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• We present a deterministic approximation algorithm for a #P-hard problem. • Our algorithm is a fully polynomial time approximation scheme (FPTAS). • To this end, the intersection volume of two cross-polytopes is also considered. • We propose an FPTAS also for the intersection volume of cross-polytopes. • We prove that computing the intersection volume is, surprisingly, still #P-hard. Given an n -dimensional convex body by a membership oracle in general, it is known that any polynomial-time deterministic algorithm cannot approximate its volume within ratio (n / log ⁡ n) n. There is a substantial progress on randomized approximation such as Markov chain Monte Carlo for a high-dimensional volume, and for many #P-hard problems, while only a few #P-hard problems are known to yield deterministic approximation. Motivated by the problem of deterministically approximating the volume of a V -polytope, that is a polytope with a small number of vertices and (possibly) exponentially many facets, this paper investigates the problem of computing the volume of a "knapsack dual polytope," which is known to be #P-hard due to Khachiyan (1989) [16]. We reduce an approximate volume of a knapsack dual polytope to that of the intersection of two cross-polytopes in a short distance, and give FPTASs for those volume computations. Interestingly, computing the volume of the intersection of two cross-polytopes (i.e., L 1 -balls) is #P-hard, unlike the cases of L ∞ -balls or L 2 -balls. [ABSTRACT FROM AUTHOR]

This paper is devoted to the distributed complexity of finding an approximation of the maximum cut (MaxCut) in graphs. A classical algorithm consists in letting each vertex choose its side of the cut uniformly at random. This does not require any communication and achieves an approximation ratio of at least 1 2 in expectation. When the graph is d -regular and triangle-free, a slightly better approximation ratio can be achieved with a randomized algorithm running in a single round. Here, we investigate the round complexity of deterministic distributed algorithms for MaxCut in regular graphs. We first prove that if G is d -regular, with d even and fixed, no deterministic algorithm running in a constant number of rounds can achieve a constant approximation ratio. We then give a simple one-round deterministic algorithm achieving an approximation ratio of 1 d for d -regular graphs when d is odd. We show that this is best possible in several ways, and in particular no deterministic algorithm with approximation ratio 1 d + ϵ (with ϵ > 0) can run in a constant number of rounds. We also prove results of a similar flavor for the MaxDiCut problem in regular oriented graphs, where we want to maximize the number of arcs oriented from the left part to the right part of the cut. [ABSTRACT FROM AUTHOR]

Given a social network G and a positive integer k , the influence maximization problem seeks for k nodes in G that can influence the largest number of nodes. This problem has found important applications, and a large amount of works have been devoted to identifying the few most influential users. But most of existing works only focus on the diffusion of a single idea or product in social networks. However, in reality, one company may produce multiple kinds of products and one user may also have multiple adoptions. For multiple kinds of different products with different activation costs and profits, it is crucial for the company to distribute the limited budget among multiple products in order to achieve profit maximization. The Profit Maximization with Multiple Adoptions (PM2A) problem aims to seek for a seed set within the budget to maximize the overall profit. In this paper, a Randomized Modified Greedy (RMG) algorithm based on the Reverse Influence Sampling (RIS) technique is presented for the PM2A problem, which could achieve a (1 − 1 / e − ε) -approximate solution with high probability and is also the best performance ratio of the PM2A problem. Comprehensive experiments on three real-world social networks are conducted, and the results demonstrate that our RMG algorithm outperforms the algorithm proposed in [16] and other heuristics in terms of profit maximization, and could better allocate the budget. [ABSTRACT FROM AUTHOR]

