1,831 research outputs found
Parameterized Approximation Schemes using Graph Widths
Combining the techniques of approximation algorithms and parameterized
complexity has long been considered a promising research area, but relatively
few results are currently known. In this paper we study the parameterized
approximability of a number of problems which are known to be hard to solve
exactly when parameterized by treewidth or clique-width. Our main contribution
is to present a natural randomized rounding technique that extends well-known
ideas and can be used for both of these widths. Applying this very generic
technique we obtain approximation schemes for a number of problems, evading
both polynomial-time inapproximability and parameterized intractability bounds
A Time Hierarchy Theorem for the LOCAL Model
The celebrated Time Hierarchy Theorem for Turing machines states, informally,
that more problems can be solved given more time. The extent to which a time
hierarchy-type theorem holds in the distributed LOCAL model has been open for
many years. It is consistent with previous results that all natural problems in
the LOCAL model can be classified according to a small constant number of
complexities, such as , etc.
In this paper we establish the first time hierarchy theorem for the LOCAL
model and prove that several gaps exist in the LOCAL time hierarchy.
1. We define an infinite set of simple coloring problems called Hierarchical
-Coloring}. A correctly colored graph can be confirmed by simply
checking the neighborhood of each vertex, so this problem fits into the class
of locally checkable labeling (LCL) problems. However, the complexity of the
-level Hierarchical -Coloring problem is ,
for . The upper and lower bounds hold for both general graphs
and trees, and for both randomized and deterministic algorithms.
2. Consider any LCL problem on bounded degree trees. We prove an
automatic-speedup theorem that states that any randomized -time
algorithm solving the LCL can be transformed into a deterministic -time algorithm. Together with a previous result, this establishes that on
trees, there are no natural deterministic complexities in the ranges
--- or ---.
3. We expose a gap in the randomized time hierarchy on general graphs. Any
randomized algorithm that solves an LCL problem in sublogarithmic time can be
sped up to run in time, which is the complexity of the distributed
Lovasz local lemma problem, currently known to be and
Algorithms for Fast Aggregated Convergecast in Sensor Networks
Fast and periodic collection of aggregated data
is of considerable interest for mission-critical and continuous
monitoring applications in sensor networks. In the many-to-one
communication paradigm, referred to as convergecast, we focus
on applications wherein data packets are aggregated at each hop
en-route to the sink along a tree-based routing topology, and
address the problem of minimizing the convergecast schedule
length by utilizing multiple frequency channels. The primary
hindrance in minimizing the schedule length is the presence of
interfering links. We prove that it is NP-complete to determine
whether all the interfering links in an arbitrary network can
be removed using at most a constant number of frequencies.
We give a sufficient condition on the number of frequencies for
which all the interfering links can be removed, and propose a
polynomial time algorithm that minimizes the schedule length
in this case. We also prove that minimizing the schedule length
for a given number of frequencies on an arbitrary network is
NP-complete, and describe a greedy scheme that gives a constant
factor approximation on unit disk graphs. When the routing tree
is not given as an input to the problem, we prove that a constant
factor approximation is still achievable for degree-bounded trees.
Finally, we evaluate our algorithms through simulations and
compare their performance under different network parameters
Parameterized complexity of the MINCCA problem on graphs of bounded decomposability
In an edge-colored graph, the cost incurred at a vertex on a path when two
incident edges with different colors are traversed is called reload or
changeover cost. The "Minimum Changeover Cost Arborescence" (MINCCA) problem
consists in finding an arborescence with a given root vertex such that the
total changeover cost of the internal vertices is minimized. It has been
recently proved by G\"oz\"upek et al. [TCS 2016] that the problem is FPT when
parameterized by the treewidth and the maximum degree of the input graph. In
this article we present the following results for the MINCCA problem:
- the problem is W[1]-hard parameterized by the treedepth of the input graph,
even on graphs of average degree at most 8. In particular, it is W[1]-hard
parameterized by the treewidth of the input graph, which answers the main open
problem of G\"oz\"upek et al. [TCS 2016];
- it is W[1]-hard on multigraphs parameterized by the tree-cutwidth of the
input multigraph;
- it is FPT parameterized by the star tree-cutwidth of the input graph, which
is a slightly restricted version of tree-cutwidth. This result strictly
generalizes the FPT result given in G\"oz\"upek et al. [TCS 2016];
- it remains NP-hard on planar graphs even when restricted to instances with
at most 6 colors and 0/1 symmetric costs, or when restricted to instances with
at most 8 colors, maximum degree bounded by 4, and 0/1 symmetric costs.Comment: 25 pages, 11 figure
The Complexity of Distributed Edge Coloring with Small Palettes
The complexity of distributed edge coloring depends heavily on the palette
size as a function of the maximum degree . In this paper we explore the
complexity of edge coloring in the LOCAL model in different palette size
regimes.
1. We simplify the \emph{round elimination} technique of Brandt et al. and
prove that -edge coloring requires
time w.h.p. and time deterministically, even on trees.
The simplified technique is based on two ideas: the notion of an irregular
running time and some general observations that transform weak lower bounds
into stronger ones.
2. We give a randomized edge coloring algorithm that can use palette sizes as
small as , which is a natural barrier for
randomized approaches. The running time of the algorithm is at most
, where is the complexity of a
permissive version of the constructive Lovasz local lemma.
3. We develop a new distributed Lovasz local lemma algorithm for
tree-structured dependency graphs, which leads to a -edge
coloring algorithm for trees running in time. This algorithm
arises from two new results: a deterministic -time LLL algorithm for
tree-structured instances, and a randomized -time graph
shattering method for breaking the dependency graph into independent -size LLL instances.
4. A natural approach to computing -edge colorings (Vizing's
theorem) is to extend partial colorings by iteratively re-coloring parts of the
graph. We prove that this approach may be viable, but in the worst case
requires recoloring subgraphs of diameter . This stands
in contrast to distributed algorithms for Brooks' theorem, which exploit the
existence of -length augmenting paths
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