A 6.55 factor primal-dual approximation algorithm for the connected facility location problem

In the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ??V, a set of demands (i.e., clients) $\mathcal{D}\subseteq VIn the connected facility location (ConFL) problem, we are given a graph G=(V,E) with nonnegative edge cost c _e on the edges, a set of facilities ℱ⊆V, a set of demands (i.e., clients) D í V\mathcal{D}\subseteq V , and a parameter M≥1. Each facility i has a nonnegative opening cost f _i and each client j has d _j units of demand. Our objective is to open some facilities, say F⊆ℱ, assign each demand j to some open facility i(j)∈F and connect all open facilities using a Steiner tree T such that the total cost, which is ?_{i ? F}f_i+?_{j ? D}d_jc_i(j)j+M?_{e ? T}c_e\sum_{i\in F}f_{i}+\sum_{j\in \mathcal{D}}d_{j}c_{i(j)j}+M\sum_{e\in T}c_{e} , is minimized. We present a primal-dual 6.55-approximation algorithm for the ConFL problem which improves the previous primal-dual 8.55-approximation algorithm given by Swamy and Kumar (Algorithmica 40:245–269, 2004).     

On Ring Grooming in optical networks

An instance I of Ring Grooming consists of m sets A ₁,A ₂,…, A _m from the universe {0, 1,…, n − 1} and an integer g ≥ 2. The unrestricted variant of Ring Grooming, referred to as Unrestricted Ring Grooming, seeks a partition {P ₁ , P ₂, …,P _k } of {1, 2, …, m} such that for each 1 ≤ i ≤ k and is minimized. The restricted variant of Ring Grooming, referred to as Restricted Ring Grooming, seeks a partition of {1,2,…,m} such that | P _i | ≤ g for each and is minimized. If g = 2, we provide an optimal polynomial-time algorithm for both variants. If g > 2, we prove that both both variants are NP-hard even with fixed g. When g is a power of two, we propose an approximation algorithm called iterative matching. Its approximation ratio is exactly 1.5 when g = 4, at most 2.5 when g = 8, and at most in general while it is conjectured to be at most . The iterative matching algorithm is also extended for Unrestricted Ring Grooming with arbitrary g, and a loose upper bound on its approximation ratio is . In addition, set-cover based approximation algorithms have been proposed for both Unrestricted Ring Grooming and Restricted Ring Grooming. They have approximation ratios of at most 1 + log g, but running time in polynomial of m ^g . Work supported by a DIMACS postdoctoral fellowship.     

On a hyperplane arrangement problem and tighter analysis of an error-tolerant pooling design

In this paper, we formulate and investigate the following problem: given integers d,k and r where k>r≥1,d≥2, and a prime power q, arrange d hyperplanes on to maximize the number of r-dimensional subspaces of each of which belongs to at least one of the hyperplanes. The problem is motivated by the need to give tighter bounds for an error-tolerant pooling design based on finite vector spaces. This work is partially supported by NSF CAREER Award CCF-0347565.     

On the total {k}-domination number of Cartesian products of graphs

Let γ _t ^{k}(G) denote the total {k}-domination number of graph G, and let denote the Cartesian product of graphs G and H. In this paper, we show that for any graphs G and H without isolated vertices, . As a corollary of this result, we have for all graphs G and H without isolated vertices, which is given by Pak Tung Ho (Util. Math., 2008, to appear) and first appeared as a conjecture proposed by Henning and Rall (Graph. Comb. 21:63–69, 2005). The work was supported by NNSF of China (No. 10701068 and No. 10671191).     

