94 research outputs found
Bi-Criteria and Approximation Algorithms for Restricted Matchings
In this work we study approximation algorithms for the \textit{Bounded Color
Matching} problem (a.k.a. Restricted Matching problem) which is defined as
follows: given a graph in which each edge has a color and a profit
, we want to compute a maximum (cardinality or profit)
matching in which no more than edges of color are
present. This kind of problems, beside the theoretical interest on its own
right, emerges in multi-fiber optical networking systems, where we interpret
each unique wavelength that can travel through the fiber as a color class and
we would like to establish communication between pairs of systems. We study
approximation and bi-criteria algorithms for this problem which are based on
linear programming techniques and, in particular, on polyhedral
characterizations of the natural linear formulation of the problem. In our
setting, we allow violations of the bounds and we model our problem as a
bi-criteria problem: we have two objectives to optimize namely (a) to maximize
the profit (maximum matching) while (b) minimizing the violation of the color
bounds. We prove how we can "beat" the integrality gap of the natural linear
programming formulation of the problem by allowing only a slight violation of
the color bounds. In particular, our main result is \textit{constant}
approximation bounds for both criteria of the corresponding bi-criteria
optimization problem
On unrooted and root-uncertain variants of several well-known phylogenetic network problems
The hybridization number problem requires us to embed a set of binary rooted
phylogenetic trees into a binary rooted phylogenetic network such that the
number of nodes with indegree two is minimized. However, from a biological
point of view accurately inferring the root location in a phylogenetic tree is
notoriously difficult and poor root placement can artificially inflate the
hybridization number. To this end we study a number of relaxed variants of this
problem. We start by showing that the fundamental problem of determining
whether an \emph{unrooted} phylogenetic network displays (i.e. embeds) an
\emph{unrooted} phylogenetic tree, is NP-hard. On the positive side we show
that this problem is FPT in reticulation number. In the rooted case the
corresponding FPT result is trivial, but here we require more subtle
argumentation. Next we show that the hybridization number problem for unrooted
networks (when given two unrooted trees) is equivalent to the problem of
computing the Tree Bisection and Reconnect (TBR) distance of the two unrooted
trees. In the third part of the paper we consider the "root uncertain" variant
of hybridization number. Here we are free to choose the root location in each
of a set of unrooted input trees such that the hybridization number of the
resulting rooted trees is minimized. On the negative side we show that this
problem is APX-hard. On the positive side, we show that the problem is FPT in
the hybridization number, via kernelization, for any number of input trees.Comment: 28 pages, 8 Figure
PTAS for Ordered Instances of Resource Allocation Problems
We consider the problem of fair allocation of indivisible goods where
we are given a set I of m indivisible resources (items) and a set P of n customers (players) competing for the resources. Each resource
j in I has a same value vj > 0 for a subset of customers interested in j and it has no value for other customers. The goal is to find a feasible allocation of the resources to the interested customers such that in the Max-Min scenario (also known as Santa Claus problem) the minimum utility (sum of the resources) received by each of the customers is as high as possible and in the Min-Max case (also known as R||C_max problem), the maximum utility is as low as possible.
In this paper we are interested in instances of the problem that admit a PTAS. These instances are not only of theoretical interest but also have practical applications. For the Max-Min allocation problem, we start with instances of the problem that can be viewed as a convex bipartite graph; there exists an ordering of the resources such that each customer is interested (has positive evaluation) in a set of consecutive resources and we demonstrate a PTAS. For the Min-Max allocation problem, we obtain a PTAS for instances in which there is an ordering of the customers (machines) and each resource (job) is adjacent to a consecutive set of customers (machines).
