5 research outputs found
Algorithms for the minimum sum coloring problem: a review
The Minimum Sum Coloring Problem (MSCP) is a variant of the well-known vertex
coloring problem which has a number of AI related applications. Due to its
theoretical and practical relevance, MSCP attracts increasing attention. The
only existing review on the problem dates back to 2004 and mainly covers the
history of MSCP and theoretical developments on specific graphs. In recent
years, the field has witnessed significant progresses on approximation
algorithms and practical solution algorithms. The purpose of this review is to
provide a comprehensive inspection of the most recent and representative MSCP
algorithms. To be informative, we identify the general framework followed by
practical solution algorithms and the key ingredients that make them
successful. By classifying the main search strategies and putting forward the
critical elements of the reviewed methods, we wish to encourage future
development of more powerful methods and motivate new applications
Any-Order Online Interval Selection
We consider the problem of online interval scheduling on a single machine,
where intervals arrive online in an order chosen by an adversary, and the
algorithm must output a set of non-conflicting intervals. Traditionally in
scheduling theory, it is assumed that intervals arrive in order of increasing
start times. We drop that assumption and allow for intervals to arrive in any
possible order. We call this variant any-order interval selection (AOIS). We
assume that some online acceptances can be revoked, but a feasible solution
must always be maintained. For unweighted intervals and deterministic
algorithms, this problem is unbounded. Under the assumption that there are at
most different interval lengths, we give a simple algorithm that achieves a
competitive ratio of and show that it is optimal amongst deterministic
algorithms, and a restricted class of randomized algorithms we call memoryless,
contributing to an open question by Adler and Azar 2003; namely whether a
randomized algorithm without access to history can achieve a constant
competitive ratio. We connect our model to the problem of call control on the
line, and show how the algorithms of Garay et al. 1997 can be applied to our
setting, resulting in an optimal algorithm for the case of proportional
weights. We also discuss the case of intervals with arbitrary weights, and show
how to convert the single-length algorithm of Fung et al. 2014 into a classify
and randomly select algorithm that achieves a competitive ratio of 2k. Finally,
we consider the case of intervals arriving in a random order, and show that for
single-lengthed instances, a one-directional algorithm (i.e. replacing
intervals in one direction), is the only deterministic memoryless algorithm
that can possibly benefit from random arrivals. Finally, we briefly discuss the
case of intervals with arbitrary weights.Comment: 19 pages, 11 figure
On Conceptually Simple Algorithms for Variants of Online Bipartite Matching
We present a series of results regarding conceptually simple algorithms for
bipartite matching in various online and related models. We first consider a
deterministic adversarial model. The best approximation ratio possible for a
one-pass deterministic online algorithm is , which is achieved by any
greedy algorithm. D\"urr et al. recently presented a -pass algorithm called
Category-Advice that achieves approximation ratio . We extend their
algorithm to multiple passes. We prove the exact approximation ratio for the
-pass Category-Advice algorithm for all , and show that the
approximation ratio converges to the inverse of the golden ratio
as goes to infinity. The convergence is
extremely fast --- the -pass Category-Advice algorithm is already within
of the inverse of the golden ratio.
We then consider a natural greedy algorithm in the online stochastic IID
model---MinDegree. This algorithm is an online version of a well-known and
extensively studied offline algorithm MinGreedy. We show that MinDegree cannot
achieve an approximation ratio better than , which is guaranteed by any
consistent greedy algorithm in the known IID model.
Finally, following the work in Besser and Poloczek, we depart from an
adversarial or stochastic ordering and investigate a natural randomized
algorithm (MinRanking) in the priority model. Although the priority model
allows the algorithm to choose the input ordering in a general but well defined
way, this natural algorithm cannot obtain the approximation of the Ranking
algorithm in the ROM model
On sum coloring and sum multi-coloring for restricted families of graphs
AbstractWe consider the sum coloring (chromatic sum) problem and the sum multi-coloring problem for restricted families of graphs. In particular, we consider the graph classes of proper intersection graphs of axis-parallel rectangles, proper interval graphs, and unit disk graphs. All the above-mentioned graph classes belong to a more general graph class of (k+1)-clawfree graphs (respectively, for k=4,2,5).We prove that sum coloring is NP-hard for penny graphs and unit square graphs which implies NP-hardness for unit disk graphs and proper intersection graphs of axis-parallel rectangles. We show a 2-approximation algorithm for unit square graphs, with the assumption that the geometric representation of the graph is given. For sum multi-coloring, we confirm that the greedy first-fit coloring, after ordering vertices by their demands, achieves a k-approximation for the preemptive version of sum multi-coloring on (k+1)-clawfree graphs. Finally, we study priority algorithms as a model for greedy algorithms for the sum coloring problem and the sum multi-coloring problem. We show various inapproximation results under several natural input representations