1,648 research outputs found
On the Implicit Graph Conjecture
The implicit graph conjecture states that every sufficiently small,
hereditary graph class has a labeling scheme with a polynomial-time computable
label decoder. We approach this conjecture by investigating classes of label
decoders defined in terms of complexity classes such as P and EXP. For
instance, GP denotes the class of graph classes that have a labeling scheme
with a polynomial-time computable label decoder. Until now it was not even
known whether GP is a strict subset of GR. We show that this is indeed the case
and reveal a strict hierarchy akin to classical complexity. We also show that
classes such as GP can be characterized in terms of graph parameters. This
could mean that certain algorithmic problems are feasible on every graph class
in GP. Lastly, we define a more restrictive class of label decoders using
first-order logic that already contains many natural graph classes such as
forests and interval graphs. We give an alternative characterization of this
class in terms of directed acyclic graphs. By showing that some small,
hereditary graph class cannot be expressed with such label decoders a weaker
form of the implicit graph conjecture could be disproven.Comment: 13 pages, MFCS 201
Tree Projections and Structural Decomposition Methods: Minimality and Game-Theoretic Characterization
Tree projections provide a mathematical framework that encompasses all the
various (purely) structural decomposition methods that have been proposed in
the literature to single out classes of nearly-acyclic (hyper)graphs, such as
the tree decomposition method, which is the most powerful decomposition method
on graphs, and the (generalized) hypertree decomposition method, which is its
natural counterpart on arbitrary hypergraphs. The paper analyzes this
framework, by focusing in particular on "minimal" tree projections, that is, on
tree projections without useless redundancies. First, it is shown that minimal
tree projections enjoy a number of properties that are usually required for
normal form decompositions in various structural decomposition methods. In
particular, they enjoy the same kind of connection properties as (minimal) tree
decompositions of graphs, with the result being tight in the light of the
negative answer that is provided to the open question about whether they enjoy
a slightly stronger notion of connection property, defined to speed-up the
computation of hypertree decompositions. Second, it is shown that tree
projections admit a natural game-theoretic characterization in terms of the
Captain and Robber game. In this game, as for the Robber and Cops game
characterizing tree decompositions, the existence of winning strategies implies
the existence of monotone ones. As a special case, the Captain and Robber game
can be used to characterize the generalized hypertree decomposition method,
where such a game-theoretic characterization was missing and asked for. Besides
their theoretical interest, these results have immediate algorithmic
applications both for the general setting and for structural decomposition
methods that can be recast in terms of tree projections
Communicating the sum of sources over a network
We consider the network communication scenario, over directed acyclic
networks with unit capacity edges in which a number of sources each
holding independent unit-entropy information wish to communicate the sum
to a set of terminals . We show that in the case in which
there are only two sources or only two terminals, communication is possible if
and only if each source terminal pair is connected by at least a
single path. For the more general communication problem in which there are
three sources and three terminals, we prove that a single path connecting the
source terminal pairs does not suffice to communicate . We then
present an efficient encoding scheme which enables the communication of
for the three sources, three terminals case, given that each source
terminal pair is connected by {\em two} edge disjoint paths.Comment: 12 pages, IEEE JSAC: Special Issue on In-network
Computation:Exploring the Fundamental Limits (to appear
Topological Additive Numbering of Directed Acyclic Graphs
We propose to study a problem that arises naturally from both Topological
Numbering of Directed Acyclic Graphs, and Additive Coloring (also known as
Lucky Labeling). Let be a digraph and a labeling of its vertices with
positive integers; denote by the sum of labels over all neighbors of
each vertex . The labeling is called \emph{topological additive
numbering} if for each arc of the digraph. The problem
asks to find the minimum number for which has a topological additive
numbering with labels belonging to , denoted by
.
We characterize when a digraph has topological additive numberings, give a
lower bound for , and provide an integer programming formulation for
our problem, characterizing when its coefficient matrix is totally unimodular.
We also present some families for which can be computed in
polynomial time. Finally, we prove that this problem is \np-Hard even when its
input is restricted to planar bipartite digraphs
Distributed Broadcasting and Mapping Protocols in Directed Anonymous Networks
We initiate the study of distributed protocols over directed anonymous networks that are not necessarily strongly connected. In such networks, nodes are aware only of their incoming and outgoing edges, have no unique identity, and have no knowledge of the network topology or even bounds on its parameters, like the number of nodes or the network diameter. Anonymous networks are of interest in various settings such as wireless ad-hoc networks and peer to peer networks. Our goal is to create distributed protocols that reduce the uncertainty by distributing the knowledge of the network topology to all the nodes.
We consider two basic protocols: broadcasting and unique label assignment. These two protocols enable a complete mapping of the network and can serve as key building blocks in more advanced protocols. We develop distributed asynchronous protocols as well as derive lower bounds on their communication complexity, total bandwidth complexity, and node label complexity. The resulting lower bounds are sometimes surprisingly high, exhibiting the complexity of topology extraction in directed anonymous networks
Regular Languages meet Prefix Sorting
Indexing strings via prefix (or suffix) sorting is, arguably, one of the most
successful algorithmic techniques developed in the last decades. Can indexing
be extended to languages? The main contribution of this paper is to initiate
the study of the sub-class of regular languages accepted by an automaton whose
states can be prefix-sorted. Starting from the recent notion of Wheeler graph
[Gagie et al., TCS 2017]-which extends naturally the concept of prefix sorting
to labeled graphs-we investigate the properties of Wheeler languages, that is,
regular languages admitting an accepting Wheeler finite automaton.
Interestingly, we characterize this family as the natural extension of regular
languages endowed with the co-lexicographic ordering: when sorted, the strings
belonging to a Wheeler language are partitioned into a finite number of
co-lexicographic intervals, each formed by elements from a single Myhill-Nerode
equivalence class. Moreover: (i) We show that every Wheeler NFA (WNFA) with
states admits an equivalent Wheeler DFA (WDFA) with at most
states that can be computed in time. This is in sharp contrast with
general NFAs. (ii) We describe a quadratic algorithm to prefix-sort a proper
superset of the WDFAs, a -time online algorithm to sort acyclic
WDFAs, and an optimal linear-time offline algorithm to sort general WDFAs. By
contribution (i), our algorithms can also be used to index any WNFA at the
moderate price of doubling the automaton's size. (iii) We provide a
minimization theorem that characterizes the smallest WDFA recognizing the same
language of any input WDFA. The corresponding constructive algorithm runs in
optimal linear time in the acyclic case, and in time in the
general case. (iv) We show how to compute the smallest WDFA equivalent to any
acyclic DFA in nearly-optimal time.Comment: added minimization theorems; uploaded submitted version; New version
with new results (W-MH theorem, linear determinization), added author:
Giovanna D'Agostin
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