We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first ri...
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We show that for the anti-ferromagnetic Ising model on the Bethe lattice, weak spatial mixing implies strong spatial mixing. As a by-product of our analysis, we obtain what is to the best of our knowledge the first rigorous proof of the uniqueness threshold for the anti-ferromagnetic Ising model (with non-zero external field) on the Bethe lattice. Following a method due to Weitz [15], we then use the equivalence between weak and strong spatial mixing to give a deterministic fully polynomial time approximation scheme for the partition function of the anti-ferromagnetic Ising model with arbitrary field on graphs of degree at most , throughout the uniqueness region of the Gibbs measure on the infinite -regular tree. By a standard correspondence, our results translate to arbitrary two-state anti-ferromagnetic spin systems with soft constraints. Subsequent to a preliminary version of this paper, Sly and Sun [13] have shown that our results are optimal in the sense that, under standard complexity theoretic assumptions, there does not exist a fully polynomial time approximation scheme for the partition function of such spin systems on graphs of maximum degree for parameters outside the uniqueness region. Taken together, the results of [13] and of this paper therefore indicate a tight relationship between complexity theory and phase transition phenomena in two-state anti-ferromagnetic spin systems.
Kernelization algorithms in the context of Parameterized Complexity are often based on a combination of reduction rules and combinatorial insights. We will expose in this paper a similar strategy for obtaining polynom...
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We consider a purchase/inventory control problem in which the purchase price and demand are stochastic, a common situation encountered by firms that replenish in a foreign currency or from commodity markets. More spec...
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We consider a purchase/inventory control problem in which the purchase price and demand are stochastic, a common situation encountered by firms that replenish in a foreign currency or from commodity markets. More specifically, we assume that the demand follows a Poisson arrival process and that the log-price evolves according to a general Wiener process. Under these circumstances, the optimal policy is a state dependent base-stock policy that can be described as a series of threshold prices. An iterative procedure for determining the optimal thresholds has been derived earlier but, even for the simplest price process, the solution quickly becomes numerically intractable. To deal with this, we propose an approximation that allows us to derive simple heuristics for finding thresholds that are close to optimal. For certain price processes the heuristics are just a series of closed-form expressions. The computational complexity is reduced significantly, and the numerical study shows that the new heuristics perform considerably better than earlier suggested heuristics.
Many large-scale real-world networks are well-known to have the power law distribution in their degree sequences: the number of vertices with degree i is proportional to i-βfor some constant β. It is a common ...
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The relay node placement problem in wireless sensor network (WSN) aims at deploying the minimum number of relay nodes over the network so that each sensor can communicate with at least one relay node. When the deploye...
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The relay node placement problem in wireless sensor network (WSN) aims at deploying the minimum number of relay nodes over the network so that each sensor can communicate with at least one relay node. When the deployed relay nodes are homogeneous and their communication ranges are circular, one way to solve the WSN relay node placement problem is to solve the minimum geometric disk cover (MGDC) problem first and place the relay nodes at the centers of the covering disks and then, if necessary, deploy additional relay nodes to meet the connection requirement of relay nodes. It is known that the MGDC problem is NP-complete. A novel linear time approximation algorithm for the MGDC problem is proposed, which identifies covering disks using the regular hexagon tessellation of the plane with bounded area. The approximation ratio of the proposed algorithmis (5+epsilon), where 0 < epsilon <= 15. Experimental results show that the worst case is rare, and on average the proposed algorithm uses less than 1.7 times the optimal disks of the MGDC problem. In cases where quick deployment is necessary, this study provides a fast 7-approximation algorithm which uses on average less than twice the optimal number of relay nodes in the simulation.
Given n robots and n target points on the plane, the minimum set cover formation (SCF) problem requires the robots to form a set cover by the minimum number of robots. In previous formation problems by mobile robots, ...
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ISBN:
(纸本)9783319144726;9783319144719
Given n robots and n target points on the plane, the minimum set cover formation (SCF) problem requires the robots to form a set cover by the minimum number of robots. In previous formation problems by mobile robots, such as gathering and pattern formation, the problems consist only of the mobile robots, and there are no points fixed in the environment. In addition, the problems do not require a control of the number of robots constructing the formation. In this paper, we first introduce the formation problem in which robots move so that they achieve a desired deployment with the minimum number of robots for a given set of positions of fixed points. Since the minimum set cover problem with disks in the centralized settings is NP-hard, our goal is to propose approximation algorithms for the minimum SCF problem. First, we show a minimal SCF algorithm from any initial configuration in the asynchronous system. Moreover, we propose an 8-approximation SCF algorithm in the semi-synchronous system for an initial configuration with a low symmetricity. This approximation algorithm achieves 2(1 + 1/l)(2) approximation ratio for an initial configuration with the lowest symmetricity (l >= 1).
