Minor-Obstructions for Apex-Pseudoforests

A graph is called a pseudoforest if none of its connected components contains more than one cycle. A graph is an apex-pseudoforest if it can become a pseudoforest by removing one of its vertices. We identify 33 graphs that form the minor-obstruction set of the class of apex-pseudoforests, i.e., the set of all minor-minimal graphs that are not apex-pseudoforests

Decompositions and Algorithms for the Disjoint Paths Problem in Planar Graphs

Στο πρόβλημα των Διακεκριμενων Μονοπατιων μας ζητείται να εξετάσουμε, δεδομένου ενός γραφήματος G και ενος συνόλου k ζευγών τερματικών,αν τα ζεύγη των τερματικών μπορούν να συνδεθούν με διακεκριμένα μονοπάτια. Στα "Graph Minors", μια σειρά 23 εργασιών μεταξύ 1984 και 2011, οι Neil Robertson και Paul D. Seymour, ανάμεσα σε άλλα σπουδαία αποτελέσματα που επηρέασαν βαθιά την Θεωρία Γραφημάτων, παρουσίασαν έναν f(k)*n^3 αλγόριθμο για το πρόβλημα των Διακεκριμενων Μονοπατιων. Για να το καταφέρουν αυτό, εισήγαγαν την "τεχνκή της άσχετης κορυφής" σύμφωνα με την οποία σε κάθε στιγμιότυπο δεντροπλάτους μεγαλύτερου του g(k) υπάρχει μια "άσχετη" κορυφή της οποίας η αφαίρεση δημιουργεί ένα ισοδύναμο στιγμιότυπο του προβλήματος. Εδώ μελετάμε το πρόβλημα σε επίπεδα γραφήματα και αποδεικνύουμε ότι για κάθε σταθερό k κάθε στιγμιότυπο του προβλήματος των Διακεκριμενων Μονοπατιων σε επιπεδα γραφηματα μπορεί να μετασχηματιστεί σε ένα ισοδύναμο που έχει φραγμένο δενδροπλάτος, αφαιρώντας ταυτόχρονα ένα σύνολο κορυφών από το δεδομένο επίπεδο γράφημα. Ως συνέπεια αυτού, το πρόβλημα των Διακεκριμένων Μονοπατιών σε επίπεδα γραφήματα μπορεί να λυθεί σε γραμμικό χρόνο για κάθε σταθερό πλήθος τερματικών.> In the Disjoint Paths Problem, given a graph G and a set of k pairs of terminals, we ask whether the pairs of terminals can be linked by pairwise disjoint paths. > In the Graph Minors series of 23 papers between 1984 and 2011, Neil Robertson and Paul D. Seymour, among other great results that heavily influenced Graph Theory, provided an f(k)\cdot n^{3} algorithm for the Disjoint Paths Problem. To achieve this, they introduced the irrelevant vertex technique according to which in every instance of treewidth greater than g(k) there is an “irrelevant” vertex whose removal creates an equivalent instance of the problem. > > We study the problem in planar graphs and we prove that for every fixed k every instance of the Planar Disjoint Paths Problem can be transformed to an equivalent one that has bounded treewidth, by simultaneously discarding a set of vertices of the given planar graph. As a consequence the Planar Disjoint Paths Problem can be solved in linear time for every fixed number of terminals

A more accurate view of the Flat Wall Theorem

We introduce a supporting combinatorial framework for the Flat Wall Theorem. In particular, we suggest two variants of the theorem and we introduce a new, more versatile, concept of wall homogeneity as well as the notion of regularity in flat walls. All proposed concepts and results aim at facilitating the use of the irrelevant vertex technique in future algorithmic applications.Comment: arXiv admin note: text overlap with arXiv:2004.1269

Branchwidth is (1,g)-self-dual

A graph parameter is self-dual in some class of graphs embeddable in some surface if its value does not change in the dual graph by more than a constant factor. We prove that the branchwidth of connected hypergraphs without bridges and loops that are embeddable in some surface of Euler genus at most g is an (1,g)-self-dual parameter. This is the first proof that branchwidth is an additively self-dual width parameter.Comment: 10 page

Fixed-Parameter Tractability of Maximum Colored Path and Beyond

We introduce a general method for obtaining fixed-parameter algorithms for problems about finding paths in undirected graphs, where the length of the path could be unbounded in the parameter. The first application of our method is as follows. We give a randomized algorithm, that given a colored $n$-vertex undirected graph, vertices $s$ and $t$, and an integer $k$, finds an $(s,t)$-path containing at least $k$ different colors in time $2^k n^{O(1)}$. This is the first FPT algorithm for this problem, and it generalizes the algorithm of Bj\"orklund, Husfeldt, and Taslaman [SODA 2012] on finding a path through $k$ specified vertices. It also implies the first $2^k n^{O(1)}$ time algorithm for finding an $(s,t)$-path of length at least $k$. Our method yields FPT algorithms for even more general problems. For example, we consider the problem where the input consists of an $n$-vertex undirected graph $G$, a matroid $M$ whose elements correspond to the vertices of $G$ and which is represented over a finite field of order $q$, a positive integer weight function on the vertices of $G$, two sets of vertices $S,T \subseteq V(G)$, and integers $p,k,w$, and the task is to find $p$ vertex-disjoint paths from $S$ to $T$ so that the union of the vertices of these paths contains an independent set of $M$ of cardinality $k$ and weight $w$, while minimizing the sum of the lengths of the paths. We give a $2^{p+O(k^2 \log (q+k))} n^{O(1)} w$ time randomized algorithm for this problem.Comment: 50 pages, 16 figure

Shortest Cycles With Monotone Submodular Costs

We introduce the following submodular generalization of the Shortest Cycle problem. For a nonnegative monotone submodular cost function $f$ defined on the edges (or the vertices) of an undirected graph $G$, we seek for a cycle $C$ in $G$ of minimum cost $\textsf{OPT}=f(C)$. We give an algorithm that given an $n$-vertex graph $G$, parameter $\varepsilon > 0$, and the function $f$ represented by an oracle, in time $n^{\mathcal{O}(\log 1/\varepsilon)}$ finds a cycle $C$ in $G$ with $f(C)\leq (1+\varepsilon)\cdot \textsf{OPT}$. This is in sharp contrast with the non-approximability of the closely related Monotone Submodular Shortest $(s,t)$-Path problem, which requires exponentially many queries to the oracle for finding an $n^{2/3-\varepsilon}$-approximation [Goel et al., FOCS 2009]. We complement our algorithm with a matching lower bound. We show that for every $\varepsilon > 0$, obtaining a $(1+\varepsilon)$-approximation requires at least $n^{\Omega(\log 1/ \varepsilon)}$ queries to the oracle. When the function $f$ is integer-valued, our algorithm yields that a cycle of cost $\textsf{OPT}$ can be found in time $n^{\mathcal{O}(\log \textsf{OPT})}$. In particular, for $\textsf{OPT}=n^{\mathcal{O}(1)}$ this gives a quasipolynomial-time algorithm computing a cycle of minimum submodular cost. Interestingly, while a quasipolynomial-time algorithm often serves as a good indication that a polynomial time complexity could be achieved, we show a lower bound that $n^{\mathcal{O}(\log n)}$ queries are required even when $\textsf{OPT} = \mathcal{O}(n)$.Comment: 17 pages, 1 figure. Accepted to SODA 202