Evaluating a weighted graph polynomial for graphs of bounded tree-width

We show that for any $k$ there is a polynomial time algorithm to evaluate the weighted graph polynomial $U$ of any graph with tree-width at most $k$ at any point. For a graph with $n$ vertices, the algorithm requires $O(a_k n^{2k+3})$ arithmetical operations, where $a_k$ depends only on $k$

Obstructions for Matroids of Path-Width at most k and Graphs of Linear Rank-Width at most k

International audienceEvery minor-closed class of matroids of bounded branch-width can be characterized by a minimal list of excluded minors, but unlike graphs, this list could be infinite in general. However, for each fixed finite field $\mathbb F$, the list contains only finitely many $\mathbb F$-representable matroids, due to the well-quasi-ordering of $\mathbb F$-representable matroids of bounded branch-width under taking matroid minors [J. F. Geelen, A. M. H. Gerards, and G. Whittle (2002)]. But this proof is non-constructive and does not provide any algorithm for computing these $\mathbb F$-representable excluded minors in general. We consider the class of matroids of path-width at most $k$ for fixed $k$. We prove that for a finite field $\mathbb F$, every $\mathbb F$-representable excluded minor for the class of matroids of path-width at most~$k$ has at most $2^{|\mathbb{F}|^{O(k^2)}}$ elements. We can therefore compute, for any integer $k$ and a fixed finite field $\mathbb F$, the set of $\mathbb F$-representable excluded minors for the class of matroids of path-width $k$, and this gives as a corollary a polynomial-time algorithm for checking whether the path-width of an $\mathbb F$-represented matroid is at most $k$. We also prove that every excluded pivot-minor for the class of graphs having linear rank-width at most $k$ has at most $2^{2^{O(k^2)}}$ vertices, which also results in a similar algorithmic consequence for linear rank-width of graphs

The Tutte polynomial for matroids of bounded branch-width

It is a classical result of Jaeger, Vertigan and Welsh that evaluating the Tutte polynomial of a graph is #P-hard in all but a few special points. On the other hand, several papers in the past few years have shown that the Tutte polynomial of a graph can be efficiently computed for graphs of bounded tree-width. In this paper we present a recursive formula computing the Tutte polynomial of a matroid $M$ represented over a finite field (which includes all graphic matroids), using a so called parse tree of a branch-decomposition of $M$. This formula provides an algorithm computing the Tutte polynomial for a representable matroid of bounded branch-width in polynomial time with a fixed exponent

Διδιαστατότητα: Θεωρία και Αλγοριθμικές Εφαρμογές

Πολλά συνδυαστικά υπολογιστικά προβλήματα είναι στη γενική μορφή τους δύσβατα, υπό την έννοια πως ακόμη και για εισόδους μέτριου μεγέθους, η εύρεση μιας ακριβούς και βέλτιστης λύσης είναι μάλλον ανέφικτη, δεδομένου ότι συνήθως απαιτεί την κλήση αλγορίθμων, των οποίων η χρονική πολυπλοκότητα είναι εκθετική ως προς το μέγεθος του προβλήματος. Συχνά τα προβλήματα αυτά μπορούν να οριστούν σε γραφήματα. Πρόσθετες δομικές ιδιότητες ενός γραφήματος, όπως η εμβαπτισιμότητα σε κάποια επιφάνεια, παρέχουν μια λαβή για το σχεδιασμό αποδοτικότερων αλγορίθμων. Η θεωρία της διδιαστατότητας αναπτύχθηκα στα πλαίσια της Παραμετρικής Πολυπλοκότητας και, βασιζόμενη στα αποτελέσματα της θεωρίας των Ελασσόνων Γραφημάτων, παρέχει ένας μετα-αλγοριθμικό πλαίσιο για την αντιμετώπιση ενός συνόλου προβλημάτων σε πλατύ φάσμα κλάσεων γραφημάτων, πιο συγκεκριμένα σε όλες τις γενικεύσεις γραφημάτων εμβαπτίσιμων σε κάποια επιφάνεια. Στη διδακτορική διατριβή αυτή θεωρούμε ζητήματα συνδυαστικής φύσης σχετικά με την εφαρμογή της θεωρίας της Διδιαστατότητας και τις δυνατότητες επέκτασης του πεδίου εφαρμογής της.Many combinatorial computational problems are considered in their general form intractable, in the sense that even for modest size problems, providing an exact optimal solution is practically infeasible, as it typically involves the use of algorithms whose running time is exponential in the size of the problem. Often these problems can be modeled by graphs. Then, additional structural properties of a graph, such as surface embeddability, can provide a handle for the design of more ecient algorithms. The theory of Bidimensionality, dened in the context of Parameterized Complexity, builds on the celebrated results of Graph Minor theory and establishes a meta algorithmic framework for addressing problems in a broad range of graph classes, namely all generalizations of graphs embeddable on some surface. In this doctoral thesis we explore topics of combinatorial nature related to the implementation of the theory of Bidimensionality and to the possibilities of the extension of its applicability range