Abstract: In this paper, we consider a restricted variant of the scheduling problem, where the machines are the strategic players. For this multi-parameter mechanism design problem, the only known truthful mechanisms use task independent allocation algorithms and only have approximation ratio [N. Nisan, A. Ronen. Algorithmic mechanism design (extended abstract), in: STOC’99: Proceedings of the thirty-first annual ACM symposium on Theory of computing, ACM, New York, NY, USA, 1999. pp. 129–140; A. Mu’alem, M. Schapira, Setting lower bounds on truthfulness: Extended abstract, in: SODA’07: Proceedings of the Eighteenth Annual ACM-SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, USA, 2007, pp. 1143–1152; P. Lu, C. Yu, An improved randomized truthful mechanism for scheduling unrelated machines, in: 25th International Symposium on Theoretical Aspects of Computer Science, STACS, 2008, pp. 527–538; P. Lu, C. Yu, Randomized truthful mechanisms for scheduling unrelated machines, in: C.H. Papadimitriou, S. Zhang (Eds.), Proceedings of WINE, in: Lecture Notes in Computer Science, vol. 5385, Springer, 2008, pp. 402–413]. Lavi and Swamy first use the cycle monotone condition and design a 3-approximation truthful mechanism for a two value variant in [R. Lavi, C. Swamy, Truthful mechanism design for multi-dimensional scheduling via cycle monotonicity, in: EC’07: Proceedings of the 8th ACM conference on Electronic commerce, ACM, New York, NY, USA, 2007, pp. 252–261], where the processing time of task on machine , say , can only be either a lower value or a higher value . We consider a generalized variant, where lies in and is a parameter satisfying some condition. We consider two special cases, case A when , and case B when ,, and give randomized truthful mechanisms with approximation ratio for both cases. Based on these two cases’ results, we are also able to deal with the general case of our two-range-values scheduling problem. We use a combination of two mechanisms, which is also a novel method in mechanism design for scheduling problems, and finally we give a randomized truthful mechanism with approximation ratio . Although the generalization seems a little incremental, we actually use a very novel technique in the key step of proving truthfulness for case A, as well as a new mechanism scheme for case B. Besides, the results in this paper are the first truthful mechanisms with constant approximation ratios when a machine (player) can report infinitely possible values, which is quite different from the two value variant, in which only finite values are available. Furthermore, together with Lavi and Swamy’s work, our results suggest that such a task-dependent approach can really do much better for the scheduling unrelated machines problem. [Copyright &y& Elsevier]

Motivated by the reconstruction of phylogenetic tree in biology, we study the full Steiner tree problem in this paper. Given a complete graph G=(V,E) with a length function on E and a proper subset R⊂V, the problem is to find a full Steiner tree of minimum length in G, which is a kind of Steiner tree with all the vertices of R as its leaves. In this paper, we show that this problem is NP-complete and MAX SNP-hard, even when the lengths of the edges are restricted to either 1 or 2. For the instances with lengths either 1 or 2, we give a 8/5-approximation algorithm to find an approximate solution for the problem. [Copyright &y& Elsevier]

In this paper, we introduce obnoxious effects into obnoxious facility location games on a line, where each agent i has a private location x i on a closed interval [ 0 , 1 ] and one facility will be built on a location y in the interval according to the bids of all the agents. In addition, there are two thresholds d 1 and d 2 in the utility function of each agent, where 0 ≤ d 1 ≤ d 2 ≤ 1 . Denote d ( y , x i ) = | y − x i | to be the distance between agent i and the facility on the location y . The utility function of agent i is 0 if d ( y , x i ) is at most d 1 ; 1 if d ( y , x i ) is at least d 2 ; otherwise a linear increasing function between 0 and 1. Each agent attempts to get the largest utility while the social welfare is to maximize the sum of all the agents' utilities. The classic obnoxious facility game [4,11] is a special case of our problem when d 1 = 0 and d 2 = 1 . In this work, we first study the hardness of approximate mechanism design on this generalized problem, which states that our problem cannot admit any deterministic strategy-proof mechanism with bounded approximation ratio if d 1 ≥ 1 2 . Then we limit the thresholds to some ranges, both deterministic and randomized strategy-proof mechanisms are studied, and the approximation ratios vary with the specific values of d 1 and d 2 . [ABSTRACT FROM AUTHOR]

Being the unbalanced version of the famous Min s – t Cut problem, the Min k -Size s – t Cut problem asks to find a k -size s – t cut with the minimum capacity, where a k -size s – t cut means an s – t cut with its s -side having size at most k . This problem is fundamental and has extensive applications, especially in community identification in social and information networks. It is known that the Min k -Size s – t Cut problem is NP-hard and can be approximated within O ( log ⁡ n ) , where n is the number of vertices in the input graph. In this paper, we give a new approximation algorithm for the Min k -Size s – t Cut problem based on the parametric flow technique. The algorithm is very simple and has only three lines to state. Its approximation ratio is k + 1 k + 1 − k ⁎ , where k ⁎ is the size of the s -side of an optimal solution. [ABSTRACT FROM AUTHOR]

