Polynomial Time Approximation Scheme for the Rectilinear Steiner Arborescence Problem

Given a set N of n terminals in the first quadrant of the Euclidean plane E ², find a minimum length directed tree rooted at the origin o, connecting to all terminals in N, and consisting of only horizontal and vertical arcs oriented from left to right or from bottom to top. This problem is called rectilinear Steiner arborescence problem, which has been proved to be NP-complete recently (Shi and Su, 11th ACM-SIAM Symposium on Discrete Algorithms (SODA), January 2000, to appear). In this paper, we present a polynomial time approximation scheme for this problem.     

(1+ε)-competitive algorithm for online OVSF code assignment with resource augmentation

This paper studies the online Orthogonal Variable Spreading Factor (OVSF) code assignment problem with resource augmentation introduced by Erlebach et al. (in STACS 2004. LNCS, vol. 2996, pp. 270–281, 2004). We propose a (1+1/α)-competitive algorithm with help of (1+?α?)lg^? h trees for the height h of the OVSF code tree and any α≥1. In other words, it is a (1+ε)-competitive algorithm with help of (1+?1/ε?)lg^? h trees for any constant 0<ε≤1. In the case of α=1 (or ε=1), we obtain a 2-competitive algorithm with 2lg^? h trees, which substantially improves the previous resource of 3h/8+2 trees shown by Chan et al. (COCOON 2009. LNCS, vol. 5609, pp. 358–367, 2009). In another aspect, if it is not necessary to bound the incurred cost for individual requests to a constant, an amortized (4/3+δ)-competitive algorithm with (11/4+4/(3δ)) trees for any 0<δ≤4/3 is also designed in Chan et al. (COCOON 2009. LNCS, vol. 5609, pp. 358–367, 2009). The algorithm in this paper gives us a new trade-off between the competitive ratio and the resource augmentation when α≥3 (or ε≤1/3), although the incurred cost for individual requests is bounded to a constant.     

Chromatic number of distance graphs generated by the sets {2,3,x,y}

Let D be a set of positive integers. The distance graph generated by D has all integers ? as the vertex set; two vertices are adjacent whenever their absolute difference falls in D. We completely determine the chromatic number for the distance graphs generated by the sets D={2,3,x,y} for all values x and y. The methods we use include the density of sequences with missing differences and the parameter involved in the so called “lonely runner conjecture”. Previous results on this problem include: For x and y being prime numbers, this problem was completely solved by Voigt and Walther (Discrete Appl. Math. 51:197–209, 1994); and other results for special integers of x and y were obtained by Kemnitz and Kolberg (Discrete Math. 191:113–123, 1998) and by Voigt and Walther (Discrete Math. 97:395–397, 1991).     

Some results on the target set selection problem

In this paper we consider a fundamental problem in the area of viral marketing, called Target Set Selection problem. We study the problem when the underlying graph is a block-cactus graph, a chordal graph or a Hamming graph. We show that if G is a block-cactus graph, then the Target Set Selection problem can be solved in linear time, which generalizes Chen’s result (Discrete Math. 23:1400–1415, 2009) for trees, and the time complexity is much better than the algorithm in Ben-Zwi et al. (Discrete Optim., 2010) (for bounded treewidth graphs) when restricted to block-cactus graphs. We show that if the underlying graph G is a chordal graph with thresholds θ(v)≤2 for each vertex v in G, then the problem can be solved in linear time. For a Hamming graph G having thresholds θ(v)=2 for each vertex v of G, we precisely determine an optimal target set S for (G,θ). These results partially answer an open problem raised by Dreyer and Roberts (Discrete Appl. Math. 157:1615–1627, 2009).     

Equivalence of two conjectures on equitable coloring of graphs

A graph G is said to be equitably k-colorable if there exists a proper k-coloring of G such that the sizes of any two color classes differ by at most one. Let Δ(G) denote the maximum degree of a vertex in G. Two Brooks-type conjectures on equitable Δ(G)-colorability have been proposed in Chen and Yen (Discrete Math., 2011) and Kierstead and Kostochka (Combinatorica 30:201–216, 2010) independently. We prove the equivalence of these conjectures.     

Are there more almost separable partitions than separable partitions?

