55 research outputs found
Odd-Minors I: Excluding small parity breaks
Given a graph class~, the -blind-treewidth of a
graph~ is the smallest integer~ such that~ has a tree-decomposition
where every bag whose torso does not belong to~ has size at
most~. In this paper we focus on the class~ of bipartite graphs
and the class~ of planar graphs together with the odd-minor
relation. For each of the two parameters, -blind-treewidth and
-blind-treewidth, we prove an analogue of the
celebrated Grid Theorem under the odd-minor relation. As a consequence we
obtain FPT-approximation algorithms for both parameters. We then provide
FPT-algorithms for \textsc{Maximum Independent Set} on graphs of bounded
-blind-treewidth and \textsc{Maximum Cut} on graphs of bounded
-blind-treewidth
Packing Directed Cycles Quarter- and Half-Integrally
The celebrated Erd\H{o}s-P\'osa theorem states that every undirected graph
that does not admit a family of vertex-disjoint cycles contains a feedback
vertex set (a set of vertices hitting all cycles in the graph) of size . After being known for long as Younger's conjecture, a similar
statement for directed graphs has been proven in 1996 by Reed, Robertson,
Seymour, and Thomas. However, in their proof, the dependency of the size of the
feedback vertex set on the size of vertex-disjoint cycle packing is not
elementary.
We show that if we compare the size of a minimum feedback vertex set in a
directed graph with the quarter-integral cycle packing number, we obtain a
polynomial bound. More precisely, we show that if in a directed graph there
is no family of cycles such that every vertex of is in at most four of
the cycles, then there exists a feedback vertex set in of size .
Furthermore, a variant of our proof shows that if in a directed graph there
is no family of cycles such that every vertex of is in at most two of
the cycles, then there exists a feedback vertex set in of size .
On the way there we prove a more general result about quarter-integral
packing of subgraphs of high directed treewidth: for every pair of positive
integers and , if a directed graph has directed treewidth
, then one can find in a family of
subgraphs, each of directed treewidth at least , such that every vertex of
is in at most four subgraphs.Comment: Accepted to European Symposium on Algorithms (ESA '19
Chordless Cycle Packing Is Fixed-Parameter Tractable
A chordless cycle or hole in a graph G is an induced cycle of length at least 4. In the Hole Packing problem, a graph G and an integer k is given, and the task is to find (if exists) a set of k pairwise vertex-disjoint chordless cycles. Our main result is showing that Hole Packing is fixed-parameter tractable (FPT), that is, can be solved in time f(k)n^O(1) for some function f depending only on k
A domination algorithm for -instances of the travelling salesman problem
We present an approximation algorithm for -instances of the
travelling salesman problem which performs well with respect to combinatorial
dominance. More precisely, we give a polynomial-time algorithm which has
domination ratio . In other words, given a
-edge-weighting of the complete graph on vertices, our
algorithm outputs a Hamilton cycle of with the following property:
the proportion of Hamilton cycles of whose weight is smaller than that of
is at most . Our analysis is based on a martingale approach.