Abstract: In the context of designing a scalable overlay network to support decentralized topic-based pub/sub communication, the Minimum Topic-Connected Overlay problem (Min-TCO in short) has been investigated: given a set of topics and a collection of users together with the lists of topics they are interested in, the aim is to connect these users to a network by a minimum number of edges such that every graph induced by users interested in a common topic is connected. It is known that Min-TCO is -hard and approximable within in polynomial time. In this paper, we further investigate the problem and some of its special instances. We give various hardness results for instances where the number of topics in which a user is interested in is bounded by a constant, and also for the instances where the number of users interested in a common topic is a constant. For the latter case, we present a first constant approximation algorithm. We also present some polynomial-time algorithms for very restricted instances of Min-TCO. [Copyright &y& Elsevier]

Abstract: We consider the offline sorting buffer problem. The input is a sequence of items of different types. All items must be processed one by one by a server. The server is equipped with a random-access buffer of limited capacity which can be used to rearrange items. The problem is to design a scheduling strategy that decides upon the order in which items from the buffer are sent to the server. Each type change incurs unit cost, and thus, the objective is to minimize the total number of type changes for serving the entire sequence. This problem is motivated by various applications in manufacturing processes and computer science, and it has attracted significant attention in the last few years. The main focus has been on online competitive algorithms. Surprisingly little is known on the basic offline problem. In this paper, we show that the sorting buffer problem with uniform cost is NP-hard and, thus, close one of the most fundamental questions for the offline problem. On the positive side, we give an -approximation algorithm when the scheduler is given a buffer only slightly larger than double the original size. We also sketch a fast dynamic programming algorithm for the special case of buffer size 2. [Copyright &y& Elsevier]

Abstract: Given an underlying communication network represented as an edge-weighted graph , a source node , a set of destination nodes , and a capacity which is a positive integer, the capacitated multicast tree routing problem asks for a minimum cost routing scheme for source to send data to all destination nodes, under the constraint that in each routing tree at most destination nodes are allowed to receive the data copies. The cost of the routing scheme is the sum of the costs of all individual routing trees therein. Improving on our previous approximation algorithm for the problem, we present a new algorithm which achieves a worst case performance ratio of , where denotes the best known approximation ratio for the Steiner minimum tree problem. Since is about at the writing of the paper, the ratio achieved by our new algorithm is less than . In comparison, the previously best ratio was . [Copyright &y& Elsevier]

Abstract: The minimum clique partitioning problem in weighted interval graphs (MCPI) is defined as follows. Given an interval graph with nonnegative node weights, the problem is to partition the nodes into a set of cliques such that the sum of node weights in each clique is no more than a given bound. The objective of the problem is to minimize the number of cliques. Recently, Chen et al. [M. Chen, J. Li, J. Li, W. Li, and L. Wang, Some approximation algorithms for the clique partitioning problem in weighted interval graphs, Theoretical Computer Science 381 (2007), 124–133] proposed three approximation algorithms having constant factors 3, 2.5 and 2, and a linear time optimal algorithm for the case with identical weights. In this paper, we show that their factor 2 algorithm does not achieve the expected approximation ratio and the linear time algorithm cannot give an optimal solution for the identical weights case. We also develop an approximation algorithm with factor 2 for the variable weights case and an exact algorithm for the identical weights case. [Copyright &y& Elsevier]

Abstract: In this paper, we study the approximability properties of symmetric TSP under an approximation measure called the differential ratio. More precisely, we improve up to (for any ) the best differential ratio of known so far, given in Hassin and Khuller, [R. Hassin, S. Khuller, z-approximations, J. Algorithms, 41 (2) (2001) 429–442]. [Copyright &y& Elsevier]

