15 research outputs found
On the Approximability and Hardness of the Minimum Connected Dominating Set with Routing Cost Constraint
In the problem of minimum connected dominating set with routing cost
constraint, we are given a graph , and the goal is to find the
smallest connected dominating set of such that, for any two
non-adjacent vertices and in , the number of internal nodes on the
shortest path between and in the subgraph of induced by is at most times that in . For general graphs, the only
known previous approximability result is an -approximation algorithm
() for by Ding et al. For any constant , we
give an -approximation
algorithm. When , we give an -approximation
algorithm. Finally, we prove that, when , unless , for any constant , the problem admits no
polynomial-time -approximation algorithm, improving
upon the bound by Du et al. (albeit under a stronger hardness
assumption)
Almost Stable Matchings by Truncating the Gale–Shapley Algorithm
We show that the ratio of matched individuals to blocking pairs grows linearly with the number of propose–accept rounds executed by the Gale–Shapley algorithm for the stable marriage problem. Consequently, the participants can arrive at an almost stable matching even without full information about the problem instance; for each participant, knowing only its local neighbourhood is enough. In distributed-systems parlance, this means that if each person has only a constant number of acceptable partners, an almost stable matching emerges after a constant number of synchronous communication rounds. We apply our results to give a distributed (2 + ε)-approximation algorithm for maximum-weight matching in bicoloured graphs and a centralised randomised constant-time approximation scheme for estimating the size of a stable matching.Peer reviewe
Approximating set multi-covers
Johnson and Lov\'asz and Stein proved independently that any hypergraph
satisfies , where is the transversal
number, is its fractional version, and denotes the
maximum degree. We prove for
the -fold transversal number . Similarly to Johnson, Lov\'asz and
Stein, we also show that this bound can be achieved non-probabilistically,
using a greedy algorithm.
As a combinatorial application, we prove an estimate on how fast
converges to . As a geometric application, we obtain an upper
bound on the minimal density of an -fold covering of the -dimensional
Euclidean space by translates of any convex body.Comment: THE TITLE CHANGED! This is the final version. 7 page
Approximate Clustering via Metric Partitioning
In this paper we consider two metric covering/clustering problems -
\textit{Minimum Cost Covering Problem} (MCC) and -clustering. In the MCC
problem, we are given two point sets (clients) and (servers), and a
metric on . We would like to cover the clients by balls centered at
the servers. The objective function to minimize is the sum of the -th
power of the radii of the balls. Here is a parameter of the
problem (but not of a problem instance). MCC is closely related to the
-clustering problem. The main difference between -clustering and MCC is
that in -clustering one needs to select balls to cover the clients.
For any \eps > 0, we describe quasi-polynomial time (1 + \eps)
approximation algorithms for both of the problems. However, in case of
-clustering the algorithm uses (1 + \eps)k balls. Prior to our work, a
and a approximation were achieved by
polynomial-time algorithms for MCC and -clustering, respectively, where is an absolute constant. These two problems are thus interesting examples of
metric covering/clustering problems that admit (1 + \eps)-approximation
(using (1+\eps)k balls in case of -clustering), if one is willing to
settle for quasi-polynomial time. In contrast, for the variant of MCC where
is part of the input, we show under standard assumptions that no
polynomial time algorithm can achieve an approximation factor better than
for .Comment: 19 page
Fault Tolerant Max-Cut
In this work, we initiate the study of fault tolerant Max-Cut, where given an edge-weighted undirected graph G = (V,E), the goal is to find a cut S ? V that maximizes the total weight of edges that cross S even after an adversary removes k vertices from G. We consider two types of adversaries: an adaptive adversary that sees the outcome of the random coin tosses used by the algorithm, and an oblivious adversary that does not. For any constant number of failures k we present an approximation of (0.878-?) against an adaptive adversary and of ?_{GW}? 0.8786 against an oblivious adversary (here ?_{GW} is the approximation achieved by the random hyperplane algorithm of [Goemans-Williamson J. ACM `95]). Additionally, we present a hardness of approximation of ?_{GW} against both types of adversaries, rendering our results (virtually) tight.
The non-linear nature of the fault tolerant objective makes the design and analysis of algorithms harder when compared to the classic Max-Cut. Hence, we employ approaches ranging from multi-objective optimization to LP duality and the ellipsoid algorithm to obtain our results