6,517 research outputs found
Algorithms and Hardness for Diameter in Dynamic Graphs
The diameter, radius and eccentricities are natural graph parameters. While these problems have been studied extensively, there are no known dynamic algorithms for them beyond the ones that follow from trivial recomputation after each update or from solving dynamic All-Pairs Shortest Paths (APSP), which is very computationally intensive. This is the situation for dynamic approximation algorithms as well, and even if only edge insertions or edge deletions need to be supported.
This paper provides a comprehensive study of the dynamic approximation of Diameter, Radius and Eccentricities, providing both conditional lower bounds, and new algorithms whose bounds are optimal under popular hypotheses in fine-grained complexity. Some of the highlights include:
- Under popular hardness hypotheses, there can be no significantly better fully dynamic approximation algorithms than recomputing the answer after each update, or maintaining full APSP.
- Nearly optimal partially dynamic (incremental/decremental) algorithms can be achieved via efficient reductions to (incremental/decremental) maintenance of Single-Source Shortest Paths. For instance, a nearly (3/2+epsilon)-approximation to Diameter in directed or undirected n-vertex, m-edge graphs can be maintained decrementally in total time m^{1+o(1)}sqrt{n}/epsilon^2. This nearly matches the static 3/2-approximation algorithm for the problem that is known to be conditionally optimal
On the Hardness of Partially Dynamic Graph Problems and Connections to Diameter
Conditional lower bounds for dynamic graph problems has received a great deal
of attention in recent years. While many results are now known for the
fully-dynamic case and such bounds often imply worst-case bounds for the
partially dynamic setting, it seems much more difficult to prove amortized
bounds for incremental and decremental algorithms. In this paper we consider
partially dynamic versions of three classic problems in graph theory. Based on
popular conjectures we show that:
-- No algorithm with amortized update time exists for
incremental or decremental maximum cardinality bipartite matching. This
significantly improves on the bound for sparse graphs
of Henzinger et al. [STOC'15] and bound of Kopelowitz,
Pettie and Porat. Our linear bound also appears more natural. In addition, the
result we present separates the node-addition model from the edge insertion
model, as an algorithm with total update time exists for the
former by Bosek et al. [FOCS'14].
-- No algorithm with amortized update time exists for
incremental or decremental maximum flow in directed and weighted sparse graphs.
No such lower bound was known for partially dynamic maximum flow previously.
Furthermore no algorithm with amortized update time
exists for directed and unweighted graphs or undirected and weighted graphs.
-- No algorithm with amortized update time exists
for incremental or decremental -approximating the diameter
of an unweighted graph. We also show a slightly stronger bound if node
additions are allowed. [...]Comment: To appear at ICALP'16. Abstract truncated to fit arXiv limit
Fully polynomial FPT algorithms for some classes of bounded clique-width graphs
Parameterized complexity theory has enabled a refined classification of the
difficulty of NP-hard optimization problems on graphs with respect to key
structural properties, and so to a better understanding of their true
difficulties. More recently, hardness results for problems in P were achieved
using reasonable complexity theoretic assumptions such as: Strong Exponential
Time Hypothesis (SETH), 3SUM and All-Pairs Shortest-Paths (APSP). According to
these assumptions, many graph theoretic problems do not admit truly
subquadratic algorithms, nor even truly subcubic algorithms (Williams and
Williams, FOCS 2010 and Abboud, Grandoni, Williams, SODA 2015). A central
technique used to tackle the difficulty of the above mentioned problems is
fixed-parameter algorithms for polynomial-time problems with polynomial
dependency in the fixed parameter (P-FPT). This technique was introduced by
Abboud, Williams and Wang in SODA 2016 and continued by Husfeldt (IPEC 2016)
and Fomin et al. (SODA 2017), using the treewidth as a parameter. Applying this
technique to clique-width, another important graph parameter, remained to be
done. In this paper we study several graph theoretic problems for which
hardness results exist such as cycle problems (triangle detection, triangle
counting, girth, diameter), distance problems (diameter, eccentricities, Gromov
hyperbolicity, betweenness centrality) and maximum matching. We provide
hardness results and fully polynomial FPT algorithms, using clique-width and
some of its upper-bounds as parameters (split-width, modular-width and
-sparseness). We believe that our most important result is an -time algorithm for computing a maximum matching where
is either the modular-width or the -sparseness. The latter generalizes
many algorithms that have been introduced so far for specific subclasses such
as cographs, -lite graphs, -extendible graphs and -tidy
graphs. Our algorithms are based on preprocessing methods using modular
decomposition, split decomposition and primeval decomposition. Thus they can
also be generalized to some graph classes with unbounded clique-width
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Variants of Plane Diameter Completion
The {\sc Plane Diameter Completion} problem asks, given a plane graph and
a positive integer , if it is a spanning subgraph of a plane graph that
has diameter at most . We examine two variants of this problem where the
input comes with another parameter . In the first variant, called BPDC,
upper bounds the total number of edges to be added and in the second, called
BFPDC, upper bounds the number of additional edges per face. We prove that
both problems are {\sf NP}-complete, the first even for 3-connected graphs of
face-degree at most 4 and the second even when on 3-connected graphs of
face-degree at most 5. In this paper we give parameterized algorithms for both
problems that run in steps.Comment: Accepted in IPEC 201
Notes on complexity of packing coloring
A packing -coloring for some integer of a graph is a mapping
such that any two vertices of color
are in distance at least . This concept
is motivated by frequency assignment problems. The \emph{packing chromatic
number} of is the smallest such that there exists a packing
-coloring of .
Fiala and Golovach showed that determining the packing chromatic number for
chordal graphs is \NP-complete for diameter exactly 5. While the problem is
easy to solve for diameter 2, we show \NP-completeness for any diameter at
least 3. Our reduction also shows that the packing chromatic number is hard to
approximate within for any .
In addition, we design an \FPT algorithm for interval graphs of bounded
diameter. This leads us to exploring the problem of finding a partial coloring
that maximizes the number of colored vertices.Comment: 9 pages, 2 figure
Spanning trees short or small
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number of nodes are required to be connected in the solution. A
prototypical example is the MST problem in which we require a tree of
minimum weight spanning at least nodes in an edge-weighted graph. We show
that the MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio for the
general edge-weighted case and for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding -trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.Comment: 27 page
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