In this paper, we study a problem of finding a vehicle scheduling to process a set of jobs which are located in an asymmetric metric space. Each job has a positive handling time , a time window , and a benefit . We consider the following two problems: MAX-VSP asks to find a schedule for a single vehicle to process a subset of jobs with the maximum benefit; and MIN-VSP asks to find a schedule to process all given jobs with the minimum number of vehicles. We first give an time algorithm that delivers a -approximate solution to MAX-VSP, where and is the maximum number of jobs that can be processed by the vehicle after processing a job and before visiting the processed job again by deadline . We then present an time algorithm that delivers a -approximate solution to MIN-VSP, where is the th harmonic number. [Copyright &y& Elsevier]

Abstract: Let be a connected graph such that each edge is weighted by nonnegative real . Let be a vertex designated as a source, be a positive integer, and be a set of terminals. The capacitated multicast tree routing problem (CMTR) asks to find a partition of and a set of trees of such that consists of at most terminals and each spans . The objective is to minimize , where denotes the sum of weights of all edges in . In this paper, we propose a -approximation algorithm to the CMTR, where is the best achievable approximation ratio for the Steiner tree problem. [Copyright &y& Elsevier]

In wireless sensor networks, a virtual backbone has been proposed as the routing infrastructure to alleviate the broadcasting storm problem and perform some other tasks such as area monitoring. Previous work in this area has mainly focused on how to set up a small virtual backbone for high efficiency, which is modelled as the minimum Connected Dominating Set (CDS) problem. In this paper we consider how to establish a small virtual backbone to balance efficiency and fault tolerance. This problem can be formalized as the minimum-connected-tuple dominating set problem, which is a general version of minimum CDS problem with and . We propose three centralized algorithms with small approximation ratios for small and improve the current best results for small . [Copyright &y& Elsevier]

Abstract: There are various techniques for reconstructing phylogenetic trees from data, and in this context the problem of determining how distant two such trees are from each other arises naturally. Various metrics for measuring the distance between two phylogenies have been defined. Another way of comparing two trees and is to compute the so called maximum agreement forest of these trees. Informally, the number of components of an agreement forest tells how many edges from each of and need to be cut so that the resulting forests agree, after performing all forced edge contractions. This problem is -hard even when the input trees have maximum degree 2. Hein et al. [J. Hein, T. Jiang, L. Wang, K. Zhang, On the complexity of comparing evolutionary trees, Discrete Applied Mathematics 71 (1996) 153–169] presented an approximation algorithm for it, claimed to have performance ratio 3. We show that the performance ratio of the algorithm proposed by Hein et al. is 4, and we also present two new 3-approximation algorithms for this problem. We show how to modify one of the algorithms into a -approximation algorithm for trees with bounded degree , . Finally, we report on some computational experiments comparing the performance of the algorithms presented in this paper. [Copyright &y& Elsevier]

Abstract: For two vertex clusters of a tree metric, we show that the sum of the average intracluster distances is always less than or equal to twice of the average intercluster distance. We show the feature in a more general form of weighted distance. This feature provides a 2-approximation algorithm for the minimum average intercluster distance spanning tree problem, which is a generalization of the minimum routing cost spanning tree or minimum average distance spanning tree problem. The results in this paper can be further generalized to more than two clusters. [Copyright &y& Elsevier]

Abstract: This paper explores three concepts: the -center problem, some of its variants, and asymmetry. The -center problem is fundamental in location theory. Variants of -center may more accurately model real-life problems than the original formulation. Asymmetry is a significant impediment to approximation in many graph problems, such as -center, facility location, -median, and the TSP. We give an -approximation algorithm for the asymmetric weighted -center problem. Here, the vertices have weights and we are given a total budget for opening centers. In the p-neighbor variant each vertex must have (unweighted) centers nearby: we give an -bicriteria algorithm using centers, for small . Finally, we show the following three versions of the asymmetric -center problem to be inapproximable: priority -center, k-supplier, and outliers with forbidden centers. [Copyright &y& Elsevier]

