475 research outputs found
Improved Approximation Algorithms for (Budgeted) Node-weighted Steiner Problems
Moss and Rabani[12] study constrained node-weighted Steiner tree problems
with two independent weight values associated with each node, namely, cost and
prize (or penalty). They give an O(log n)-approximation algorithm for the
prize-collecting node-weighted Steiner tree problem (PCST). They use the
algorithm for PCST to obtain a bicriteria (2, O(log n))-approximation algorithm
for the Budgeted node-weighted Steiner tree problem. Their solution may cost up
to twice the budget, but collects a factor Omega(1/log n) of the optimal prize.
We improve these results from at least two aspects.
Our first main result is a primal-dual O(log h)-approximation algorithm for a
more general problem, prize-collecting node-weighted Steiner forest, where we
have (h) demands each requesting the connectivity of a pair of vertices. Our
algorithm can be seen as a greedy algorithm which reduces the number of demands
by choosing a structure with minimum cost-to-reduction ratio. This natural
style of argument (also used by Klein and Ravi[10] and Guha et al.[8]) leads to
a much simpler algorithm than that of Moss and Rabani[12] for PCST.
Our second main contribution is for the Budgeted node-weighted Steiner tree
problem, which is also an improvement to [12] and [8]. In the unrooted case, we
improve upon an O(log^2(n))-approximation of [8], and present an O(log
n)-approximation algorithm without any budget violation. For the rooted case,
where a specified vertex has to appear in the solution tree, we improve the
bicriteria result of [12] to a bicriteria approximation ratio of (1+eps, O(log
n)/(eps^2)) for any positive (possibly subconstant) (eps). That is, for any
permissible budget violation (1+eps), we present an algorithm achieving a
tradeoff in the guarantee for prize. Indeed, we show that this is almost tight
for the natural linear-programming relaxation used by us as well as in [12].Comment: To appear in ICALP 201
Connectivity Oracles for Graphs Subject to Vertex Failures
We introduce new data structures for answering connectivity queries in graphs
subject to batched vertex failures. A deterministic structure processes a batch
of failed vertices in time and thereafter
answers connectivity queries in time. It occupies space . We develop a randomized Monte Carlo version of our data structure
with update time , query time , and space
for any failure bound . This is the first connectivity oracle for
general graphs that can efficiently deal with an unbounded number of vertex
failures.
We also develop a more efficient Monte Carlo edge-failure connectivity
oracle. Using space , edge failures are processed in time and thereafter, connectivity queries are answered in
time, which are correct w.h.p.
Our data structures are based on a new decomposition theorem for an
undirected graph , which is of independent interest. It states that
for any terminal set we can remove a set of
vertices such that the remaining graph contains a Steiner forest for with
maximum degree
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
A Tight Bound for Shortest Augmenting Paths on Trees
The shortest augmenting path technique is one of the fundamental ideas used
in maximum matching and maximum flow algorithms. Since being introduced by
Edmonds and Karp in 1972, it has been widely applied in many different
settings. Surprisingly, despite this extensive usage, it is still not well
understood even in the simplest case: online bipartite matching problem on
trees. In this problem a bipartite tree is being revealed
online, i.e., in each round one vertex from with its incident edges
arrives. It was conjectured by Chaudhuri et. al. [K. Chaudhuri, C. Daskalakis,
R. D. Kleinberg, and H. Lin. Online bipartite perfect matching with
augmentations. In INFOCOM 2009] that the total length of all shortest
augmenting paths found is . In this paper, we prove a tight upper bound for the total length of shortest augmenting paths for
trees improving over bound [B. Bosek, D. Leniowski, P.
Sankowski, and A. Zych. Shortest augmenting paths for online matchings on
trees. In WAOA 2015].Comment: 22 pages, 10 figure
Parameterized (in)approximability of subset problems
We discuss approximability and inapproximability in FPT-time for a large
class of subset problems where a feasible solution is a subset of the input
data and the value of is . The class handled encompasses many
well-known graph, set, or satisfiability problems such as Dominating Set,
Vertex Cover, Set Cover, Independent Set, Feedback Vertex Set, etc. In a first
time, we introduce the notion of intersective approximability that generalizes
the one of safe approximability and show strong parameterized inapproximability
results for many of the subset problems handled. Then, we study approximability
of these problems with respect to the dual parameter where is the
size of the instance and the standard parameter. More precisely, we show
that under such a parameterization, many of these problems, while
W[]-hard, admit parameterized approximation schemata.Comment: 7 page
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