A partition of a set of n points in d-dimensional space into p parts is called an (almost) separable partition if the convex hulls formed by the parts are (almost) pairwise disjoint. These two partition classes are the most encountered ones in clustering and other partition problems for high-dimensional points and their usefulness depends critically on the issue whether their numbers are small. The problem of bounding separable partitions has been well studied in the literature (Alon and Onn in Discrete Appl. Math. 91:39–51, 1999; Barnes et al. in Math. Program. 54:69–86, 1992; Harding in Proc. Edinb. Math. Soc. 15:285–289, 1967; Hwang et al. in SIAM J. Optim. 10:70–81, 1999; Hwang and Rothblum in J. Comb. Optim. 21:423–433, 2011a). In this paper, we prove that for d≤2 or p≤2, the maximum number of almost separable partitions is equal to the maximum number of separable partitions.     

A primal-dual approximation algorithm for the Asymmetric Prize-Collecting TSP

We present a primal-dual ?log(n)?-approximation algorithm for the version of the asymmetric prize collecting traveling salesman problem, where the objective is to find a directed tour that visits a subset of vertices such that the length of the tour plus the sum of penalties associated with vertices not in the tour is as small as possible. The previous algorithm for the problem (V.H. Nguyen and T.T Nguyen in Int. J. Math. Oper. Res. 4(3):294–301, 2012) which is not combinatorial, is based on the Held-Karp relaxation and heuristic methods such as the Frieze et al.’s heuristic (Frieze et al. in Networks 12:23–39, 1982) or the recent Asadpour et al.’s heuristic for the ATSP (Asadpour et al. in 21st ACM-SIAM symposium on discrete algorithms, 2010). Depending on which of the two heuristics is used, it gives respectively 1+?log(n)? and $3+ 8\frac{\log(n)}{\log(\log(n))}$ as an approximation ratio. Our algorithm achieves an approximation ratio of ?log(n)? which is weaker than $3+ 8\frac{\log(n)}{\log(\log(n))}$ but represents the first combinatorial approximation algorithm for the Asymmetric Prize-Collecting TSP.     

Shifting strategy for geometric graphs without?geometry

We give a simple framework which is an alternative to the celebrated and widely used shifting strategy of Hochbaum and Maass (J. ACM 32(1):103?C136, 1985) which has yielded efficient algorithms with good approximation bounds for numerous optimization problems in low-dimensional Euclidean space. Our framework does not require the input graph/metric to have a geometric realization??it only requires that the input graph satisfy some weak property referred to as growth boundedness. Growth bounded graphs form an important graph class that has been used to model wireless networks. We show how to apply the framework to obtain a polynomial time approximation scheme (PTAS) for the maximum (weighted) independent set problem on this important graph class; the problem is W[1]-complete. Via a more sophisticated application of our framework, we show how to obtain a PTAS for the maximum (weighted) independent set for intersection graphs of (low-dimensional) fat objects that are expressed without geometry. Erlebach et al. (SIAM J. Comput. 34(6):1302?C1323, 2005) and Chan (J. Algorithms 46(2):178?C189, 2003) independently gave a PTAS for maximum weighted independent set problem for intersection graphs of fat geometric objects, say ball graphs, which required a geometric representation of the input. Our result gives a positive answer to a question of Erlebach et al. (SIAM J. Comput. 34(6):1302?C1323, 2005) who asked if a PTAS for this problem can be obtained without access to a geometric representation.     

