25 research outputs found
Robust Design of Single-Commodity Networks
The results in the present work were obtained in a collaboration with Eduardo Ălvarez-
Miranda, Valentina Cacchiani, Tim Dorneth, Michael JĂŒnger, Frauke Liers, Andrea Lodi
and Tiziano Parriani.
The subject of this thesis is a robust network design problem, i.e., a problem of the type
âdimension a network such that it has sufficient capacity in all likely scenarios.â In our case,
we model the network with an undirected graph in which each scenario defines a supply or
demand for each node. We say that a flow in the network is feasible for a scenario if it can
balance out its supplies and demands. A scenario polytope B defines which scenarios are
relevant. The task is now to find integer capacities that minimize the total installation costs
while allowing for a feasible flow in each scenario. This problem is called Single-Commodity
Robust Network Design Problem (sRND) and was introduced by Buchheim, Liers and SanitĂ
(INOC 2011). The problem contains the Steiner Tree Problem (given an undirected graph
and a terminal set, find a minimum cost subtree that connects all terminals) and therefore
is N P-hard. The problem is also a natural extension of minimum cost flows.
The network design literature treats the case that the scenario polytope B is given as
the finite set of its extreme points (finite case) and that it is given as the feasible region
of finitely many linear inequalities (polyhedral case). Both descriptions are equivalent,
however, an efficient transformation is not possible in general.
Buchheim, Liers and SanitĂ (INOC 2011) propose a Branch-and-Cut algorithm for the
finite case. In this case, there exists a canonical problem formulation as a mixed integer
linear program (MIP). It contains a set of flow variables for every scenario. Buchheim, Liers
and SanitĂ enhance the formulation with general cutting planes that are called target cuts.
The first part of the dissertation considers the problem variant where every scenario has
exactly two terminal nodes. If the underlying network is a complete, unweighted graph,
then this problem is the Network Synthesis Problem as defined by Chien (IBM Journal of
R&D 1960). There exist polynomial time algorithms by Gomory and Hu (SIAM J. of Appl.
Math 1961) and by Kabadi, Yan, Du and Nair (SIAM J. on Discr. Math.) for this special
case. However, these algorithms are based on the fact that complete graphs are Hamiltonian.
The result of this part is a similar algorithm for hypercube graphs that assumes a special
distribution of the supplies and demands. These graphs are also Hamiltonian.
The second part of the thesis discusses the structure of the polyhedron of feasible sRND
solutions. Here, the first result is a new MIP-based capacity formulation for the sRND
problem. The size of this formulation is independent of the number of extreme points
of B and therefore, it is also suited for the polyhedral case. The formulation uses so-called
cut-set inequalities that are known in similar form from other network design problems. By
adapting a proof by Mattia (Computational Optimization and Applications 2013), we show
that cut-set inequalities induce facets of the sRND polyhedron. To obtain a better linear
programming relaxation of the capacity formulation, we interpret certain general mixed
integer cuts as 3-partition inequalities and show that these inequalities induce facets as well.
The capacity formulation has exponential size and we therefore need a separation algorithm
for cut-set inequalities. In the finite case, we reduce the cut-set separation problem to
a minimum cut problem that can be solved in polynomial time. In the polyhedral case,
however, the separation problem is N P-hard, even if we assume that the scenario polytope
is basically a cube. Such a scenario polytope is called Hose polytope. Nonetheless, we can
solve the separation problem in practice: We show a MIP based separation procedure for
the Hose scenario polytope. Additionally, the thesis presents two separation methods for
3-partition inequalities. These methods are independent of the encoding of the scenario
polytope. Additionally, we present several rounding heuristics.
The result is a Branch-and-Cut algorithm for the capacity formulation. We analyze the
algorithm in the last part of the thesis. There, we show experimentally that the algorithm
works in practice, both in the finite and in the polyhedral case. As a reference point, we
use a CPLEX implementation of the flow based formulation and the computational results by
Buchheim, Liers and SanitĂ . Our experiments show that the new Branch-and-Cut algorithm
is an improvement over the existing approach. Here, the algorithm excels on problem
instances with many scenarios. In particular, we can show that the MIP separation of the
cut-set inequalities is practical
Scheduling Series-Parallel Orders Subject to 0/1-Communication Delays
We consider the problem P}&;| prec},cij&;{0,1}|Îș of scheduling jobs with arbitrary processing times on sufficiently many parallel processors subject to series-parallel precedence constraints and 0/1-communication delays in order to minimize a regular performance measure Îș. Such schedules without processor restrictions are used for generating approximate solutions for a restricted number of processors
Further Exploiting c-Closure for FPT Algorithms and Kernels for Domination Problems
For a positive integer c, a graph G is said to be c-closed if every pair of nonadjacent vertices in G have at most c - 1 neighbors in common. The closure of a graph G, denoted by cl(G), is the least positive integer c for which G is c-closed. The class of c-closed graphs was introduced by J. Fox, T. Roughgarden, C. Seshadhri, F. Wei, and N. Wein [Proceedings of the International Colloquium on Automata, Languages, and Programming (2018), 55; SIAM J. Comput., 49 (2020), pp. 448-464]. T. Koana, C. Komusiewicz, and F. Sommer [Proceedings of the European Symposium on Algorithms (2020), 65; SIAM J. Discrete Math., 36 (2022), pp. 2798-2821] started the study of using cl(G) as an additional structural parameter to design kernels for problems that are W-hard under standard parameterizations. In particular, they studied problems such as Independent Set, Induced Matching, Irredundant Set, and (Threshold) Dominating Set and showed that each of these problems admits a polynomial kernel when parameterized either by k + c or by k for each fixed value of c. Here, k is the solution size and c = cl(G). The work of Koana et al. left several questions open, one of which was whether the Perfect Code problem admits a fixed-parameter tractable (FPT) algorithm and a polynomial kernel on c-closed graphs. In this paper, among other results, we answer this question in the affirmative. Inspired by the FPT algorithm for Perfect Code, we further explore two more domination problems on the graphs of bounded closure. The other problems that we study are Connected Dominating Set and Partial Dominating Set. We show that Perfect Code and Connected Dominating Set are fixed-parameter tractable when parameterized by k + cl(G), whereas Partial Dominating Set, parameterized by k, is W[1]-hard even when cl(G) = 2. We also show that for each fixed c, Perfect Code admits a polynomial kernel on the class of c-closed graphs. And we observe that Connected Dominating Set has no polynomial kernel even on 2-closed graphs unless NP â co-NP/poly
LIPIcs, Volume 261, ICALP 2023, Complete Volume
LIPIcs, Volume 261, ICALP 2023, Complete Volum
LIPIcs, Volume 251, ITCS 2023, Complete Volume
LIPIcs, Volume 251, ITCS 2023, Complete Volum