53 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
A Strongly Polynomial-Time Algorithm for Weighted General Factors with Three Feasible Degrees
General factors are a generalization of matchings. Given a graph with a
set of feasible degrees, called a degree constraint, for each vertex
of , the general factor problem is to find a (spanning) subgraph of
such that for every of . When all
degree constraints are symmetric -matroids, the problem is solvable in
polynomial time. The weighted general factor problem is to find a general
factor of the maximum total weight in an edge-weighted graph. Strongly
polynomial-time algorithms are only known for weighted general factor problems
that are reducible to the weighted matching problem by gadget constructions.
In this paper, we present the first strongly polynomial-time algorithm for a
type of weighted general factor problems with real-valued edge weights that is
provably not reducible to the weighted matching problem by gadget
constructions
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