162 research outputs found
On the approximability and exact algorithms for vector domination and related problems in graphs
We consider two graph optimization problems called vector domination and
total vector domination. In vector domination one seeks a small subset S of
vertices of a graph such that any vertex outside S has a prescribed number of
neighbors in S. In total vector domination, the requirement is extended to all
vertices of the graph. We prove that these problems (and several variants
thereof) cannot be approximated to within a factor of clnn, where c is a
suitable constant and n is the number of the vertices, unless P = NP. We also
show that two natural greedy strategies have approximation factors ln D+O(1),
where D is the maximum degree of the input graph. We also provide exact
polynomial time algorithms for several classes of graphs. Our results extend,
improve, and unify several results previously known in the literature.Comment: In the version published in DAM, weaker lower bounds for vector
domination and total vector domination were stated. Being these problems
generalization of domination and total domination, the lower bounds of 0.2267
ln n and (1-epsilon) ln n clearly hold for both problems, unless P = NP or NP
\subseteq DTIME(n^{O(log log n)}), respectively. The claims are corrected in
the present versio
Graph Algorithms and Applications
The mixture of data in real-life exhibits structure or connection property in nature. Typical data include biological data, communication network data, image data, etc. Graphs provide a natural way to represent and analyze these types of data and their relationships. Unfortunately, the related algorithms usually suffer from high computational complexity, since some of these problems are NP-hard. Therefore, in recent years, many graph models and optimization algorithms have been proposed to achieve a better balance between efficacy and efficiency. This book contains some papers reporting recent achievements regarding graph models, algorithms, and applications to problems in the real world, with some focus on optimization and computational complexity
Revisiting Area Convexity: Faster Box-Simplex Games and Spectrahedral Generalizations
We investigate different aspects of area convexity [Sherman '17], a
mysterious tool introduced to tackle optimization problems under the
challenging geometry. We develop a deeper understanding of its
relationship with more conventional analyses of extragradient methods
[Nemirovski '04, Nesterov '07]. We also give improved solvers for the
subproblems required by variants of the [Sherman '17] algorithm, designed
through the lens of relative smoothness [Bauschke-Bolte-Teboulle '17,
Lu-Freund-Nesterov '18].
Leveraging these new tools, we give a state-of-the-art first-order algorithm
for solving box-simplex games (a primal-dual formulation of
regression) in a matrix with bounded rows, using matrix-vector queries. As a consequence, we obtain improved
complexities for approximate maximum flow, optimal transport, min-mean-cycle,
and other basic combinatorial optimization problems. We also develop a
near-linear time algorithm for a matrix generalization of box-simplex games,
capturing a family of problems closely related to semidefinite programs
recently used as subroutines in robust statistics and numerical linear algebra
- …