We consider approximation algorithms for non-uniform buy-at-bulk network design problems. The first non-trivial approximation algorithm for this problem is due to Charikar and Karagiozova (STOC' 05);for an instanc...
详细信息
ISBN:
(纸本)0769527205
We consider approximation algorithms for non-uniform buy-at-bulk network design problems. The first non-trivial approximation algorithm for this problem is due to Charikar and Karagiozova (STOC' 05);for an instance on h pairs their algorithm has an approximation guarantee of exp(O(root log h log log h)) for the uniform-demand case, and log D (.) exp(O(root log h log log h)) for the general demand case, where D is the total demand. We improve upon this result, by presenting the first poly-logarithmic approximation for this problem. The ratio we obtain is O(log(3) h (.) min{log D, gamma(h(2))}) where h is the number of pairs and gamma(n) is the worst case distortion in embedding the metric induced by a n vertex graph into a distribution over its spanning trees. Using the best known upper bound on gamma(n) we obtain an O(min{log(3) h(.)log D, log(5) h log log h}) ratio approximation. We also give poly-logarithmic approximations for some variants of the singe-source problem that we need for the multicoinntodity problem.
In the Priority Steiner Tree (PST) problem, we are given an undirected graph G = (V, E) with a source s is an element of V and terminals T subset of V\{s}, where each terminal v is an element of T requires a nonnegati...
详细信息
ISBN:
(纸本)9783030895433;9783030895426
In the Priority Steiner Tree (PST) problem, we are given an undirected graph G = (V, E) with a source s is an element of V and terminals T subset of V\{s}, where each terminal v is an element of T requires a nonnegative priority P(v). The goal is to compute a minimum weight Steiner tree containing edges of varying rates such that the path from s to each terminal v consists of edges of rate greater than or equal to P(v). The PST problem with k priorities admits a min{2 ln vertical bar T vertical bar + 2, k rho}-approximation [Charikar et al., 2004], and is hard to approximate with ratio c log log n for some constant c [Chuzhoy et al., 2008]. In this paper, we first strengthen the analysis provided by [Charikar et al., 2004] for the (2 ln vertical bar T vertical bar + 2)-approximation to show an approximation ratio of inverted right perpendicular log(2) vertical bar T vertical bar inverted left pependicular + 1 <= 1.443 ln vertical bar T vertical bar + 2, then provide a very simple, parallelizable algorithm which achieves the same approximation ratio. We then consider a more difficult node-weighted version of the PST problem, and provide a (2 ln vertical bar T vertical bar + 2)-approximation using extensions of the spider decomposition by [Klein & Ravi, 1995]. This is the first result for the PST problem in node-weighted graphs. Moreover, the approximation ratios for all above algorithms are tight.
We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to ...
详细信息
ISBN:
(数字)9783662476727
ISBN:
(纸本)9783662476727;9783662476710
We study the problem of computing maximin share guarantees, a recently introduced fairness notion. Given a set of n agents and a set of goods, the maximin share of a single agent is the best that she can guarantee to herself, if she would be allowed to partition the goods in any way she prefers, into n bundles, and then receive her least desirable bundle. The objective then in our problem is to find a partition, so that each agent is guaranteed her maximin share. In settings with indivisible goods, such allocations are not guaranteed to exist, hence, we resort to approximation algorithms. Our main result is a 2/3-approximation, that runs in polynomial time for any number of agents. This improves upon the algorithm of Procaccia and Wang [14], which also produces a 2/3-approximation but runs in polynomial time only for a constant number of agents. We then investigate the intriguing case of 3 agents, for which it is already known that exact maximin share allocations do not always exist. We provide a 6/7-approximation algorithm for this case, improving on the currently known ratio of 3/4. Finally, we undertake a probabilistic analysis. We prove that in randomly generated instances, with high probability there exists a maximin share allocation. This can be seen as a justification of the experimental evidence reported in [5,14], that maximin share allocations exist almost always.
In this paper we consider the following variant of clustering or laying out problems of graphs: Given a directed acyclic graph (DAG for short) and an integer B, the objective is to find a mapping of its nodes into blo...
详细信息
ISBN:
(纸本)9783319951652;9783319951645
In this paper we consider the following variant of clustering or laying out problems of graphs: Given a directed acyclic graph (DAG for short) and an integer B, the objective is to find a mapping of its nodes into blocks of size at most B that minimizes the maximum number of external arcs during traversals of the acyclic structure by following paths from the roots to the leaves. An external arc is defined as an arc connecting two distinct blocks. This paper focuses on the case B = 2. Even if B = 2 and the height of the DAG is three, it is known that the problem is NP-hard, and furthermore, there is no 3/2 - epsilon factor approximation algorithm for B = 2 and a small positive epsilon unless P = NP. On the other hand, the best approximation ratio previously shown is 3. In this paper we improve the approximation ratio into strictly smaller than 2. Also, we investigate the relationship between the height of input DAGs and the inapproximability, since the above inapproximability bound 3/2 - epsilon is shown only for DAGs of height 3.
