233 research outputs found
Functional description of MINTO, a mixed integer optimizer
MINTO is a software system that solves mixed-integer linear programs by a branch-and-bound algorithm with linear programming relaxations. It also provides automatic constraint classification, preprocessing, primal heuristics and constraint generation. Moreover, the user can enrich the basic algorithm by providing a variety of specialized application routines that can customize MINTO to achieve maximum efficiency for a problem class. This paper documents MINTO by specifying what it is capable of doing and how to use it
Budget-restricted utility games with ordered strategic decisions
We introduce the concept of budget games. Players choose a set of tasks and
each task has a certain demand on every resource in the game. Each resource has
a budget. If the budget is not enough to satisfy the sum of all demands, it has
to be shared between the tasks. We study strategic budget games, where the
budget is shared proportionally. We also consider a variant in which the order
of the strategic decisions influences the distribution of the budgets. The
complexity of the optimal solution as well as existence, complexity and quality
of equilibria are analyzed. Finally, we show that the time an ordered budget
game needs to convergence towards an equilibrium may be exponential
Approximation Algorithms for Connected Maximum Cut and Related Problems
An instance of the Connected Maximum Cut problem consists of an undirected
graph G = (V, E) and the goal is to find a subset of vertices S V
that maximizes the number of edges in the cut \delta(S) such that the induced
graph G[S] is connected. We present the first non-trivial \Omega(1/log n)
approximation algorithm for the connected maximum cut problem in general graphs
using novel techniques. We then extend our algorithm to an edge weighted case
and obtain a poly-logarithmic approximation algorithm. Interestingly, in stark
contrast to the classical max-cut problem, we show that the connected maximum
cut problem remains NP-hard even on unweighted, planar graphs. On the positive
side, we obtain a polynomial time approximation scheme for the connected
maximum cut problem on planar graphs and more generally on graphs with bounded
genus.Comment: 17 pages, Conference version to appear in ESA 201
Thresholded Covering Algorithms for Robust and Max-Min Optimization
The general problem of robust optimization is this: one of several possible
scenarios will appear tomorrow, but things are more expensive tomorrow than
they are today. What should you anticipatorily buy today, so that the
worst-case cost (summed over both days) is minimized? Feige et al. and
Khandekar et al. considered the k-robust model where the possible outcomes
tomorrow are given by all demand-subsets of size k, and gave algorithms for the
set cover problem, and the Steiner tree and facility location problems in this
model, respectively.
In this paper, we give the following simple and intuitive template for
k-robust problems: "having built some anticipatory solution, if there exists a
single demand whose augmentation cost is larger than some threshold, augment
the anticipatory solution to cover this demand as well, and repeat". In this
paper we show that this template gives us improved approximation algorithms for
k-robust Steiner tree and set cover, and the first approximation algorithms for
k-robust Steiner forest, minimum-cut and multicut. All our approximation ratios
(except for multicut) are almost best possible.
As a by-product of our techniques, we also get algorithms for max-min
problems of the form: "given a covering problem instance, which k of the
elements are costliest to cover?".Comment: 24 page
Constrained Non-Monotone Submodular Maximization: Offline and Secretary Algorithms
Constrained submodular maximization problems have long been studied, with
near-optimal results known under a variety of constraints when the submodular
function is monotone. The case of non-monotone submodular maximization is less
understood: the first approximation algorithms even for the unconstrainted
setting were given by Feige et al. (FOCS '07). More recently, Lee et al. (STOC
'09, APPROX '09) show how to approximately maximize non-monotone submodular
functions when the constraints are given by the intersection of p matroid
constraints; their algorithm is based on local-search procedures that consider
p-swaps, and hence the running time may be n^Omega(p), implying their algorithm
is polynomial-time only for constantly many matroids. In this paper, we give
algorithms that work for p-independence systems (which generalize constraints
given by the intersection of p matroids), where the running time is poly(n,p).
Our algorithm essentially reduces the non-monotone maximization problem to
multiple runs of the greedy algorithm previously used in the monotone case.
Our idea of using existing algorithms for monotone functions to solve the
non-monotone case also works for maximizing a submodular function with respect
to a knapsack constraint: we get a simple greedy-based constant-factor
approximation for this problem.
With these simpler algorithms, we are able to adapt our approach to
constrained non-monotone submodular maximization to the (online) secretary
setting, where elements arrive one at a time in random order, and the algorithm
must make irrevocable decisions about whether or not to select each element as
it arrives. We give constant approximations in this secretary setting when the
algorithm is constrained subject to a uniform matroid or a partition matroid,
and give an O(log k) approximation when it is constrained by a general matroid
of rank k.Comment: In the Proceedings of WINE 201
Tight Kernel Bounds for Problems on Graphs with Small Degeneracy
In this paper we consider kernelization for problems on d-degenerate graphs,
i.e. graphs such that any subgraph contains a vertex of degree at most .
This graph class generalizes many classes of graphs for which effective
kernelization is known to exist, e.g. planar graphs, H-minor free graphs, and
H-topological-minor free graphs. We show that for several natural problems on
d-degenerate graphs the best known kernelization upper bounds are essentially
tight.Comment: Full version of ESA 201
On k-Column Sparse Packing Programs
We consider the class of packing integer programs (PIPs) that are column
sparse, i.e. there is a specified upper bound k on the number of constraints
that each variable appears in. We give an (ek+o(k))-approximation algorithm for
k-column sparse PIPs, improving on recent results of and
. We also show that the integrality gap of our linear programming
relaxation is at least 2k-1; it is known that k-column sparse PIPs are
-hard to approximate. We also extend our result (at the loss
of a small constant factor) to the more general case of maximizing a submodular
objective over k-column sparse packing constraints.Comment: 19 pages, v3: additional detail
On the equivalence of strong formulations for capacitated multi-level lot sizing problems with setup times
Several mixed integer programming formulations have been proposed for modeling capacitated multi-level lot sizing problems with setup times. These formulations include the so-called facility location formulation, the shortest route formulation, and the inventory and lot sizing formulation with (l,S) inequalities. In this paper, we demonstrate the equivalence of these formulations when the integrality requirement is relaxed for any subset of binary setup decision variables. This equivalence has significant implications for decomposition-based methods since same optimal solution values are obtained no matter which formulation is used. In particular, we discuss the relax-and-fix method, a decomposition-based heuristic used for the efficient solution of hard lot sizing problems. Computational tests allow us to compare the effectiveness of different formulations using benchmark problems. The choice of formulation directly affects the required computational effort, and our results therefore provide guidelines on choosing an effective formulation during the development of heuristic-based solution procedures
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