19 research outputs found
Exact Algorithms for B-Bandwidth Problem with Restricted B
The B-BANDWIDTH problem is a decision problem whether the bandwidth of a given graph is smaller than B, and it is NP-complete even if the graph is a small graph class of trees. Cygan and Pilipczuk proposed exponential time and space algorithms for B-BANDWIDTH with n/3 ≤ B where n is the number of vertices. In this paper, we propose two algorithms for the B-BANDWIDTH problem with n/4 ≤ B < n/3. These algorithms are extension of Cygan and Pilipczuk algorithms with restricted B. One of the algorithms takes O∗(4.5n) time and O∗(1.5n) space when n/4 ≤ B < n / 2 log2 1.5, and the other takes O∗(4.77n) time and O∗(1.59n) space when n / 2 log2 1.5 ≤ B < n/3. Our algorithms are fastest O∗(2n) space algorithms for n/4 ≤B < n/3.The 17th Korea-Japan Joint Workshop on Algorithms and Computation, July 13-15, 2014, Okinawa, Japa
Revisiting Interval Graphs for Network Science
The vertices of an interval graph represent intervals over a real line where
overlapping intervals denote that their corresponding vertices are adjacent.
This implies that the vertices are measurable by a metric and there exists a
linear structure in the system. The generalization is an embedding of a graph
onto a multi-dimensional Euclidean space and it was used by scientists to study
the multi-relational complexity of ecology. However the research went out of
fashion in the 1980s and was not revisited when Network Science recently
expressed interests with multi-relational networks known as multiplexes. This
paper studies interval graphs from the perspective of Network Science
A Fast Minimum Degree Algorithm and Matching Lower Bound
The minimum degree algorithm is one of the most widely-used heuristics for
reducing the cost of solving large sparse systems of linear equations. It has
been studied for nearly half a century and has a rich history of bridging
techniques from data structures, graph algorithms, and scientific computing. In
this paper, we present a simple but novel combinatorial algorithm for computing
an exact minimum degree elimination ordering in time, which improves on
the best known time complexity of and offers practical improvements
for sparse systems with small values of . Our approach leverages a careful
amortized analysis, which also allows us to derive output-sensitive bounds for
the running time of , where is
the number of unique fill edges and original edges that the algorithm
encounters and is the maximum degree of the input graph.
Furthermore, we show there cannot exist an exact minimum degree algorithm
that runs in time, for any , assuming
the strong exponential time hypothesis. This fine-grained reduction goes
through the orthogonal vectors problem and uses a new low-degree graph
construction called -fillers, which act as pathological inputs and cause any
minimum degree algorithm to exhibit nearly worst-case performance. With these
two results, we nearly characterize the time complexity of computing an exact
minimum degree ordering.Comment: 17 page
A Polynomial-time Algorithm for Outerplanar Diameter Improvement
The Outerplanar Diameter Improvement problem asks, given a graph and an
integer , whether it is possible to add edges to in a way that the
resulting graph is outerplanar and has diameter at most . We provide a
dynamic programming algorithm that solves this problem in polynomial time.
Outerplanar Diameter Improvement demonstrates several structural analogues to
the celebrated and challenging Planar Diameter Improvement problem, where the
resulting graph should, instead, be planar. The complexity status of this
latter problem is open.Comment: 24 page
Polynomial kernels for Proper Interval Completion and related problems
Given a graph G = (V,E) and a positive integer k, the Proper Interval
Completion problem asks whether there exists a set F of at most k pairs of (V
\times V)\E such that the graph H = (V,E \cup F) is a proper interval graph.
The Proper Interval Completion problem finds applications in molecular biology
and genomic research. First announced by Kaplan, Tarjan and Shamir in FOCS '94,
this problem is known to be FPT, but no polynomial kernel was known to exist.
We settle this question by proving that Proper Interval Completion admits a
kernel with at most O(k^5) vertices. Moreover, we prove that a related problem,
the so-called Bipartite Chain Deletion problem, admits a kernel with at most
O(k^2) vertices, completing a previous result of Guo
Unit Interval Editing is Fixed-Parameter Tractable
Given a graph~ and integers , , and~, the unit interval
editing problem asks whether can be transformed into a unit interval graph
by at most vertex deletions, edge deletions, and edge
additions. We give an algorithm solving this problem in time , where , and denote respectively
the numbers of vertices and edges of . Therefore, it is fixed-parameter
tractable parameterized by the total number of allowed operations.
Our algorithm implies the fixed-parameter tractability of the unit interval
edge deletion problem, for which we also present a more efficient algorithm
running in time . Another result is an -time algorithm for the unit interval vertex deletion problem,
significantly improving the algorithm of van 't Hof and Villanger, which runs
in time .Comment: An extended abstract of this paper has appeared in the proceedings of
ICALP 2015. Update: The proof of Lemma 4.2 has been completely rewritten; an
appendix is provided for a brief overview of related graph classe