Previously, the best result in this direction was a polynomial-time algorithm
with domination ratio for arbitrary edge-weights. We also prove a
hardness result showing that, if the Exponential Time Hypothesis holds, there
exists a constant such that cannot be replaced by in the result above.Comment: 29 pages (final version to appear in Random Structures and
Algorithms
Παραμετρικοί - Προσεγγιστικοί Αλγόριθμοι και Ιδιότητες Erdős-Pósa σε Γραφήματα
Η διατριβή αυτή επικεντρώνεται στη μελέτη παραμετρικών προσεγγιστικών αλγορίθμων που σχετίζονται με γραφήματα κολοκύθες. Χρησιμοποιώντας μία γενικευμένη προσέγγιση (που μπορεί να επεκταθεί και σε πιο γενικές κλάσεις γραφημάτων) σχεδιάζουμε αλγορίθμους που ανιχνεύουν μοντέλα κολοκυθών και που χτυπούν μοντέλα κολοκυθών σε μεγάλα γραφήματα. Στηριζόμενοι σε αυτούς τους αλγορίθμους αποδεικνύουμε ιδιότητες τύπου Erdős-Pósa ως προς κορυφές και ακμές για τις κλάσεις των κολοκυθών και των διπλών κολοκυθών· για την πρώτη βελτιώνουμε υπάρχοντα αποτελέσματα ενώ για τη δεύτερη παρέχουμε τα πρώτα του είδους τους. Στην πορεία προς τούτο, γενικεύουμε προηγούμενα αποτέλεσματα που παρέχουν συνθήκες οι οποίες εξαναγκάζουν την ύπαρξη μιας ελάσσονος κλίκας εκθετικού μεγέθους μέσα σε ένα μεγαλύτερο γράφημα-φορέα.This thesis is centred around the study of parameterized approximation algorithms related to pumpkin graphs. Using a generalised approach (which could be expanded to other graph classes) we design algorithms that find pumpkin models and hit pumpkin models in large graphs. Based on these algorithms we prove Erdős-Pósa- like results, both for vertices and edges, for the classes of pumpkins and double pumpkins; for the former we improve previous results and for the latter we provide the first such results. As a necessary step in our process, but of independent value, we generalise previous results that provide conditions which force the existence of a clique of exponential size as a minor inside a larger host-graph
Parameterization Above a Multiplicative Guarantee
Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (at most) k+g(I). Here, g(I) is usually a lower bound (resp. upper bound) on the maximum (resp. minimum) size of a solution. Since its introduction in 1999 for Max SAT and Max Cut (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research.
We highlight a multiplicative form of parameterization above a guarantee: Given an instance (I,k) of some (parameterized) problem ? with a guarantee g(I), decide whether I admits a solution of size at least (resp. at most) k ? g(I). In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, and provide a parameterized algorithm for this problem. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ?>0, multiplicative parameterization above g(I)^(1+?) of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of algorithms for other problems parameterized multiplicatively above girth
Fast Strategies in Waiter-Client Games on
Waiter-Client games are played on some hypergraph , where
denotes the family of winning sets. For some bias , during
each round of such a game Waiter offers to Client elements of , of
which Client claims one for himself while the rest go to Waiter. Proceeding
like this Waiter wins the game if she forces Client to claim all the elements
of any winning set from . In this paper we study fast strategies
for several Waiter-Client games played on the edge set of the complete graph,
i.e. , in which the winning sets are perfect matchings, Hamilton
cycles, pancyclic graphs, fixed spanning trees or factors of a given graph.Comment: 38 page
Hitting Meets Packing: How Hard Can it Be?
We study a general family of problems that form a common generalization of
classic hitting (also referred to as covering or transversal) and packing
problems. An instance of X-HitPack asks: Can removing k (deletable) vertices of
a graph G prevent us from packing vertex-disjoint objects of type X?
This problem captures a spectrum of problems with standard hitting and packing
on opposite ends. Our main motivating question is whether the combination
X-HitPack can be significantly harder than these two base problems. Already for
a particular choice of X, this question can be posed for many different
complexity notions, leading to a large, so-far unexplored domain in the
intersection of the areas of hitting and packing problems.
On a high-level, we present two case studies: (1) X being all cycles, and (2)
X being all copies of a fixed graph H. In each, we explore the classical
complexity, as well as the parameterized complexity with the natural parameters
k+l and treewidth. We observe that the combined problem can be drastically
harder than the base problems: for cycles or for H being a connected graph with
at least 3 vertices, the problem is \Sigma_2^P-complete and requires
double-exponential dependence on the treewidth of the graph (assuming the
Exponential-Time Hypothesis). In contrast, the combined problem admits
qualitatively similar running times as the base problems in some cases,
although significant novel ideas are required. For example, for X being all
cycles, we establish a 2^poly(k+l)n^O(1) algorithm using an involved branching
method. Also, for X being all edges (i.e., H = K_2; this combines Vertex Cover
and Maximum Matching) the problem can be solved in time 2^\poly(tw)n^O(1) on
graphs of treewidth tw. The key step enabling this running time relies on a
combinatorial bound obtained from an algebraic (linear delta-matroid)
representation of possible matchings
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