37 research outputs found
On Generalizations of Network Design Problems with Degree Bounds
Iterative rounding and relaxation have arguably become the method of choice
in dealing with unconstrained and constrained network design problems. In this
paper we extend the scope of the iterative relaxation method in two directions:
(1) by handling more complex degree constraints in the minimum spanning tree
problem (namely, laminar crossing spanning tree), and (2) by incorporating
`degree bounds' in other combinatorial optimization problems such as matroid
intersection and lattice polyhedra. We give new or improved approximation
algorithms, hardness results, and integrality gaps for these problems.Comment: v2, 24 pages, 4 figure
Distributed Connectivity Decomposition
We present time-efficient distributed algorithms for decomposing graphs with
large edge or vertex connectivity into multiple spanning or dominating trees,
respectively. As their primary applications, these decompositions allow us to
achieve information flow with size close to the connectivity by parallelizing
it along the trees. More specifically, our distributed decomposition algorithms
are as follows:
(I) A decomposition of each undirected graph with vertex-connectivity
into (fractionally) vertex-disjoint weighted dominating trees with total weight
, in rounds.
(II) A decomposition of each undirected graph with edge-connectivity
into (fractionally) edge-disjoint weighted spanning trees with total
weight , in
rounds.
We also show round complexity lower bounds of
and
for the above two decompositions,
using techniques of [Das Sarma et al., STOC'11]. Moreover, our
vertex-connectivity decomposition extends to centralized algorithms and
improves the time complexity of [Censor-Hillel et al., SODA'14] from
to near-optimal .
As corollaries, we also get distributed oblivious routing broadcast with
-competitive edge-congestion and -competitive
vertex-congestion. Furthermore, the vertex connectivity decomposition leads to
near-time-optimal -approximation of vertex connectivity: centralized
and distributed . The former moves
toward the 1974 conjecture of Aho, Hopcroft, and Ullman postulating an
centralized exact algorithm while the latter is the first distributed vertex
connectivity approximation
Optimal Random Matchings, Tours, and Spanning Trees in Hierarchically Separated Trees
We derive tight bounds on the expected weights of several combinatorial
optimization problems for random point sets of size distributed among the
leaves of a balanced hierarchically separated tree. We consider {\it
monochromatic} and {\it bichromatic} versions of the minimum matching, minimum
spanning tree, and traveling salesman problems. We also present tight
concentration results for the monochromatic problems.Comment: 24 pages, to appear in TC
Probabilistic methods in combinatorial and stochastic optimization
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2005.Includes bibliographical references (leaves 103-106).(cont.) Packing/Covering problems, we prove upper and lower bounds on the adaptivity gap depending on the dimension. We also design polynomial-time algorithms achieving near-optimal approximation guarantees with respect to the adaptive optimum. Finally, we prove complexity-theoretic results regarding optimal adaptive policies. These results are based on a connection between adaptive policies and Arthur-Merlin games which yields PSPACE-hardness results for numerous questions regarding adaptive policies.In this thesis we study a variety of combinatorial problems with inherent randomness. In the first part of the thesis, we study the possibility of covering the solutions of an optimization problem on random subgraphs. The motivation for this approach is a situation where an optimization problem needs to be solved repeatedly for random instances. Then we seek a pre-processing stage which would speed-up subsequent queries by finding a fixed sparse subgraph covering the solution for a random subgraph with high probability. The first problem that we investigate is the minimum spanning tree. Our essential result regarding this problem is that for every graph with edge weights, there is a set of O(n log n) edges which contains the minimum spanning tree of a random subgraph with high probability. More generally, we extend this result to matroids. Further, we consider optimization problems based on the shortest path metric and we find covering sets of size 0(n(Ì1+2/c) log2Ì n) that approximate the shortest path metric of a random vertex-induced subgraph within a constant factor of c with high probability. In the second part, we turn to a model of stochastic optimization, where a solution is built sequentially by selecting a collection of "items". We distinguish between adaptive and non-adaptive strategies, where adaptivity means being able to perceive the precise characteristics of chosen items and use this knowledge in subsequent decisions. The benefit of adaptivity is our central concept which we investigate for a variety of specific problems. For the Stochastic Knapsack problem, we prove constant upper and lower bounds on the "adaptivity gap" between optimal adaptive and non-adaptive policies. For more general Stochasticby Jan Vondrák.Ph.D
Error-Sensitive Proof-Labeling Schemes
Proof-labeling schemes are known mechanisms providing nodes of networks with certificates that can be verified locally by distributed algorithms. Given a boolean predicate on network states, such schemes enable to check whether the predicate is satisfied by the actual state of the network, by having nodes interacting with their neighbors only. Proof-labeling schemes are typically designed for enforcing fault-tolerance, by making sure that if the current state of the network is illegal with respect to some given predicate, then at least one node will detect it. Such a node can raise an alarm, or launch a recovery procedure enabling the system to return to a legal state. In this paper, we introduce error-sensitive proof-labeling schemes. These are proof-labeling schemes which guarantee that the number of nodes detecting illegal states is linearly proportional to the edit-distance between the current state and the set of legal states. By using error-sensitive proof-labeling schemes, states which are far from satisfying the predicate will be detected by many nodes, enabling fast return to legality. We provide a structural characterization of the set of boolean predicates on network states for which there exist error-sensitive proof-labeling schemes. This characterization allows us to show that classical predicates such as, e.g., acyclicity, and leader admit error-sensitive proof-labeling schemes, while others like regular subgraphs don\u27t. We also focus on compact error-sensitive proof-labeling schemes. In particular, we show that the known proof-labeling schemes for spanning tree and minimum spanning tree, using certificates on O(log n) bits, and on O(log^2 n) bits, respectively, are error-sensitive, as long as the trees are locally represented by adjacency lists, and not just by parent pointers