In 1913, Pólya asked for which (0,1)-matrices A it is possible to create a new matrix A′ by changing some of the signs such that the permanent of A equals the determinant of A′. A combinatorial solution to this problem was found by Little in 1975; he found these matrices to be exactly the biadjacency matrices of bipartite graphs excluding K3,3 as a matching minor. Utilising ideas from graph minors theory, this characterisation was later shown to yield a polynomial time algorithm to compute the permanent of matrices which satisfy Little’s condition. By a seminal result of Valiant, computing the permanent of (0,1)-matrices in general is #P-hard; however, it can be observed that the tractability of the permanent is closely related to the exclusion of matchings minors in bipartite graphs.

By a seminal result of Valiant, computing the permanent of (0,1)-matrices is, in general, #P-hard. In 1913 Polya asked for which (0,1)-matrices A it is possible to change some signs such that the permanent of A equals the determinant of the resulting matrix. In 1975, Little showed these matrices to be exactly the biadjacency matrices of bipartite graphs excluding K3,3 as a matching minor. This was turned into a polynomial time algorithm by McCuaig, Robertson, Seymour, and Thomas in 1999. However, the relation between the exclusion of some matching minor in a bipartite graph and the tractability of the permanent extends beyond K3,3. Recently it was shown that the exclusion of any planar bipartite graph as a matching minor yields a class of bipartite graphs on which the permanent of the corresponding (0,1)-matrices can be computed efficiently. In this paper we unify the two results above into a single, more general result in the style of the celebrated structure theorem for single-crossing-minor-free graphs. We identify a class of bipartite graphs strictly generalising planar bipartite graphs and K3,3 which includes infinitely many non-Pfaffian graphs. The exclusion of any member of this class as a matching minor yields a structure that allows for the efficient evaluation of the permanent. Moreover, we show that the evaluation of the permanent remains #P-hard on bipartite graphs which exclude K5,5 as a matching minor. This establishes a first computational lower bound for the problem of counting perfect matchings on matching minor closed classes. As another application of our structure theorem, we obtain a strict generalisation of the algorithm for the k-vertex disjoint directed paths problem on digraphs of bounded directed treewidth.

The tangle tree-decomposition theorem, proved by Robertson and Seymour in their seminal graph minors series, turns out to be an extremely valuable tool in structural and algorithmic graph theory. In this paper, we prove the analogous result for digraphs, the directed tangle tree-decomposition theorem. More precisely, we introduce directed tangles and provide a directed tree-decomposition of digraphs G that distinguishes all maximal directed tangles in G. Furthermore, for any integer k, we construct a directed tree-decomposition that distinguishes all directed tangles of order k. By relaxing the bound slightly, we can make the previous result algorithmic: for fixed k, we design a polynomial-time algorithm that finds a directed tree-decomposition distinguishing all directed tangles of order 6k–1 separated by some separation of order less than k. As a direct application of the tangle tree-decomposition theorem, we prove that for every fixed k there is a polynomial-time algorithm which, on input G, and source and sink vertices (s1, t1),…, (sk, tk), either finds a family of paths P1,…, Pk such that each Pi links si to ti and every vertex of G is contained in at most two paths, or determines that there is no set of pairwise vertex-disjoint paths each connecting si to ti. This result improves previous results (with “two” replaced by “three”), and given known hardness results, our result cannot be extended to fixed parameter tractability nor fully vertex-disjoint directed paths.

Suppose ${\mathcal{F}}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion of at most $k$ edges of $G$ can result in a graph that does not contain any graph from $\mathcal{F}$ as an immersion. This problem is a close relative of the $\mathcal{F}$-Minor Deletion problem studied by Fomin et al. [ Proceedings of FOCS, IEEE, 2012, pp. 470--479], where one deletes vertices in order to remove all minor models of graphs from $\mathcal{F}$. We prove that whenever all graphs from $\mathcal{F}$ are connected and at least one graph of $\mathcal{F}$ is planar and subcubic, then the $\mathcal{F}$-Immersion Deletion problem admits a constant-factor approximation algorithm running in time $\mathcal{O}(m^3 \cdot n^3 \cdot \log m)$, a linear kernel that can be computed in time $\mathcal{O}(m^4 \cdot n^3 \cdot \log m)$, and a $\mathcal{O}(2^{\mathcal{O}(k)} + m^4 \cdot n^3 \cdot \log m)$-time fixed-parameter algorithm, where $n,m$ count the vertices and edges of the input graph. These results mirror the findings of Fomin et al., who obtained a similar set of algorithmic results for $\mathcal{F}$-Minor Deletion, under the assumption that at least one graph from $\mathcal{F}$ is planar. An important difference is that we are able to obtain a linear kernel for $\mathcal{F}$-Immersion Deletion, while the exponent of the kernel of Fomin et al. for $\mathcal{F}$-Minor Deletion depends heavily on the family $\mathcal{F}$. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ ACM Trans. Algorithms, 13 (2017), p. 35]. This reveals that the kernelization complexity of $\mathcal{F}$-Immersion Deletion is quite different from that of $\mathcal{F}$-Minor Deletion.

