35 research outputs found
Generalized roof duality and bisubmodular functions
Consider a convex relaxation of a pseudo-boolean function . We
say that the relaxation is {\em totally half-integral} if is a
polyhedral function with half-integral extreme points , and this property is
preserved after adding an arbitrary combination of constraints of the form
, , and where \gamma\in\{0, 1, 1/2} is a
constant. A well-known example is the {\em roof duality} relaxation for
quadratic pseudo-boolean functions . We argue that total half-integrality is
a natural requirement for generalizations of roof duality to arbitrary
pseudo-boolean functions. Our contributions are as follows. First, we provide a
complete characterization of totally half-integral relaxations by
establishing a one-to-one correspondence with {\em bisubmodular functions}.
Second, we give a new characterization of bisubmodular functions. Finally, we
show some relationships between general totally half-integral relaxations and
relaxations based on the roof duality.Comment: 14 pages. Shorter version to appear in NIPS 201
Polynomial combinatorial algorithms for skew-bisubmodular function minimization
Huber et al. (SIAM J Comput 43:1064–1084, 2014) introduced a concept of skew bisubmodularity, as a generalization of bisubmodularity, in their complexity dichotomy theorem for valued constraint satisfaction problems over the three-value domain, and Huber and Krokhin (SIAM J Discrete Math 28:1828–1837, 2014) showed the oracle tractability of minimization of skew-bisubmodular functions. Fujishige et al. (Discrete Optim 12:1–9, 2014) also showed a min–max theorem that characterizes the skew-bisubmodular function minimization, but devising a combinatorial polynomial algorithm for skew-bisubmodular function minimization was left open. In the present paper we give first combinatorial (weakly and strongly) polynomial algorithms for skew-bisubmodular function minimization
Mixed-Integer Programming Approaches to Generalized Submodular Optimization and its Applications
Submodularity is an important concept in integer and combinatorial
optimization. A classical submodular set function models the utility of
selecting homogenous items from a single ground set, and such selections can be
represented by binary variables. In practice, many problem contexts involve
choosing heterogenous items from more than one ground set or selecting multiple
copies of homogenous items, which call for extensions of submodularity. We
refer to the optimization problems associated with such generalized notions of
submodularity as Generalized Submodular Optimization (GSO). GSO is found in
wide-ranging applications, including infrastructure design, healthcare, online
marketing, and machine learning. Due to the often highly nonlinear (even
non-convex and non-concave) objective function and the mixed-integer decision
space, GSO is a broad subclass of challenging mixed-integer nonlinear
programming problems. In this tutorial, we first provide an overview of
classical submodularity. Then we introduce two subclasses of GSO, for which we
present polyhedral theory for the mixed-integer set structures that arise from
these problem classes. Our theoretical results lead to efficient and versatile
exact solution methods that demonstrate their effectiveness in practical
problems using real-world datasets
Half-integrality, LP-branching and FPT Algorithms
A recent trend in parameterized algorithms is the application of polytope
tools (specifically, LP-branching) to FPT algorithms (e.g., Cygan et al., 2011;
Narayanaswamy et al., 2012). However, although interesting results have been
achieved, the methods require the underlying polytope to have very restrictive
properties (half-integrality and persistence), which are known only for few
problems (essentially Vertex Cover (Nemhauser and Trotter, 1975) and Node
Multiway Cut (Garg et al., 1994)). Taking a slightly different approach, we
view half-integrality as a \emph{discrete} relaxation of a problem, e.g., a
relaxation of the search space from to such that
the new problem admits a polynomial-time exact solution. Using tools from CSP
(in particular Thapper and \v{Z}ivn\'y, 2012) to study the existence of such
relaxations, we provide a much broader class of half-integral polytopes with
the required properties, unifying and extending previously known cases.
In addition to the insight into problems with half-integral relaxations, our
results yield a range of new and improved FPT algorithms, including an
-time algorithm for node-deletion Unique Label Cover with
label set and an -time algorithm for Group Feedback Vertex
Set, including the setting where the group is only given by oracle access. All
these significantly improve on previous results. The latter result also implies
the first single-exponential time FPT algorithm for Subset Feedback Vertex Set,
answering an open question of Cygan et al. (2012).
Additionally, we propose a network flow-based approach to solve some cases of
the relaxation problem. This gives the first linear-time FPT algorithm to
edge-deletion Unique Label Cover.Comment: Added results on linear-time FPT algorithms (not present in SODA
paper
Signed ring families and signed posets
The one-to-one correspondence between finite distributive lattices and finite partially ordered sets (posets) is a well-known theorem of G. Birkhoff. This implies a nice representation of any distributive lattice by its corresponding poset, where the size of the former (distributive lattice) is often exponential in the size of the underlying set of the latter (poset). A lot of engineering and economic applications bring us distributive lattices as a ring family of sets which is closed with respect to the set union and intersection. When it comes to a ring family of sets, the underlying set is partitioned into subsets (or components) and we have a poset structure on the partition. This is a set-theoretical variant of the Birkhoff theorem revealing the correspondence between finite ring families and finite posets on partitions of the underlying sets, which was pursued by Masao Iri around 1978, especially concerned with what is called the principal partition of discrete systems such as graphs, matroids, and polymatroids. In the present paper we investigate a signed-set version of the Birkhoff-Iri decomposition in terms of signed ring family, which corresponds to Reiner's result on signed posets, a signed counterpart of the Birkhoff theorem. We show that given a signed ring family, we have a signed partition of the underlying set together with a signed poset on the signed partition which represents the given signed ring family. This representation is unique up to certain reflections