12,637 research outputs found
Design by Measure and Conquer, A Faster Exact Algorithm for Dominating Set
The measure and conquer approach has proven to be a powerful tool to analyse
exact algorithms for combinatorial problems, like Dominating Set and
Independent Set. In this paper, we propose to use measure and conquer also as a
tool in the design of algorithms. In an iterative process, we can obtain a
series of branch and reduce algorithms. A mathematical analysis of an algorithm
in the series with measure and conquer results in a quasiconvex programming
problem. The solution by computer to this problem not only gives a bound on the
running time, but also can give a new reduction rule, thus giving a new,
possibly faster algorithm. This makes design by measure and conquer a form of
computer aided algorithm design. When we apply the methodology to a Set Cover
modelling of the Dominating Set problem, we obtain the currently fastest known
exact algorithms for Dominating Set: an algorithm that uses time
and polynomial space, and an algorithm that uses time
Faster Graph Coloring in Polynomial Space
We present a polynomial-space algorithm that computes the number independent
sets of any input graph in time for graphs with maximum degree 3
and in time for general graphs, where n is the number of
vertices. Together with the inclusion-exclusion approach of Bj\"orklund,
Husfeldt, and Koivisto [SIAM J. Comput. 2009], this leads to a faster
polynomial-space algorithm for the graph coloring problem with running time
. As a byproduct, we also obtain an exponential-space
time algorithm for counting independent sets. Our main algorithm
counts independent sets in graphs with maximum degree 3 and no vertex with
three neighbors of degree 3. This polynomial-space algorithm is analyzed using
the recently introduced Separate, Measure and Conquer approach [Gaspers &
Sorkin, ICALP 2015]. Using Wahlstr\"om's compound measure approach, this
improvement in running time for small degree graphs is then bootstrapped to
larger degrees, giving the improvement for general graphs. Combining both
approaches leads to some inflexibility in choosing vertices to branch on for
the small-degree cases, which we counter by structural graph properties
Exact Algorithms for Maximum Independent Set
We show that the maximum independent set problem (MIS) on an -vertex graph
can be solved in time and polynomial space, which even is
faster than Robson's -time exponential-space algorithm
published in 1986. We also obtain improved algorithms for MIS in graphs with
maximum degree 6 and 7, which run in time of and
, respectively. Our algorithms are obtained by using fast
algorithms for MIS in low-degree graphs in a hierarchical way and making a
careful analyses on the structure of bounded-degree graphs
A Branch-and-Reduce Algorithm for Finding a Minimum Independent Dominating Set
An independent dominating set D of a graph G = (V,E) is a subset of vertices
such that every vertex in V \ D has at least one neighbor in D and D is an
independent set, i.e. no two vertices of D are adjacent in G. Finding a minimum
independent dominating set in a graph is an NP-hard problem. Whereas it is hard
to cope with this problem using parameterized and approximation algorithms,
there is a simple exact O(1.4423^n)-time algorithm solving the problem by
enumerating all maximal independent sets. In this paper we improve the latter
result, providing the first non trivial algorithm computing a minimum
independent dominating set of a graph in time O(1.3569^n). Furthermore, we give
a lower bound of \Omega(1.3247^n) on the worst-case running time of this
algorithm, showing that the running time analysis is almost tight.Comment: Full version. A preliminary version appeared in the proceedings of WG
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