7,845 research outputs found
Data Reduction for Graph Coloring Problems
This paper studies the kernelization complexity of graph coloring problems
with respect to certain structural parameterizations of the input instances. We
are interested in how well polynomial-time data reduction can provably shrink
instances of coloring problems, in terms of the chosen parameter. It is well
known that deciding 3-colorability is already NP-complete, hence parameterizing
by the requested number of colors is not fruitful. Instead, we pick up on a
research thread initiated by Cai (DAM, 2003) who studied coloring problems
parameterized by the modification distance of the input graph to a graph class
on which coloring is polynomial-time solvable; for example parameterizing by
the number k of vertex-deletions needed to make the graph chordal. We obtain
various upper and lower bounds for kernels of such parameterizations of
q-Coloring, complementing Cai's study of the time complexity with respect to
these parameters.
Our results show that the existence of polynomial kernels for q-Coloring
parameterized by the vertex-deletion distance to a graph class F is strongly
related to the existence of a function f(q) which bounds the number of vertices
which are needed to preserve the NO-answer to an instance of q-List-Coloring on
F.Comment: Author-accepted manuscript of the article that will appear in the FCT
2011 special issue of Information & Computatio
Track Allocation in Freight-Train Classification with Mixed Tracks
We consider the process of forming outbound trains from cars of inbound trains at rail-freight
hump yards. Given the arrival and departure times as well as the composition of the trains, we
study the problem of allocating classification tracks to outbound trains such that every outbound
train can be built on a separate classification track. We observe that the core problem can be
formulated as a special list coloring problem in interval graphs, which is known to be NP-complete.
We focus on an extension where individual cars of different trains can temporarily be stored on
a special subset of the tracks. This problem induces several new variants of the list-coloring
problem, in which the given intervals can be shortened by cutting off a prefix of the interval. We
show that in case of uniform and sufficient track lengths, the corresponding coloring problem can
be solved in polynomial time, if the goal is to minimize the total cost associated with cutting off
prefixes of the intervals. Based on these results, we devise two heuristics as well as an integer
program to tackle the problem. As a case study, we consider a real-world problem instance from
the Hallsberg Rangerbangård hump yard in Sweden. Planning over horizons of seven days, we
obtain feasible solutions from the integer program in all scenarios, and from the heuristics in
most scenarios
Reverse Nearest Neighbor Heat Maps: A Tool for Influence Exploration
We study the problem of constructing a reverse nearest neighbor (RNN) heat
map by finding the RNN set of every point in a two-dimensional space. Based on
the RNN set of a point, we obtain a quantitative influence (i.e., heat) for the
point. The heat map provides a global view on the influence distribution in the
space, and hence supports exploratory analyses in many applications such as
marketing and resource management. To construct such a heat map, we first
reduce it to a problem called Region Coloring (RC), which divides the space
into disjoint regions within which all the points have the same RNN set. We
then propose a novel algorithm named CREST that efficiently solves the RC
problem by labeling each region with the heat value of its containing points.
In CREST, we propose innovative techniques to avoid processing expensive RNN
queries and greatly reduce the number of region labeling operations. We perform
detailed analyses on the complexity of CREST and lower bounds of the RC
problem, and prove that CREST is asymptotically optimal in the worst case.
Extensive experiments with both real and synthetic data sets demonstrate that
CREST outperforms alternative algorithms by several orders of magnitude.Comment: Accepted to appear in ICDE 201
Boundaries of Amplituhedra and NMHV Symbol Alphabets at Two Loops
In this sequel to arXiv:1711.11507 we classify the boundaries of amplituhedra
relevant for determining the branch points of general two-loop amplitudes in
planar super-Yang-Mills theory. We explain the connection to
on-shell diagrams, which serves as a useful cross-check. We determine the
branch points of all two-loop NMHV amplitudes by solving the Landau equations
for the relevant configurations and are led thereby to a conjecture for the
symbol alphabets of all such amplitudes.Comment: 42 pages, 6 figures, 8 tables; v2: minor corrections and improvement
Hump Yard Track Allocation with Temporary Car Storage
In rail freight operation, freight cars need to be separated and reformed into new trains at
hump yards. The classification procedure is complex and hump yards constitute bottlenecks
in the rail freight network, often causing outbound trains to be delayed. One of the problems
is that planning for the allocation of tracks at hump yards is difficult, given that the planner
has limited resources (tracks, shunting engines, etc.) and needs to foresee the future capacity
requirements when planning for the current inbound trains. In this paper, we consider
the problem of allocating classification tracks in a rail freight hump yard for arriving and
departing trains with predetermined arrival and departure times. The core problem can be
formulated as a special list coloring problem. We focus on an extension where individual
cars can temporarily be stored on a special subset of the tracks. An extension where individual
cars can temporarily be stored on a special subset of the tracks is also considered. We
model the problem using mixed integer programming, and also propose several heuristics
that can quickly give feasible track allocations. As a case study, we consider a real-world
problem instance from the Hallsberg Rangerbangård hump yard in Sweden. Planning over
horizons over two to four days, we obtain feasible solutions from both the exact and heuristic
approaches that allow all outgoing trains to leave on time
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