Abstract: For MAX SAT, which is a well-known NP-hard problem, many approximation algorithms have been proposed. Two types of best approximation algorithms for MAX SAT were proposed by Asano and Williamson: one with best proven performance guarantee 0.7846 and the other with performance guarantee 0.8331 if the performance guarantee of the Zwick''s algorithm for MAX SAT is as conjectured. Both algorithms are based on their sharpened analysis of Goemans and Williamson''s LP-relaxation for MAX SAT. In this paper, we present an improved analysis which is simpler than the previous analysis. Furthermore, algorithms based on this analysis will play a role as a better building block in designing an improved approximation algorithm for MAX SAT. Actually we show an example that algorithms based on this analysis lead to approximation algorithms with performance guarantee 0.7877 and conjectured performance guarantee 0.8353 which are slightly better than the best known corresponding performance guarantees 0.7846 and 0.8331, respectively. [Copyright &y& Elsevier]

Abstract: Given a set of positive numbers, the max–min partition problem asks for a k-partition such that the minimum part is maximized. The min–ratio partition problem has the similar definition but the objective is to minimize the ratio of the maximum to the minimum parts. In this paper, we analyze the performances of the longest processing time (LPT) algorithm for the two problems. We show that the tight bounds of the LPT are, respectively and . [Copyright &y& Elsevier]

Abstract: Given a graph, the Hamiltonian path completion problem is to find an augmenting edge set such that the augmented graph has a Hamiltonian path. In this paper, we show that the Hamiltonian path completion problem will unlikely have any constant ratio approximation algorithm unless NP = P. This problem remains hard to approximate even when the given subgraph is a tree. Moreover, if the edge weights are restricted to be either 1 or 2, the Hamiltonian path completion problem on a tree is still NP-hard. Then it is observed that this problem is strongly NP-hard, so it does not have any fully polynomial-time approximation scheme (FPTAS) unless NP=P. When the given tree is a k-tree, we give an approximation algorithm with performance ratio 1.5. [Copyright &y& Elsevier]

Abstract: It has been a challenging open problem whether there is a polynomial time approximation algorithm for the VERTEX COVER problem whose approximation ratio is bounded by a constant less than 2. In this paper, we study the VERTEX COVER problem on graphs with perfect matching (shortly, VC-PM). We show that if the VC-PM problem has a polynomial time approximation algorithm with approximation ratio bounded by a constant less than 2, then so does the VERTEX COVER problem on general graphs. Approximation algorithms for VC-PM are developed, which induce improvements over previously known algorithms on sparse graphs. For example, for graphs of average degree 5, the approximation ratio of our algorithm is 1.414, compared with the previously best ratio 1.615 by Halldórsson and Radhakrishnan. [Copyright &y& Elsevier]

Abstract: Given a bipartite graph , a 2-layered drawing consists of placing nodes in the first node set V on a straight line and placing nodes in the second node set W on a parallel line . For a given ordering of nodes in W on , the one-sided crossing minimization problem asks to find an ordering of nodes in V on so that the number of arc crossings is minimized. A well-known lower bound LB on the minimum number of crossings is obtained by summing up over all node pairs , where denotes the number of crossings generated by arcs incident to u and when u precedes in an ordering. In this paper, we prove that there always exists a solution whose crossing number is at most if the minimum degree of a node in V is at least 5. [Copyright &y& Elsevier]

Abstract: In this paper we investigate the problem of finding a 2-connected spanning subgraph of minimal cost in a complete and weighted graph G. This problem is known to be APX-hard, for both the edge and the vertex connectivity case. Here we prove that the APX-hardness still holds even if one restricts the edge costs to an interval , for an arbitrary small . This result implies the first explicit lower bound on the approximability of the general version (i.e., for arbitrary graphs) of the problem. On the other hand, if the input graph satisfies the sharpened -triangle inequality, then a -approximation algorithm is designed. This ratio tends to 1 with tending to , and it improves the previous known bound of , holding for graphs satisfying the triangle inequality, as soon as . Furthermore, a generalized problem of increasing to 2 the edge-connectivity of any spanning subgraph of G by means of a set of edges of minimum cost is considered. This problem is known to admit a 2-approximation algorithm. Here we show that whenever the input graph satisfies the sharpened -triangle inequality with , then this ratio can be improved to . [Copyright &y& Elsevier]