558 research outputs found
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
When the Cut Condition is Enough: A Complete Characterization for Multiflow Problems in Series-Parallel Networks
Let be a supply graph and a demand graph defined on the
same set of vertices. An assignment of capacities to the edges of and
demands to the edges of is said to satisfy the \emph{cut condition} if for
any cut in the graph, the total demand crossing the cut is no more than the
total capacity crossing it. The pair is called \emph{cut-sufficient} if
for any assignment of capacities and demands that satisfy the cut condition,
there is a multiflow routing the demands defined on within the network with
capacities defined on . We prove a previous conjecture, which states that
when the supply graph is series-parallel, the pair is
cut-sufficient if and only if does not contain an \emph{odd spindle} as
a minor; that is, if it is impossible to contract edges of and delete edges
of and so that becomes the complete bipartite graph , with
odd, and is composed of a cycle connecting the vertices of
degree 2, and an edge connecting the two vertices of degree . We further
prove that if the instance is \emph{Eulerian} --- that is, the demands and
capacities are integers and the total of demands and capacities incident to
each vertex is even --- then the multiflow problem has an integral solution. We
provide a polynomial-time algorithm to find an integral solution in this case.
In order to prove these results, we formulate properties of tight cuts (cuts
for which the cut condition inequality is tight) in cut-sufficient pairs. We
believe these properties might be useful in extending our results to planar
graphs.Comment: An extended abstract of this paper will be published at the 44th
Symposium on Theory of Computing (STOC 2012
Towards Duality of Multicommodity Multiroute Cuts and Flows: Multilevel Ball-Growing
An elementary h-route flow, for an integer h ≥ 1, is a set of h edge-disjoint paths between a source and a sink, each path carrying a unit of flow, and an h-route flow is a non-negative linear combination of elementary h-route flows. An h-route cut is a set of edges whose removal decreases the maximum h-route flow between a given source-sink pair (or between every source-sink pair in the multicommodity setting) to zero. The main result of this paper is an approximate duality theorem for multicommodity h-route cuts and flows, for h ≤ 3: The size of a minimum h-route cut is at least f/h and at most O(log 3 k·f) where f is the size of the maximum h-route flow and k is the number of commodities. The main step towards the proof of this duality is the design and analysis of a polynomial-time approximation algorithm for the minimum h-route cut problem for h = 3 that has an approximation ratio of O(log 3 k). Previously, polylogarithmic approximation was known only for h-route cuts for h ≤ 2. A key ingredient of our algorithm is a novel rounding technique that we call multilevel ball-growing. Though the proof of the duality relies on this algorithm, it is not a straightforward corollary of it as in the case of classical multicommodity flows and cuts. Similar results are shown also for the sparsest multiroute cut problem
Distances and cuts in planar graphs
AbstractWe prove the following theorem. Let G = (V, E) be a planar bipartite graph, embedded in the euclidean plane. Let O and I be two of its faces. Then there exist pairwise edge-disjoint cuts C1, …, Ct so that for each two vertices v, w with v, w ϵ O of v, w ϵ I, the distance from v to w in G is equal to the number of cuts Cj separating v and w. This theorem is dual to a theorem of Okamura on plane multicommodity flows, in the same way as a theorem of Karzanov is dual to one of Lomonosov
On the Number of Iterations for Dantzig-Wolfe Optimization and Packing-Covering Approximation Algorithms
We give a lower bound on the iteration complexity of a natural class of
Lagrangean-relaxation algorithms for approximately solving packing/covering
linear programs. We show that, given an input with random 0/1-constraints
on variables, with high probability, any such algorithm requires
iterations to compute a
-approximate solution, where is the width of the input.
The bound is tight for a range of the parameters .
The algorithms in the class include Dantzig-Wolfe decomposition, Benders'
decomposition, Lagrangean relaxation as developed by Held and Karp [1971] for
lower-bounding TSP, and many others (e.g. by Plotkin, Shmoys, and Tardos [1988]
and Grigoriadis and Khachiyan [1996]). To prove the bound, we use a discrepancy
argument to show an analogous lower bound on the support size of
-approximate mixed strategies for random two-player zero-sum
0/1-matrix games
- …