10,328 research outputs found
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
More applications of the d-neighbor equivalence: acyclicity and connectivity constraints
In this paper, we design a framework to obtain efficient algorithms for
several problems with a global constraint (acyclicity or connectivity) such as
Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree,
Longest Induced Path, and Feedback Vertex Set. We design a meta-algorithm that
solves all these problems and whose running time is upper bounded by
, , and where is respectively the clique-width,
-rank-width, rank-width and maximum induced matching width of a
given decomposition. Our meta-algorithm simplifies and unifies the known
algorithms for each of the parameters and its running time matches
asymptotically also the running times of the best known algorithms for basic
NP-hard problems such as Vertex Cover and Dominating Set. Our framework is
based on the -neighbor equivalence defined in [Bui-Xuan, Telle and
Vatshelle, TCS 2013]. The results we obtain highlight the importance of this
equivalence relation on the algorithmic applications of width measures.
We also prove that our framework could be useful for -hard problems
parameterized by clique-width such as Max Cut and Maximum Minimal Cut. For
these latter problems, we obtain , and time
algorithms where is respectively the clique-width, the
-rank-width and the rank-width of the input graph
More Applications of the d-Neighbor Equivalence: Connectivity and Acyclicity Constraints
In this paper, we design a framework to obtain efficient algorithms for several problems with a global constraint (acyclicity or connectivity) such as Connected Dominating Set, Node Weighted Steiner Tree, Maximum Induced Tree, Longest Induced Path, and Feedback Vertex Set. For all these problems, we obtain 2^O(k)* n^O(1), 2^O(k log(k))* n^O(1), 2^O(k^2) * n^O(1) and n^O(k) time algorithms parameterized respectively by clique-width, Q-rank-width, rank-width and maximum induced matching width. Our approach simplifies and unifies the known algorithms for each of the parameters and match asymptotically also the running time of the best algorithms for basic NP-hard problems such as Vertex Cover and Dominating Set. Our framework is based on the d-neighbor equivalence defined in [Bui-Xuan, Telle and Vatshelle, TCS 2013]. The results we obtain highlight the importance and the generalizing power of this equivalence relation on width measures. We also prove that this equivalence relation could be useful for Max Cut: a W[1]-hard problem parameterized by clique-width. For this latter problem, we obtain n^O(k), n^O(k) and n^(2^O(k)) time algorithm parameterized by clique-width, Q-rank-width and rank-width
The Price of Information in Combinatorial Optimization
Consider a network design application where we wish to lay down a
minimum-cost spanning tree in a given graph; however, we only have stochastic
information about the edge costs. To learn the precise cost of any edge, we
have to conduct a study that incurs a price. Our goal is to find a spanning
tree while minimizing the disutility, which is the sum of the tree cost and the
total price that we spend on the studies. In a different application, each edge
gives a stochastic reward value. Our goal is to find a spanning tree while
maximizing the utility, which is the tree reward minus the prices that we pay.
Situations such as the above two often arise in practice where we wish to
find a good solution to an optimization problem, but we start with only some
partial knowledge about the parameters of the problem. The missing information
can be found only after paying a probing price, which we call the price of
information. What strategy should we adopt to optimize our expected
utility/disutility?
A classical example of the above setting is Weitzman's "Pandora's box"
problem where we are given probability distributions on values of
independent random variables. The goal is to choose a single variable with a
large value, but we can find the actual outcomes only after paying a price. Our
work is a generalization of this model to other combinatorial optimization
problems such as matching, set cover, facility location, and prize-collecting
Steiner tree. We give a technique that reduces such problems to their non-price
counterparts, and use it to design exact/approximation algorithms to optimize
our utility/disutility. Our techniques extend to situations where there are
additional constraints on what parameters can be probed or when we can
simultaneously probe a subset of the parameters.Comment: SODA 201
On Structural Parameterizations of Hitting Set: Hitting Paths in Graphs Using 2-SAT
Hitting Set is a classic problem in combinatorial optimization. Its input
consists of a set system F over a finite universe U and an integer t; the
question is whether there is a set of t elements that intersects every set in
F. The Hitting Set problem parameterized by the size of the solution is a
well-known W[2]-complete problem in parameterized complexity theory. In this
paper we investigate the complexity of Hitting Set under various structural
parameterizations of the input. Our starting point is the folklore result that
Hitting Set is polynomial-time solvable if there is a tree T on vertex set U
such that the sets in F induce connected subtrees of T. We consider the case
that there is a treelike graph with vertex set U such that the sets in F induce
connected subgraphs; the parameter of the problem is a measure of how treelike
the graph is. Our main positive result is an algorithm that, given a graph G
with cyclomatic number k, a collection P of simple paths in G, and an integer
t, determines in time 2^{5k} (|G| +|P|)^O(1) whether there is a vertex set of
size t that hits all paths in P. It is based on a connection to the 2-SAT
problem in multiple valued logic. For other parameterizations we derive
W[1]-hardness and para-NP-completeness results.Comment: Presented at the 41st International Workshop on Graph-Theoretic
Concepts in Computer Science, WG 2015. (The statement of Lemma 4 was
corrected in this update.
Hardness of Vertex Deletion and Project Scheduling
Assuming the Unique Games Conjecture, we show strong inapproximability
results for two natural vertex deletion problems on directed graphs: for any
integer and arbitrary small , the Feedback Vertex Set
problem and the DAG Vertex Deletion problem are inapproximable within a factor
even on graphs where the vertices can be almost partitioned into
solutions. This gives a more structured and therefore stronger UGC-based
hardness result for the Feedback Vertex Set problem that is also simpler
(albeit using the "It Ain't Over Till It's Over" theorem) than the previous
hardness result.
In comparison to the classical Feedback Vertex Set problem, the DAG Vertex
Deletion problem has received little attention and, although we think it is a
natural and interesting problem, the main motivation for our inapproximability
result stems from its relationship with the classical Discrete Time-Cost
Tradeoff Problem. More specifically, our results imply that the deadline
version is NP-hard to approximate within any constant assuming the Unique Games
Conjecture. This explains the difficulty in obtaining good approximation
algorithms for that problem and further motivates previous alternative
approaches such as bicriteria approximations.Comment: 18 pages, 1 figur
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