1,808 research outputs found
Interval scheduling and colorful independent sets
Numerous applications in scheduling, such as resource allocation or steel
manufacturing, can be modeled using the NP-hard Independent Set problem (given
an undirected graph and an integer k, find a set of at least k pairwise
non-adjacent vertices). Here, one encounters special graph classes like 2-union
graphs (edge-wise unions of two interval graphs) and strip graphs (edge-wise
unions of an interval graph and a cluster graph), on which Independent Set
remains NP-hard but admits constant-ratio approximations in polynomial time. We
study the parameterized complexity of Independent Set on 2-union graphs and on
subclasses like strip graphs. Our investigations significantly benefit from a
new structural "compactness" parameter of interval graphs and novel problem
formulations using vertex-colored interval graphs. Our main contributions are:
1. We show a complexity dichotomy: restricted to graph classes closed under
induced subgraphs and disjoint unions, Independent Set is polynomial-time
solvable if both input interval graphs are cluster graphs, and is NP-hard
otherwise.
2. We chart the possibilities and limits of effective polynomial-time
preprocessing (also known as kernelization).
3. We extend Halld\'orsson and Karlsson (2006)'s fixed-parameter algorithm
for Independent Set on strip graphs parameterized by the structural parameter
"maximum number of live jobs" to show that the problem (also known as Job
Interval Selection) is fixed-parameter tractable with respect to the parameter
k and generalize their algorithm from strip graphs to 2-union graphs.
Preliminary experiments with random data indicate that Job Interval Selection
with up to fifteen jobs and 5*10^5 intervals can be solved optimally in less
than five minutes.Comment: This revision does not contain Theorem 7 of the first revision, whose
proof contained an erro
Parameterized complexity of machine scheduling: 15 open problems
Machine scheduling problems are a long-time key domain of algorithms and
complexity research. A novel approach to machine scheduling problems are
fixed-parameter algorithms. To stimulate this thriving research direction, we
propose 15 open questions in this area whose resolution we expect to lead to
the discovery of new approaches and techniques both in scheduling and
parameterized complexity theory.Comment: Version accepted to Computers & Operations Researc
Finding Diverse Trees, Paths, and More
Mathematical modeling is a standard approach to solve many real-world
problems and {\em diversity} of solutions is an important issue, emerging in
applying solutions obtained from mathematical models to real-world problems.
Many studies have been devoted to finding diverse solutions. Baste et al.
(Algorithms 2019, IJCAI 2020) recently initiated the study of computing diverse
solutions of combinatorial problems from the perspective of fixed-parameter
tractability. They considered problems of finding solutions that maximize
some diversity measures (the minimum or sum of the pairwise Hamming distances
among them) and gave some fixed-parameter tractable algorithms for the diverse
version of several well-known problems, such as {\sc Vertex Cover}, {\sc
Feedback Vertex Set}, {\sc -Hitting Set}, and problems on bounded-treewidth
graphs. In this work, we investigate the (fixed-parameter) tractability of
problems of finding diverse spanning trees, paths, and several subgraphs. In
particular, we show that, given a graph and an integer , the problem of
computing spanning trees of maximizing the sum of the pairwise Hamming
distances among them can be solved in polynomial time. To the best of the
authors' knowledge, this is the first polynomial-time solvable case for finding
diverse solutions of unbounded size.Comment: 15 page
On the Fine-Grained Parameterized Complexity of Partial Scheduling to Minimize the Makespan
We study a natural variant of scheduling that we call partial scheduling: In this variant an instance of a scheduling problem along with an integer k is given and one seeks an optimal schedule where not all, but only k jobs, have to be processed.
Specifically, we aim to determine the fine-grained parameterized complexity of partial scheduling problems parameterized by k for all variants of scheduling problems that minimize the makespan and involve unit/arbitrary processing times, identical/unrelated parallel machines, release/due dates, and precedence constraints. That is, we investigate whether algorithms with runtimes of the type f(k)n^?(1) or n^?(f(k)) exist for a function f that is as small as possible.
Our contribution is two-fold: First, we categorize each variant to be either in ?, NP-complete and fixed-parameter tractable by k, or ?[1]-hard parameterized by k. Second, for many interesting cases we further investigate the run time on a finer scale and obtain run times that are (almost) optimal assuming the Exponential Time Hypothesis. As one of our main technical contributions, we give an ?(8^k k(|V|+|E|)) time algorithm to solve instances of partial scheduling problems minimizing the makespan with unit length jobs, precedence constraints and release dates, where G = (V,E) is the graph with precedence constraints
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