L(d,1)-labelings of the edge-path-replacement of a graph

For two positive integers j and k with j≥k, an L(j,k)-labeling of a graph G is an assignment of nonnegative integers to V(G) such that the difference between labels of adjacent vertices is at least j, and the difference between labels of vertices that are distance two apart is at least k. The span of an L(j,k)-labeling of a graph G is the difference between the maximum and minimum integers used by it. The L(j,k)-labelings-number of G is the minimum span over all L(j,k)-labelings of G. This paper focuses on L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G. Note that G(P ₃) is the incidence graph of G. L(d,1)-labelings of the edge-path-replacement G(P _k ) of a graph, called (d,1)-total labeling of G, was introduced in 2002 by Havet and Yu (Workshop graphs and algorithms, 2003; Discrete Math 308:493–513, 2008). Havet and Yu (Discrete Math 308:498–513, 2008) obtained the bound $\Delta+ d-1\leq\lambda^{T}_{d}(G)\leq2\Delta+ d-1$ and conjectured $\lambda^{T}_{d}(G)\leq\Delta+2d-1$ . In (Lü in J Comb Optim, to appear; Zhejiang University, submitted), we worked on L(2,1)-labelings-number and L(1,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G, and obtained that λ(G(P _k ))≤Δ+2 for k≥5, and conjecture λ(G(P ₄))≤Δ+2 for any graph G with maximum degree Δ. In this paper, we will study L(d,1)-labelings-number of the edge-path-replacement G(P _k ) of a graph G for d≥3 and k≥4.     

An improved distributed data aggregation scheduling in wireless sensor networks

This paper focuses on the distributed data aggregation collision-free scheduling problem, which is one of very important issues in wireless sensor networks. Bo et al. (Proc. IEEE INFOCOM, 2009) proposed an approximate distributed algorithm for the problem and Xu et al. (Proc. ACM FOWANC, 2009) proposed a centralized algorithm and its distributed implementation to generate a collision-free scheduling for the problem, which are the only two existing distributed algorithms. Unfortunately, there are a few mistakes in their performance analysis in Bo et al. (Proc. IEEE INFOCOM, 2009), and the distributed algorithm can not get the same latency as the centralized algorithm because the distributed implementation was not an accurate implementation of the centralized algorithm (Xu et al. in Proc. ACM FOWANC, 2009). According to those, we propose an improved distributed algorithm to generate a collision-free schedule for data aggregation in wireless sensor networks. Not an arbitrary tree in Bo et al. (Proc. IEEE INFOCOM, 2009) but a breadth first search tree (BFS) rooted at the sink node is adopted, the bounded latency 61R+5Δ?67 of the schedule is obtained, where R is the radius of the network with respect to the sink node and Δ is the maximum node degree. We also correct the latency bound of the schedule in Bo et al. (Proc. IEEE INFOCOM, 2009) as 61D+5Δ?67, where D is a diameter of the network and prove that our algorithm is more efficient than the algorithm (Bo et al. in Proc. IEEE INFOCOM, 2009). We also give a latency bound for the distributed implementation in Xu et al. (Proc. ACM FOWANC, 2009).     

On minimum balanced bipartitions of triangle-free graphs

A balanced bipartition of a graph G is a partition of V(G) into two subsets V ₁ and V ₂ that differ in cardinality by at most 1. A minimum balanced bipartition of G is a balanced bipartition V ₁, V ₂ of G minimizing e(V ₁,V ₂), where e(V ₁,V ₂) is the number of edges joining V ₁ and V ₂ and is usually referred to as the size of the bipartition. In this paper, we show that every 2-connected graph G admits a balanced bipartition V ₁,V ₂ such that the subgraphs of G induced by V ₁ and by V ₂ are both connected. This yields a good upper bound to the size of minimum balanced bipartition of sparse graphs. We also present two upper bounds to the size of minimum balanced bipartitions of triangle-free graphs which sharpen the corresponding bounds of Fan et al. (Discrete Math. 312:1077–1083, 2012).     

Improved approximation algorithms for the max-bisection and the disjoint 2-catalog segmentation problems

We consider the max-bisection problem and the disjoint 2-catalog segmentation problem, two well-known NP-hard combinatorial optimization problems. For the first problem, we apply the semidefinite programming (SDP) relaxation and the RPR² technique of Feige and Langberg (J. Algorithms 60:1–23, 2006) to obtain a performance curve as a function of the ratio of the optimal SDP value over the total weight through finer analysis under the assumption of convexity of the RPR² function. This ratio is shown to be in the range of [0.5,1]. The performance curve implies better approximation performance when this ratio is away from 0.92, corresponding to the lowest point on this curve with the currently best approximation ratio of 0.7031 due to Feige and Langberg (J. Algorithms 60:1–23, 2006). For the second problem, similar technique results in an approximation ratio of 0.7469, improving the previously best known result 0.7317 due to Wu et al. (J. Ind. Manag. Optim. 8:117–126, 2012).     

