioned main directions, we have obtained major progress during the grant, fulfilling all or almost all objectives. Furthermore, in the first mentioned direction we have went far beyond the original plan of work.
We highlight main results and achievements in the three aforementioned main directions.

(1) Within the systematic study of graph classes, we have proven that for all classes with one forbidden induced subgraph (which are the "basic" graph classes in the graph class taxonomy) whenever Maximum Independent Set is not known to be NP-hard, it admits a quasi-polynomial time approximation scheme. While not being the Holy Grail "polynomial-time algorithm", the result shows that Maximum Independent Set is much simpler in these graph classes than in full generality. The wide applicability of our result came as a surprise. It is worth noting that the result came from a unique blend of algorithmic methods and graph-theoretical understanding (in particular, the "Three-in-a-Tree" theorem of Chudnovsky and Seymour). Furthermore, for many of the considered graph classes, we obtained a fast quasi-polynomial-time algorithm (without any approximation). Interestingly, the main tool in the latter result is the Gyarfas' path argument, a tool originating from the theory of chi-boundedness. Finally, we generalized many of the results not only to the Maximum Independent Set problem, but to a wide range of computational problems that include Feedback Vertex Set.

On the graph-theoretic side, we have proven that a graph class excluding some caterpilar (a tree with all branching vertices on one path) and its complement as induced subgraphs have the so-called strong Erdos-Hajnal property, that is, satisfy the famous Erdos-Hajnal conjecture in a particularly strong form. This result significantly expands the set of known graph classes that satisfy the Erdos-Hajnal property.

(2) We achieved here our main goal: We showed fixed-parameter tractable (i.e. polynomial with fixed degree of the polynomial running time bound) algorithm for a wide range of sparse graph classes (so-called classes closed under taking minors), solving a long-standing open problem.

(3) We introduced a new technique, called flow-augmentation, that essentially reduces the considered problem to the case where the sought cut is a minimum-cardinality cut. This allowed not only to solve almost all open problems in the area, but also found rich unexpected applications in the area of Constraint Satisfaction Problems. Interestingly, the new developments did not allow us to make any progress on one of the open problems in the area: better fixed-parameter or preprocessing algorithms for the classic Directed Feedback Vertex Set problem.

The major results in all directions have been presented and major theoretical computer science conferences.
Within the first main direction, we have reached far beyond the initial plan. With the current state-of-the-art, we conjecture that indeed Maximum Independent Set is polynomial-time solvable in all graph classes with one forbidden induced subgraph, where it is not known to be NP-hard, and, furthermore, in many of this graph classes the same applies to a wide range of problems including Feedback Vertex Set. Furthermore, we expect this question to be resolved in the next few years, based on the progress obtained through the duration of the project.

The progress in the second direction fully fulfilled our initial plan and we resolved a long-standing open problem in the literature.

The flow-augmentation technique turned out not only to be the missing algorithmic piece in the parameterized landscape of directed graph separation problems, but also found applications in the area of Constraint Satisfaction Problems. We obtained a fully parameterized complexity dichotomy for MinCSP problems over boolean domain and latest results indicate that a similar dichotomy within temporal problems (point line and Allen algebras) can be within reach in the foreseeable future.