Next we show that our method for the Max-Min scenario, can be extended to a broader class of bipartite graphs where the resources can be viewed as a tree and each customer is interested in a sub-tree of a bounded number of leaves of this tree (e.g. a sub-path)
Treewidth of display graphs: bounds, brambles and applications
Phylogenetic trees and networks are leaf-labelled graphs used to model evolution. Display graphs are created by identifying common leaf labels in two or more phylogenetic trees or networks. The treewidth of such graphs is bounded as a function of many common dissimilarity measures between phylogenetic trees and this has been leveraged in fixed parameter tractability results. Here we further elucidate the properties of display graphs and their interaction with treewidth. We show that it is NP-hard to recognize display graphs, but that display graphs of bounded treewidth can be recognized in linear time. Next we show that if a phylogenetic network displays (i.e. topologically embeds) a phylogenetic tree, the treewidth of their display graph is bounded by a function of the treewidth of the original network (and also by various other parameters). In fact, using a bramble argument we show that this treewidth bound is sharp up to an additive term of 1. We leverage this bound to give an FPT algorithm, parameterized by treewidth, for determining whether a network displays a tree, which is an intensively-studied problem in the field. We conclude with a discussion on the future use of display graphs and treewidth in phylogenetics
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
Relaxed Agreement Forests
There are multiple factors which can cause the phylogenetic inference process
to produce two or more conflicting hypotheses of the evolutionary history of a
set X of biological entities. That is: phylogenetic trees with the same set of
leaf labels X but with distinct topologies. This leads naturally to the goal of
quantifying the difference between two such trees T_1 and T_2. Here we
introduce the problem of computing a 'maximum relaxed agreement forest' (MRAF)
and use this as a proxy for the dissimilarity of T_1 and T_2, which in this
article we assume to be unrooted binary phylogenetic trees. MRAF asks for a
partition of the leaf labels X into a minimum number of blocks S_1, S_2, ...
S_k such that for each i, the subtrees induced in T_1 and T_2 by S_i are
isomorphic up to suppression of degree-2 nodes and taking the labels X into
account. Unlike the earlier introduced maximum agreement forest (MAF) model,
the subtrees induced by the S_i are allowed to overlap. We prove that it is
NP-hard to compute MRAF, by reducing from the problem of partitioning a
permutation into a minimum number of monotonic subsequences (PIMS).
Furthermore, we show that MRAF has a polynomial time O(log n)-approximation
algorithm where n=|X| and permits exact algorithms with single-exponential
running time. When at least one of the two input trees has a caterpillar
topology, we prove that testing whether a MRAF has size at most k can be
answered in polynomial time when k is fixed. We also note that on two
caterpillars the approximability of MRAF is related to that of PIMS. Finally,
we establish a number of bounds on MRAF, compare its behaviour to MAF both in
theory and in an experimental setting and discuss a number of open problems.Comment: 14 pages plus appendi
From big data to big information and big knowledge: The case of Earth observation data
Some particularly important rich sources of open and free big geospatial data are the Earth observation (EO) programs of various countries such as the Landsat program of the US and the Copernicus programme of the European Union. EO data is a paradigmatic case of big data and the same is true for the big information and big knowledge extracted from it. EO data (satellite images and in-situ data), and the information and knowledge extracted from it, can be utilized in many applications with financial and environmental impact in areas such as emergency management, climate change, agriculture and security
On Unrooted and Root-Uncertain Variants of Several Well-Known Phylogenetic Network Problems
International audienceThe hybridization number problem requires us to embed a set of binary rooted phylogenetic trees into a binary rooted phylogenetic network such that the number of nodes with indegree two is minimized. However, from a biological point of view accurately inferring the root location in a phylogenetic tree is notoriously difficult and poor root placement can artificially inflate the hybridization number. To thisend we study a number of relaxed variants of this problem. We start by showing that the fundamental problem of determining whether an unrooted phylogenetic network displays (i.e. embeds) an unrooted phylogenetic tree, is NP-hard. On the positive side we show that this problem is FPT in reticulation number. In the rooted case the corresponding FPT result is trivial, but here we require more subtle argumentation. Next we show that the hybridization number problem for unrooted networks (when given two unrooted trees) is equivalent to the problem of computing the tree bisection and reconnect distance of the two unrooted trees. In the third part of the paper we consider the âroot uncertainâ variant of hybridization number. Here we are free to choose the root location in each of a set of unrooted input trees such that the hybridization number of the resulting rooted trees is minimized. On the negative side we show that this problem is APX-hard. On the positive side, we show that the problem is FPT in the hybridization number, via kernelization, for any number of input trees
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