Group testing in graphs

This paper studies the group testing problem in graphs as follows. Given a graph G=(V,E), determine the minimum number t(G) such that t(G) tests are sufficient to identify an unknown edge e with each test specifies a subset X⊆V and answers whether the unknown edge e is in G[X] or not. Damaschke proved that ⌈log ₂ e(G)⌉≤t(G)≤⌈log ₂ e(G)⌉+1 for any graph G, where e(G) is the number of edges of G. While there are infinitely many complete graphs that attain the upper bound, it was conjectured by Chang and Hwang that the lower bound is attained by all bipartite graphs. Later, they proved that the conjecture is true for complete bipartite graphs. Chang and Juan verified the conjecture for bipartite graphs G with e(G)≤2⁴ or for k≥5. This paper proves the conjecture for bipartite graphs G with e(G)≤2⁵ or for k≥6. Dedicated to Professor Frank K. Hwang on the occasion of his 65th birthday. J.S.-t.J. is supported in part by the National Science Council under grant NSC89-2218-E-260-013. G.J.C. is supported in part by the National Science Council under grant NSC93-2213-E002-28. Taida Institute for Mathematical Sciences, National Taiwan University, Taipei 10617, Taiwan. National Center for Theoretical Sciences, Taipei Office.     

Optimal strategies for the one-round discrete Voronoi game on a line

The one-round discrete Voronoi game, with respect to a n-point user set  $\mathcal {U}$ , consists of two players Player 1 (P1) and Player 2 (P2). At first, P1 chooses a set $\mathcal{F}_{1}$ of m facilities following which P2 chooses another set $\mathcal{F}_{2}$ of m facilities, disjoint from  $\mathcal{F}_{1}$ , where m(=O(1)) is a positive constant. The payoff of P2 is defined as the cardinality of the set of points in $\mathcal{U}$ which are closer to a facility in $\mathcal{F}_{2}$ than to every facility in $\mathcal{F}_{1}$ , and the payoff of P1 is the difference between the number of users in $\mathcal{U}$ and the payoff of P2. The objective of both the players in the game is to maximize their respective payoffs. In this paper, we address the case where the points in $\mathcal{U}$ are located along a line. We show that if the sorted order of the points in $\mathcal{U}$ along the line is known, then the optimal strategy of P2, given any placement of facilities by P1, can be computed in O(n) time. We then prove that for m≥2 the optimal strategy of P1 in the one-round discrete Voronoi game, with the users on a line, can be computed in $O(n^{m-\lambda_{m}})$ time, where 0<λ _m <1, is a constant depending only on m.     

Finding nucleolus of flow game

We study the algorithmic issues of finding the nucleolus of a flow game. The flow game is a cooperative game defined on a network D=(V,E;ω). The player set is E and the value of a coalition S⊆E is defined as the value of a maximum flow from source to sink in the subnetwork induced by S. We show that the nucleolus of the flow game defined on a simple network (ω(e)=1 for each e∈E) can be computed in polynomial time by a linear program duality approach, settling a twenty-three years old conjecture by Kalai and Zemel. In contrast, we prove that both the computation and the recognition of the nucleolus are -hard for flow games with general capacity. Supported by NCET, NSFC (10771200), a CERG grant (CityU 1136/04E) of Hong Kong RGC, an SRG grant (7001838) of City University of Hong Kong.     

Partitioning a weighted partial order

The problem of partitioning a partially ordered set into a minimum number of chains is a well-known problem. In this paper we study a generalization of this problem, where we not only assume that the chains have bounded size, but also that a weight w _i is given for each element i in the partial order such that w _i ≤w _j if i ≺ j. The problem is then to partition the partial order into a minimum-weight set of chains of bounded size, where the weight of a chain equals the weight of the heaviest element in the chain. We prove that this problem is -hard, and we propose and analyze lower bounds for this problem. Based on these lower bounds, we exhibit a 2-approximation algorithm, and show that it is tight. We report computational results for a number of real-world and randomly generated problem instances.     

Sublinear time width-bounded separators and their application to the protein side-chain packing problem

Given d>2 and a set of n grid points Q in ℜ ^d , we design a randomized algorithm that finds a w-wide separator, which is determined by a hyper-plane, in sublinear time such that Q has at most points on either side of the hyper-plane, and at most points within distance to the hyper-plane, where c _d is a constant for fixed d. In particular, c ₃=1.209. To our best knowledge, this is the first sublinear time algorithm for finding geometric separators. Our 3D separator is applied to derive an algorithm for the protein side-chain packing problem, which improves and simplifies the previous algorithm of Xu (Research in computational molecular biology, 9th annual international conference, pp. 408–422, 2005). This research is supported by Louisiana Board of Regents fund under contract number LEQSF(2004-07)-RD-A-35. The part of this research was done while Bin Fu was associated with the Department of Computer Science, University of New Orleans, LA 70148, USA and the Research Institute for Children, 200 Henry Clay Avenue, New Orleans, LA 70118, USA.     