Given a collection of phylogenetic trees with identical leaf label-set, the Maximum Agreement Forest problem (maf) asks for a largest common subforest of these input trees. The maf problem on two binary phylogenetic t...
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ISBN:
(纸本)9783319087832;9783319087825
Given a collection of phylogenetic trees with identical leaf label-set, the Maximum Agreement Forest problem (maf) asks for a largest common subforest of these input trees. The maf problem on two binary phylogenetic trees has been studied extensively in the literature. In this paper, we will be focused on the maf problem on multiple (i.e., two or more) binary phylogenetic trees and present two polynomial-time approximation algorithms, one for the maf problem on multiple rooted trees, and the other for the maf problem on multiple unrooted trees. The ratio of our algorithm for the maf problem on multiple rooted trees is 3, which is an improvement over the previously best ratio 8 for the problem. Our 4-approximation algorithm for the maf problem on multiple unrooted trees is the first approximation algorithm for the problem.
Let G = G(AUB,A x B), with vertical bar A vertical bar =vertical bar B vertical bar = n, be a weighted bipartite graph, and let d(.,.) be the cost function on the edges. Let w(M) denote the weight of a matching in G, ...
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ISBN:
(纸本)9781450327107
Let G = G(AUB,A x B), with vertical bar A vertical bar =vertical bar B vertical bar = n, be a weighted bipartite graph, and let d(.,.) be the cost function on the edges. Let w(M) denote the weight of a matching in G, and M* a minimum-cost perfect matching in G. We call a perfect matching M c-approximate, for c >= 1, if w(M) <= c . w(M*). We present three approximation algorithms for computing minimum-cost perfect matchings in G. First, we consider the case when d(.,.) is a metric. For any delta > 0, we present an algorithm that, in O(n(2+delta) log n log(2) (1/delta)) time, computes a O(1/delta(alpha))-approximate matching of G, where alpha = log(3) 2 approximate to 0.631. Next, we assume the existence of a dynamic data structure for answering approximate nearest neighbor (ANN) queries under d(.,.). Given two parameters epsilon, delta is an element of (0, 1), we present an algorithm that, in O(epsilon(-2) n(1+delta)tau (n, epsilon) log(2) (n/epsilon) log(1/delta)) time, computes a O(1/delta(alpha))-approximate matching of G, where alpha = 1 + log(2)(1 + epsilon) and tau (n, epsilon) is the query and update time of an (epsilon/2)-ANN data structure. Finally, we present an algorithm that works even if d(.,.) is not a metric but admits an ANN data structure for d(.,.). In particular, we present an algorithm that computes, in O(epsilon(-1)n(3/2)tau(n, epsilon) log(4) (n/epsilon) E) log Delta) time, a (1 + epsilon)-approximate matching of A and B;here Delta is the ratio of the largest to the smallest-cost edge in G, and tau(n, epsilon) is the query and update time of an (epsilon/c)-ANN data structure for some constant c > 1. We show that our results lead to faster matching algorithms for many geometric settings.
We consider the Max-Buying Problem with Limited Supply, in which there are n items, with C-i copies of each item i, and m bidders such that every bidder b has valuation v(ib) for item i. The goal is to find a pricing ...
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ISBN:
(纸本)9783642544231
We consider the Max-Buying Problem with Limited Supply, in which there are n items, with C-i copies of each item i, and m bidders such that every bidder b has valuation v(ib) for item i. The goal is to find a pricing p and an allocation of items to bidders that maximize the profit, where every item is allocated to at most C-i bidders, every bidder receives at most one item and if a bidder b receives item i then p(i) <= vib. Briest and Krysta presented a 2-approximation for this problem and Aggarwal et al. presented a 4-approximation for the Price Ladder variant where the pricing must be non-increasing (that is, p(1) >= p(2) >= ... >= p(n)). We present a randomized e/(e-1)-approximation for the Max-Buying Problem with Limited Supply and, for every epsilon > 0, a (2 + epsilon)-approximation for the Price Ladder variant.
We consider the dynamic map labeling problem: given a set of rectangular labels on the map, the goal is to appropriately select visible ranges for all the labels such that no two consistent labels overlap at every sca...
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ISBN:
(纸本)9783319080161;9783319080154
We consider the dynamic map labeling problem: given a set of rectangular labels on the map, the goal is to appropriately select visible ranges for all the labels such that no two consistent labels overlap at every scale and the sum of total visible ranges is maximized. We propose approximation algorithms for several variants of this problem. For the simple ARO problem, we provide a 3c log n-approximation algorithm for the unit-width rectangular labels if there is a c-approximation algorithm for unit-width label placement problem in the plane;and a randomized polynomial-time O(log n log log n)-approximation algorithm for arbitrary rectangular labels. For the general ARO problem, we prove that it is NP-complete even for congruent square labels with equal selectable scale range. Moreover, we contribute 12-approximation algorithms for both arbitrary square labels and unit-width rectangular labels, and a 6-approximation algorithm for congruent square labels.
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