26 research outputs found
On total dual integrality
AbstractWe prove that each (rational) polyhedron of full dimension is determined by a unique minimal total dual integral system of linear inequalities, with integral left hand sides (thus extending a result of Giles and Pulleyblank), and we give a characterization of total dual integrality
Max-Flow on Regular Spaces
The max-flow and max-coflow problem on directed graphs is studied in the
common generalization to regular spaces, i.e., to kernels or row spaces of
totally unimodular matrices. Exhibiting a submodular structure of the family of
paths within this model we generalize the Edmonds-Karp variant of the classical
Ford-Fulkerson method and show that the number of augmentations is
quadratically bounded if augmentations are chosen along shortest possible
augmenting paths
An integer analogue of Carathéodory's theorem
AbstractWe prove a theorem on Hilbert bases analogous to Carathéodory's theorem for convex cones. The result is used to give an upper bound on the number of nonzero variables needed in optimal solutions to integer programs associated with totally dual integral systems. For integer programs arising from perfect graphs the general bounds are improved to show that if G is a perfect graph with n nodes and w is a vector of integral node weights, then there exists a minimum w-covering of the nodes that uses at most n distinct cliques
Intersecting restrictions in clutters
A clutter is intersecting if the members do not have a common element yet every two members intersect. It has been conjectured that for clutters without an intersecting minor, total primal integrality and total dual integrality of the corresponding set covering linear system must be equivalent. In this paper, we provide a polynomial characterization of clutters without an intersecting minor. One important class of intersecting clutters comes from projective planes, namely the deltas, while another comes from graphs, namely the blockers of extended odd holes. Using similar techniques, we provide a poly- nomial algorithm for finding a delta or the blocker of an extended odd hole minor in a given clutter. This result is quite surprising as the same problem is NP-hard if the input were the blocker instead of the clutter
Rerouting Flows When Links Fail
We introduce and investigate reroutable flows, a robust version of network flows in which link failures can be mitigated by rerouting the affected flow. Given a capacitated network, a path flow is reroutable if after failure of an arbitrary arc, we can reroute the interrupted flow from the tail of that arc to the sink, without modifying the flow that is not affected by the failure. Similar types of restoration, which are often termed "local", were previously investigated in the context of network design, such as min-cost capacity planning. In this paper, our interest is in computing maximum flows under this robustness assumption. An important new feature of our model, distinguishing it from existing max robust flow models, is that no flow can get lost in the network.
We also study a tightening of reroutable flows, called strictly reroutable flows, making more restrictive assumptions on the capacities available for rerouting. For both variants, we devise a reroutable-flow equivalent of an s-t-cut and show that the corresponding max flow/min cut gap is bounded by 2. It turns out that a strictly reroutable flow of maximum value can be found using a compact LP formulation, whereas the problem of finding a maximum reroutable flow is NP-hard, even when all capacities are in {1, 2}. However, the tightening can be used to get a 2-approximation for reroutable flows. This ratio is tight in general networks, but we show that in the case of unit capacities, every reroutable flow can be transformed into a strictly reroutable flow of same value. While it is NP-hard to compute a maximal integral flow even for unit capacities, we devise a surprisingly simple combinatorial algorithm that finds a half-integral strictly reroutable flow of value 1, or certifies that no such solutions exits. Finally, we also give a hardness result for the case of multiple arc failures
Arc connectivity and submodular flows in digraphs
Let be a digraph. For an integer , a -arc-connected
flip is an arc subset of such that after reversing the arcs in it the
digraph becomes (strongly) -arc-connected.
The first main result of this paper introduces a sufficient condition for the
existence of a -arc-connected flip that is also a submodular flow for a
crossing submodular function. More specifically, given some integer , suppose for all , where and denote the number of arcs
in leaving and entering , respectively. Let be a crossing
family over ground set , and let be a crossing
submodular function such that for
all . Then has a -arc-connected flip such that
for all . The result has several
applications to Graph Orientations and Combinatorial Optimization. In
particular, it strengthens Nash-Williams' so-called weak orientation theorem,
and proves a weaker variant of Woodall's conjecture on digraphs whose
underlying undirected graph is -edge-connected.
The second main result of this paper is even more general. It introduces a
sufficient condition for the existence of capacitated integral solutions to the
intersection of two submodular flow systems. This sufficient condition implies
the classic result of Edmonds and Giles on the box-total dual integrality of a
submodular flow system. It also has the consequence that in a weakly connected
digraph, the intersection of two submodular flow systems is totally dual
integral.Comment: 29 pages, 4 figure