On zero error algorithms having oracle access to one query

It is known that (Cai, 2001). The reverse direction of whether ZPP^NP is contained in remains open. We show that if the zero-error algorithm is allowed to ask only one query to the NP oracle (for any input and random string), then it can be simulated in . That is, we prove that . Next we consider whether the above result can be improved as and point out a difficulty in doing so. Via a simple proof, we observe that BPP ⊆ ZPP^NP[1] (a result implicitly proven in some prior work). Thus, achieving the above improvement would imply BPP ⊆ P^NP, settling a long standing open problem. We then argue that the above mentioned improvement can be obtained for the next level of the polynomial time hierarchy. Namely, we prove that . On the other hand, by adapting our proof of our main result it can be shown that . For the purpose of comparing these two results, we prove that . We conclude by observing that the above claims extend to the higher levels of the hierarchy: for k ≥ 2, and . Research supported in part by NSF grant CCR-0208013. A preliminary version of the paper was presented at COCOON′05 Cai and Chakaravarthy (2005). Part of the research was conducted while the author was at the University of Wisconsin, Madison.     

Domination and total domination in complementary prisms

Let G be a graph and be the complement of G. The complementary prism of G is the graph formed from the disjoint union of G and by adding the edges of a perfect matching between the corresponding vertices of G and . For example, if G is a 5-cycle, then is the Petersen graph. In this paper we consider domination and total domination numbers of complementary prisms. For any graph G, and , where γ(G) and γ _t (G) denote the domination and total domination numbers of G, respectively. Among other results, we characterize the graphs G attaining these lower bounds. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Improved algorithms for largest cardinality 2-interval pattern problem

The 2-INTERVAL PATTERN problem is to find the largest constrained pattern in a set of 2-intervals. The constrained pattern is a subset of the given 2-intervals such that any pair of them are R-comparable, where model . The problem stems from the study of general representation of RNA secondary structures. In this paper, we give three improved algorithms for different models. Firstly, an O(n{log} n +L) algorithm is proposed for the case , where is the total length of all 2-intervals (density d is the maximum number of 2-intervals over any point). This improves previous O(n ²log n) algorithm. Secondly, we use dynamic programming techniques to obtain an O(nlog n + dn) algorithm for the case R = { <, ⊏ }, which improves previous O(n ²) result. Finally, we present another algorithm for the case with disjoint support(interval ground set), which improves previous O(n ²□n) upper bound. A preliminary version of this article appears in Proceedings of the 16th Annual International Symposium on Algorithms and Computation, Springer LNCS, Vol. 3827, pp. 412–421, Hainan, China, December 19–21, 2005.     

Performance ratios of the Karmarkar-Karp differencing method

We consider the multiprocessor scheduling problem in which one must schedule n independent tasks nonpreemptively on m identical, parallel machines, such that the completion time of the last task is minimal. For this well-studied problem the Largest Differencing Method of Karmarkar and Karp outperforms other existing polynomial-time approximation algorithms from an average-case perspective. For m ≥ 3 the worst-case performance of the Largest Differencing Method has remained a challenging open problem. In this paper we show that the worst-case performance ratio is bounded between . For fixed m we establish further refined bounds in terms of n.     

Optimal st-orientations for plane triangulations

For plane triangulations, it has been proved that there exists a plane triangulation G with n vertices such that for any st-orientation of G, the length of the longest directed paths of G in the st-orientation is (Zhang and He in Lecture Notes in Computer Science, vol. 3383, pp. 425–430, 2005). In this paper, we prove the bound is optimal by showing that every plane triangulation G with n-vertices admits an st-orientation with the length of its longest directed paths bounded by . In addition, this st-orientation is constructible in linear time. A by-product of this result is that every plane graph G with n vertices admits a visibility representation with height , constructible in linear time, which is also optimal. A preliminary version of this paper was presented at AAIM 2007.     