We initiate the study of approximating the largest induced expander in a given graph G. Given a Delta-regular graph G with n vertices, the goal is to find the set with the largest induced expansion of size at least de...
详细信息
ISBN:
(纸本)9781611974782
We initiate the study of approximating the largest induced expander in a given graph G. Given a Delta-regular graph G with n vertices, the goal is to find the set with the largest induced expansion of size at least delta . n. We design a bi-criteria approximation algorithm for this problem;if the optimum has induced spectral expansion lambda our algorithm returns a lambda/log(2)delta exp(Delta/lambda)-(spectral) expander of size at least (5n (up to constants). Our proof introduces and employs a novel semidefinite programming relaxation for the largest induced expander problem. We expect to see further applications of our SDP relaxation in graph partitioning problems. In particular, because of the close connection to the small set expansion problem, one may be able to obtain new insights into the unique games problem.
This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick k centers with the minimum clustering cost such that each group is at least m...
详细信息
This paper studies the fair range clustering problem in which the data points are from different demographic groups and the goal is to pick k centers with the minimum clustering cost such that each group is at least minimally represented in the centers set and no group dominates the centers set. More precisely, given a set of n points in a metric space (P, d) where each point belongs to one of the l different demographics (i.e., P = P-1 (sic) P-2 (sic) center dot center dot center dot (sic) P-l) and a set of l intervals [alpha(1), beta(1)], center dot center dot center dot, [alpha(1), beta(l)] on desired number of centers from each group, the goal is to pick a set of k centers C with minimum l(p)-clustering cost (i.e., (Sigma(v subset of P) d(v, C)(p))(1/p)) such that for each group i is an element of l, vertical bar C n P-i vertical bar is an element of [alpha(i), beta(i)]. In particular, the fair range l(p)-clustering captures fair range k-center, k-median and k-means as its special cases. In this work, we provide an efficient constant factor approximation algorithm for the fair range l(p)-clustering for all values of p is an element of [1, infinity).
The MAXIMUM CARPOOL MATCHING problem is a star packing problem in directed graphs. Formally, given a directed graph G = (V, A), a capacity function c : V -> N, and a weight function w : A -> R, a feasible carpoo...
详细信息
ISBN:
(数字)9783319587479
ISBN:
(纸本)9783319587479;9783319587462
The MAXIMUM CARPOOL MATCHING problem is a star packing problem in directed graphs. Formally, given a directed graph G = (V, A), a capacity function c : V -> N, and a weight function w : A -> R, a feasible carpool matching is a triple (P, D, M), where P (passengers) and D (drivers) form a partition of V, and M is a subset of A boolean AND (P x D), under the constraints that for every vertex d is an element of D, deg(in)(M) (d) <= c(d), and for every vertex p is an element of P, dego(ut)(M) (p) <= 1. In the MAXIMUM CARPOOL MATCHING problem we seek for a matching (P, D, M) that maximizes the total weight of M. The problem arises when designing an online carpool service, such as Zimride [1], that tries to connect between passengers and drivers based on (arbitrary) similarity function. The problem is known to be NP-hard, even for uniform weights and without capacity constraints. We present a 3-approximation algorithm for the problem and 2-approximation algorithm for the unweighted variant of the problem.
In this paper we consider the problem of selecting a collection of source-destination paths in a capacitated network in order to maximize the sum of concave utility functions. We show that the problem is NP-complete e...
详细信息
We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the...
详细信息
ISBN:
(纸本)9781577357605
We consider optimal route planning when the objective function is a general nonlinear and non-monotonic function. Such an objective models user behavior more accurately, for example, when a user is risk-averse, or the utility function needs to capture a penalty for early arrival. It is known that as nonlinearity arises, the problem becomes NP-hard and little is known about computing optimal solutions when in addition there is no monotonicity guarantee. We show that an approximately optimal non-simple path can be efficiently computed under some natural constraints. In particular, we provide a fully polynomial approximation scheme under hop constraints. Our approximation algorithm can extend to run in pseudo-polynomial time under a more general linear constraint that sometimes is useful. As a by-product, we show that our algorithm can be applied to the problem of finding a path that is most likely to be on time for a given deadline.
暂无评论