(p,q)-total labeling of complete graphs

Given a graph G and positive integers p,q with p≥q, the (p,q)-total number $\lambda_{p,q}^{T}(G)$ of G is the width of the smallest range of integers that suffices to label the vertices and the edges of G such that the labels of any two adjacent vertices are at least q apart, the labels of any two adjacent edges are at least q apart, and the difference between the labels of a vertex and its incident edges is at least p. Havet and Yu (Discrete Math 308:496–513, 2008) first introduced this problem and determined the exact value of $\lambda_{p,1}^{T}(K_{n})$ except for even n with p+5≤n≤6p ²?10p+4. Their proof for showing that $\lambda _{p,1}^{T}(K_{n})\leq n+2p-3$ for odd n has some mistakes. In this paper, we prove that if n is odd, then $\lambda_{p}^{T}(K_{n})\leq n+2p-3$ if p=2, p=3, or $4\lfloor\frac{p}{2}\rfloor+3\leq n\leq4p-1$ . And we extend some results that were given in Havet and Yu (Discrete Math 308:496–513, 2008). Beside these, we give a lower bound for $\lambda_{p,q}^{T}(K_{n})$ under the condition that q<p<2q.     

Acyclic edge coloring of planar graphs without 4-cycles

An acyclic edge coloring of a graph G is a proper edge coloring such that no bichromatic cycles are produced. The acyclic chromatic index a′(G) of G is the smallest integer k such that G has an acyclic edge coloring using k colors. Fiam?ik (Math. Slovaca 28:139–145, 1978) and later Alon, Sudakov and Zaks (J. Graph Theory 37:157–167, 2001) conjectured that a′(G)≤Δ+2 for any simple graph G with maximum degree Δ. In this paper, we confirm this conjecture for planar graphs G with Δ≠4 and without 4-cycles.     

An improved approximation algorithm for uncapacitated facility location problem with penalties

In this paper, we consider an interesting variant of the classical facility location problem called uncapacitated facility location problem with penalties (UFLWP for short) in which each client is either assigned to an opened facility or rejected by paying a penalty. The UFLWP problem has been effectively used to model the facility location problem with outliers. Three constant approximation algorithms have been obtained (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001; Jain et al. in J. ACM 50(6):795–824, 2003; Xu and Xu in Inf. Process. Lett. 94(3):119–123, 2005), and the best known performance ratio is 2. The only known hardness result is a 1.463-inapproximability result inherited from the uncapacitated facility location problem (Guha and Khuller in J. Algorithms 31(1):228–248, 1999). In this paper, We present a 1.8526-approximation algorithm for the UFLWP problem. Our algorithm significantly reduces the gap between known performance ratio and the inapproximability result. Our algorithm first enhances the primal-dual method for the UFLWP problem (Charikar et al. in Proceedings of the Symposium on Discrete Algorithms, pp. 642–651, 2001) so that outliers can be recognized more efficiently, and then applies a local search heuristic (Charikar and Guha in Proceedings of the 39th IEEE Symposium on Foundations of Computer Science, pp. 378–388, 1999) to further reduce the cost for serving those non-rejected clients. Our algorithm is simple and can be easily implemented. The research of this work was supported in part by NSF through CAREER award CCF-0546509 and grant IIS-0713489. A preliminary version of this paper appeared in the Proceedings of the 11th Annual International Computing and Combinatorics Conference (COCOON’05).     