Finding a Noncrossing Steiner Forest in Plane Graphs Under a 2-Face Condition

Let G = (V,E) be a plane graph with nonnegative edge weights, and let be a family of k vertex sets , called nets. Then a noncrossing Steiner forest for in G is a set of k trees in G such that each tree connects all vertices, called terminals, in net N _i, any two trees in do not cross each other, and the sum of edge weights of all trees is minimum. In this paper we give an algorithm to find a noncrossing Steiner forest in a plane graph G for the case where all terminals in nets lie on any two of the face boundaries of G. The algorithm takes time if G has n vertices and each net contains a bounded number of terminals.     

Center and Distinguisher for Strings with Unbounded Alphabet

Consider two sets and of strings of length L with characters from an unbounded alphabet , i.e., the size of is not bounded by a constant and has to be taken into consideration as a parameter for input size. A closest string s* of is a string that minimizes the maximum of Hamming¹ distance(s, s*) over all string s : s . In contrast, a farthest string t* from maximizes the minimum of Hamming distance(t*,t) over all elements t: t . A distinguisher of from is a string that is close to every string in and far away from any string in . We obtain polynomial time approximation schemes to settle the above problems.     

Online tradeoff scheduling on a single machine to minimize makespan and maximum lateness

In this paper, we consider the following single machine online tradeoff scheduling problem. A set of n independent jobs arrive online over time. Each job \(J_{j}\) has a release date \(r_{j}\), a processing time \(p_{j}\) and a delivery time \(q_{j}\). The characteristics of a job are unknown until it arrives. The goal is to find a schedule that minimizes the makespan \(C_{\max } = \max _{1 \le j \le n} C_{j}\) and the maximum lateness \(L_{\max } = \max _{1 \le j \le n} L_{j}\), where \(L_{j} = C_{j} + q_{j}\). For the problem, we present a nondominated \(( \rho , 1 + \displaystyle \frac{1}{\rho } )\)-competitive online algorithm for each \(\rho \) with \( 1 \le \rho \le \displaystyle \frac{\sqrt{5} + 1}{2}\).     

Efficient point coverage in wireless sensor networks

We study minimum-cost sensor placement on a bounded 3D sensing field, R, which comprises a number of discrete points that may or may not be grid points. Suppose we have ℓ types of sensors available with different sensing ranges and different costs. We want to find, given an integer σ ≥ 1, a selection of sensors and a subset of points to place these sensors such that every point in R is covered by at least σ sensors and the total cost of the sensors is minimum. This problem is known to be NP-hard. Let k_i denote the maximum number of points that can be covered by a sensor of the ith type. We present in this paper a polynomial-time approximation algorithm for this problem with a proven approximation ratio . In applications where the distance of any two points has a fixed positive lower bound, each k_i is a constant, and so we have a polynomial-time approximation algorithms with a constant guarantee. While γ may be large, we note that it is only a worst-case upper bound. In practice the actual approximation ratio is small, even on randomly generated points that do not have a fixed positive minimum distance between them. We provide a number of numerical results for comparing approximation solutions and optimal solutions, and show that the actual approximation ratios in these examples are all less than 3, even though γ is substantially larger. This research was supported in part by NSF under grant CCF-04080261 and by NSF of China under grant 60273062.     

Augmenting a Submodular and Posi-modular Set Function by a Multigraph

Given a finite set V and a set function , we consider the problem of constructing an undirected multigraph G = (V,E) such that the cut function together has value at least 2 for all non-empty and proper subsets of V. If f is intersecting submodular and posi-modular, and satisfies the tripartite inequality, then we show that such a multigraph G with the minimum number of edges can be found in time, where is the time to compute the value of f(X) for a subset .     

The inverse 1-maxian problem with edge length modification

We consider the problem of modifying the lengths of the edges of a graph at minimum cost such that a prespecified vertex becomes a 1-maxian with respect to the new edge lengths. The inverse 1-maxian problem with edge length modification is shown to be strongly -hard and remains weakly -hard even on series-parallel graphs. Moreover, a transformation of the inverse 1-maxian problem with edge length modification on a tree to a minimum cost circulation problem is given which solves the original problem in . This research has been supported by the Austrian Science Fund (FWF) Project P18918-N18.