At the core of the Robertson-Seymour Theory of Graph Minors lies a powerful structure theorem which captures, for any fixed graph H, the common structural features of all the graphs not containing H as a minor [15]. An important step towards this structure theorem is the Flat Wall Theorem [14], which has a lot of algorithmic applications (for example, the minor-testing and the disjoint paths problem with fixed number terminals). In this paper, we prove the directed analogue of this Flat Wall Theorem. Our result builds on the recent Directed Grid Theorem by two of the authors (Kawarabayashi and Kreutzer), and we hope that this is an important and significant step toward the directed structure theorem, as with the case for the undirected graph for the graph minor project.

The notion of tree-cut width has been introduced by Wollan in [The structure of graphs not admitting a fixed immersion, Journal of Combinatorial Theory, Series B, 110:47 - 66, 2015]. It is defined via tree-cut decompositions, which are tree-like decompositions that highlight small (edge) cuts in a graph. In that sense, tree-cut decompositions can be seen as an edge-version of tree-decompositions and have algorithmic applications on problems that remain intractable on graphs of bounded treewidth. In this paper, we prove that every graph admits an optimal tree-cut decomposition that satisfies a certain Menger-like condition similar to that of the lean tree decompositions of Thomas [A Menger-like property of tree-width: The finite case, Journal of Combinatorial Theory, Series B, 48(1):67 - 76, 1990]. This allows us to give, for every k in N, an upper-bound on the number immersion-minimal graphs of tree-cut width k. Our results imply the constructive existence of a linear FPT-algorithm for tree-cut width.

Cutwidth is one of the classic layout parameters for graphs. It measures how well one can order the vertices of a graph in a linear manner, so that the maximum number of edges between any prefix and its complement suffix is minimized. As graphs of cutwidth at most k are closed under taking immersions, the results of Robertson and Seymour imply that there is a finite list of minimal immersion obstructions for admitting a cut layout of width at most k. We prove that every minimal immersion obstruction for cutwidth at most k has size at most $$2^{{O}(k^3\log k)}$$ . As an interesting algorithmic byproduct, we design a new fixed-parameter algorithm for computing the cutwidth of a graph that runs in time $$2^{{O}(k^2\log k)}\cdot n$$ , where k is the optimum width and n is the number of vertices. While being slower by a $$\log k$$ -factor in the exponent than the fastest known algorithm, given by Thilikos et al. (J Algorithms 56(1):1–24, 2005; J Algorithms 56(1):25–49, 2005), our algorithm has the advantage of being simpler and self-contained; arguably, it explains better the combinatorics of optimum-width layouts.

We prove that whenever G is a graph from a nowhere dense graph class C, and A is a subset of vertices of G, then the number of subsets of A that are realized as intersections of A with r-neighborhoods of vertices of G is at most f(r,eps)|A|^(1+eps), where r is any positive integer, eps is any positive real, and f is a function that depends only on the class C. This yields a characterization of nowhere dense classes of graphs in terms of neighborhood complexity, which answers a question posed by [Reidl et al., CoRR, 2016]. As an algorithmic application of the above result, we show that for every fixed integer r, the parameterized Distance-r Dominating Set problem admits an almost linear kernel on any nowhere dense graph class. This proves a conjecture posed by [Drange et al., STACS 2016], and shows that the limit of parameterized tractability of Distance-r Dominating Set on subgraph-closed graph classes lies exactly on the boundary between nowhere denseness and somewhere denseness.