Constant time approximation scheme for largest well predicted subset

The largest well predicted subset problem is formulated for comparison of two predicted 3D protein structures from the same sequence. A 3D protein structure is represented by an ordered point set A={a ₁,…,a _n }, where each a _i is a point in 3D space. Given two ordered point sets A={a ₁,…,a _n } and B={b ₁,b ₂,…b _n } containing n points, and a threshold d, the largest well predicted subset problem is to find the rigid body transformation T for a largest subset B _opt of B such that the distance between a _i and T(b _i ) is at most d for every b _i in B _opt . A meaningful prediction requires that the size of B _opt is at least αn for some constant α (Li et al. in CPM 2008, 2008). We use LWPS(A,B,d,α) to denote the largest well predicted subset problem with meaningful prediction. An (1+δ ₁,1?δ ₂)-approximation for LWPS(A,B,d,α) is to find a transformation T to bring a subset B′?B of size at least (1?δ ₂)|B _opt | such that for each b _i ∈B′, the Euclidean distance between the two points distance?(a _i ,T(b _i ))≤(1+δ ₁)d. We develop a constant time (1+δ ₁,1?δ ₂)-approximation algorithm for LWPS(A,B,d,α) for arbitrary positive constants δ ₁ and δ ₂. To our knowledge, this is the first constant time algorithm in this area. Li et al. (CPM 2008, 2008) showed an $O(n(\log n)^{2}/\delta_{1}^{5})$ time randomized (1+δ ₁)-distance approximation algorithm for the largest well predicted subset problem under meaningful prediction. We also study a closely related problem, the bottleneck distance problem, where we are given two ordered point sets A={a ₁,…,a _n } and B={b ₁,b ₂,…b _n } containing n points and the problem is to find the smallest d _opt such that there exists a rigid transformation T with distance(a _i ,T(b _i ))≤d _opt for every point b _i ∈B. A (1+δ)-approximation for the bottleneck distance problem is to find a transformation T, such that for each b _i ∈B, distance?(a _i ,T(b _i ))≤(1+δ)d _opt , where δ is a constant. For an arbitrary constant δ, we obtain a linear O(n/δ ⁶) time (1+δ)-algorithm for the bottleneck distance problem. The best known algorithms for both problems require super-linear time (Li et al. in CPM 2008, 2008).     

Approximation algorithm for uniform bounded facility location problem

The uniform bounded facility location problem (UBFLP) seeks for the optimal way of locating facilities to minimize total costs (opening costs plus routing costs), while the maximal routing costs of all clients are at most a given bound M. After building a mixed 0–1 integer programming model for UBFLP, we present the first constant-factor approximation algorithm with an approximation guarantee of 6.853+? for UBFLP on plane, which is composed of the algorithm by Dai and Yu (Theor. Comp. Sci. 410:756–765, 2009) and the schema of Xu and Xu (J. Comb. Optim. 17:424–436, 2008). We also provide a heuristic algorithm based on Benders decomposition to solve UBFLP on general graphes, and the computational experience shows that the heuristic works well.     

On the maximum number of fault-free mutually independent Hamiltonian cycles in the faulty hypercube

Hsieh and Yu (2007) first claimed that an injured n-dimensional hypercube Q _n contains (n?1?f)-mutually independent fault-free Hamiltonian cycles, where f≤n?2 denotes the total number of permanent edge-faults in Q _n for n≥4, and edge-faults can occur everywhere at random. Later, Kueng et al. (2009a) presented a formal proof to validate Hsieh and Yu’s argument. This paper aims to improve this mentioned result by showing that up to (n?f)-mutually independent fault-free Hamiltonian cycles can be embedded under the same condition. Let F denote the set of f faulty edges. If all faulty edges happen to be incident with an identical vertex s, i.e., the minimum degree of the survival graph Q _n ?F is equal to n?f, then Q _n ?F contains at most (n?f)-mutually independent Hamiltonian cycles starting from s. From such a point of view, the presented result is optimal. Thus, not only does our improvement increase the number of mutually independent fault-free Hamiltonian cycles by one, but also the optimality can be achieved.     

Three conjectures on the signed cycle domination in graphs

Let G=(V,E) be a graph, a function g:E→{?1,1} is said to be a signed cycle dominating function (SCDF for short) of G if ∑ _e∈E(C) g(e)≥1 holds for any induced cycle C of G. The signed cycle domination number of G is defined as γ _sc (G)=min{∑ _e∈E(G) g(e)∣g is an SCDF of G}. Xu (Discrete Math. 309:1007–1012, 2009) first researched the signed cycle domination number of graphs and raised the following conjectures: (1) Let G be a maximal planar graphs of order n≥3. Then γ _sc (G)=n?2; (2) For any graph G with δ(G)=3, γ _sc (G)≥1; (3) For any 2-connected graph G, γ _sc (G)≥1. In this paper, we present some results about these conjectures.