20,028 research outputs found
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
The Stable Roommates problem with short lists
We consider two variants of the classical Stable Roommates problem with Incomplete (but strictly ordered) preference lists (sri) that are degree constrained, i.e., preference lists are of bounded length. The first variant, egald-sri, involves finding an egalitarian stable matching in solvable instances of sri with preference lists of length at most d. We show that this problem is NP-hard even if d = 3. On the positive side we give a 2d+372d+37-approximation algorithm for d ∈{3,4,5} which improves on the known bound of 2 for the unbounded preference list case. In the second variant of sri, called d-srti, preference lists can include ties and are of length at most d. We show that the problem of deciding whether an instance of d-srti admits a stable matching is NP-complete even if d = 3. We also consider the “most stable” version of this problem and prove a strong inapproximability bound for the d = 3 case. However for d = 2 we show that the latter problem can be solved in polynomial time
The Stable Roommates problem with short lists
We consider two variants of the classical Stable Roommates problem with
Incomplete (but strictly ordered) preference lists SRI that are degree
constrained, i.e., preference lists are of bounded length. The first variant,
EGAL d-SRI, involves finding an egalitarian stable matching in solvable
instances of SRI with preference lists of length at most d. We show that this
problem is NP-hard even if d=3. On the positive side we give a
(2d+3)/7-approximation algorithm for d={3,4,5} which improves on the known
bound of 2 for the unbounded preference list case. In the second variant of
SRI, called d-SRTI, preference lists can include ties and are of length at most
d. We show that the problem of deciding whether an instance of d-SRTI admits a
stable matching is NP-complete even if d=3. We also consider the "most stable"
version of this problem and prove a strong inapproximability bound for the d=3
case. However for d=2 we show that the latter problem can be solved in
polynomial time.Comment: short version appeared at SAGT 201
Travelling on Graphs with Small Highway Dimension
We study the Travelling Salesperson (TSP) and the Steiner Tree problem (STP)
in graphs of low highway dimension. This graph parameter was introduced by
Abraham et al. [SODA 2010] as a model for transportation networks, on which TSP
and STP naturally occur for various applications in logistics. It was
previously shown [Feldmann et al. ICALP 2015] that these problems admit a
quasi-polynomial time approximation scheme (QPTAS) on graphs of constant
highway dimension. We demonstrate that a significant improvement is possible in
the special case when the highway dimension is 1, for which we present a
fully-polynomial time approximation scheme (FPTAS). We also prove that STP is
weakly NP-hard for these restricted graphs. For TSP we show NP-hardness for
graphs of highway dimension 6, which answers an open problem posed in [Feldmann
et al. ICALP 2015]
Single machine scheduling problems with uncertain parameters and the OWA criterion
In this paper a class of single machine scheduling problems is discussed. It
is assumed that job parameters, such as processing times, due dates, or weights
are uncertain and their values are specified in the form of a discrete scenario
set. The Ordered Weighted Averaging (OWA) aggregation operator is used to
choose an optimal schedule. The OWA operator generalizes traditional criteria
in decision making under uncertainty, such as the maximum, average, median or
Hurwicz criterion. It also allows us to extend the robust approach to
scheduling by taking into account various attitudes of decision makers towards
the risk. In this paper a general framework for solving single machine
scheduling problems with the OWA criterion is proposed and some positive and
negative computational results for two basic single machine scheduling problems
are provided
On the approximability of robust spanning tree problems
In this paper the minimum spanning tree problem with uncertain edge costs is
discussed. In order to model the uncertainty a discrete scenario set is
specified and a robust framework is adopted to choose a solution. The min-max,
min-max regret and 2-stage min-max versions of the problem are discussed. The
complexity and approximability of all these problems are explored. It is proved
that the min-max and min-max regret versions with nonnegative edge costs are
hard to approximate within for any unless
the problems in NP have quasi-polynomial time algorithms. Similarly, the
2-stage min-max problem cannot be approximated within unless the
problems in NP have quasi-polynomial time algorithms. In this paper randomized
LP-based approximation algorithms with performance ratio of for
min-max and 2-stage min-max problems are also proposed
Inapproximability of Combinatorial Optimization Problems
We survey results on the hardness of approximating combinatorial optimization
problems
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