The F -M inor -F ree D eletion problem asks, for a fixed set F and an input consisting of a graph G and integer k , whether k vertices can be removed from G such that the resulting graph does not contain any member of F as a minor. At FOCS 2012, Fomin et al. showed that the special case when F contains at least one planar graph has a kernel of size f ( F ) ċ k g ( F ) for some functions f and g . They left open whether this P lanar F -M inor -F ree D eletion problem has kernels whose size is uniformly polynomial, of the form f ( F ) ċ k c for some universal constant c . We prove that some P lanar F -M inor -F ree D eletion problems do not have uniformly polynomial kernels (unless NP ⊆ coNP/poly), not even when parameterized by the vertex cover number. On the positive side, we consider the problem of determining whether k vertices can be removed to obtain a graph of treedepth at most η. We prove that this problem admits uniformly polynomial kernels with O ( k 6 ) vertices for every fixed η.

We study the design of fixed-parameter algorithms for problems already known to be solvable in polynomial time. The main motivation is to get more efficient algorithms for problems with unattractive polynomial running times. Here, we focus on a fundamental graph problem: Longest Path , that is, given an undirected graph, find a maximum-length path in G . Longest Path is NP-hard in general but known to be solvable in O ( n 4 ) time on n -vertex interval graphs. We show how to solve Longest Path on Interval Graphs , parameterized by vertex deletion number k to proper interval graphs, in O ( k 9 n ) time. Notably, Longest Path is trivially solvable in linear time on proper interval graphs, and the parameter value k can be approximated up to a factor of 4 in linear time. From a more general perspective, we believe that using parameterized complexity analysis may enable a refined understanding of efficiency aspects for polynomial-time solvable problems similarly to what classical parameterized complexity analysis does for NP-hard problems.

The Plane Subgraph (resp. Topological Minor) Completion problem asks, given a (possibly disconnected) plane (multi)graph Gamma and a connected plane (multi)graph Delta, whether it is possible to add edges in Gamma without violating the planarity of its embedding so that it contains some subgraph (resp. topological minor) that is topologically isomorphic to Delta. We give FPT algorithms that solve both problems in f(|E(Delta)|)*|E(\Gamma)|^{2} steps. Moreover, for the Plane Subgraph Completion problem we show that f(k)=2^{O(k*log(k))}.

In the Tree Deletion Set problem the input is a graph $G$ together with an integer $k$. The objective is to determine whether there exists a set $S$ of at most $k$ vertices such that $G\setminus S$ is a tree. The problem is \tt NP-complete and even \tt NP-hard to approximate within any factor of $\text{OPT}^c$ for any constant $c$. In this paper we give an $\mathcal{O}(k^5)$ size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on systems of linear equations, that we use to handle the instances on which Tree Deletion Set is hard to approximate.

Tolerance graphs model interval relations in such a way that intervals can tolerate a certain amount of overlap without being in conflict. In one of the most natural generalizations of tolerance graphs with direct applications in the comparison of DNA sequences from different organisms, namely multitolerance graphs, two tolerances are allowed for each interval: one on the left side and the other on the right side. Several efficient algorithms for optimization problems that are NP-hard in general graphs have been designed for tolerance and multitolerance graphs. In spite of this progress, the complexity status of some fundamental algorithmic problems on tolerance and multitolerance graphs, such as the dominating set problem, remained unresolved until now---three decades after the introduction of tolerance graphs. In this paper we introduce two new geometric representations for tolerance and multitolerance graphs, given by points and line segments in the plane. Apart from being important on their own, these new representations prove to be a powerful tool for deriving both hardness results and polynomial time algorithms. Using them, we surprisingly prove that the dominating set problem can be solved in polynomial time on tolerance graphs and that it is APX-hard on multitolerance graphs, thus solving a longstanding open problem. This problem is the first one that has been discovered with a different complexity status in these two graph classes.

In the Tree Deletion Set problem the input is a graph G together with an integer k. The objective is to determine whether there exists a set S of at most k vertices such that G \ S is a tree. The problem is NP-complete and even NP-hard to approximate within any factor of OPT^c for any constant c. In this paper we give an O(k^5) size kernel for the Tree Deletion Set problem. An appealing feature of our kernelization algorithm is a new reduction rule, based on system of linear equations, that we use to handle the instances on which Tree Deletion Set is hard to approximate.

One of the major results of [N. Robertson and P. D. Seymour, Graph minors. XIII. The disjoint paths problem, J. Combin. Theory Ser. B, 63 (1995), pp. 65--110], also known as the weak structure theorem, reveals the local structure of graphs excluding some graph as a minor: each such graph $G$ either has small treewidth or contains the subdivision of a planar graph (a wall) that can be arranged in a flat manner inside $G$, given that some small set of vertices is removed. We prove an optimized version of that theorem where (i) the relation between the treewidth of the graph and the height of the wall is linear (thus best possible) and (ii) the number of vertices to be removed is minimized.