Some variations on constrained minimum enclosing circle problem

Given a set P of n points and a straight line L, we study three important variations of minimum enclosing circle problem as follows:

1. Computing k identical circles of minimum radius with centers on L, whose union covers all the points in P.
2. Computing the minimum radius circle centered on L that can enclose at least k points of P.
3. If each point in P is associated with one of the k given colors, then compute a minimum radius circle with center on L such that at least one point of each color lies inside it.

We propose three algorithms for Problem (i). The first one runs in O(nklogn) time and O(n) space. The second one is efficient where k?n; it runs in O(nlogn+nk+k ²log³ n) time and O(nlogn) space. The third one is based on parametric search and it runs in O(nlogn+klog⁴ n) time. For Problem (ii), the time and space complexities of the proposed algorithm are O(nk) and O(n) respectively. For Problem (iii), our proposed algorithm runs in O(nlogn) time and O(n) space.     

Vertices Contained in all or in no Minimum Paired-Dominating Set of a Tree

A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. We characterize the set of vertices of a tree that are contained in all, or in no, minimum paired-dominating sets of the tree. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

On the complexity and approximation of non-unique probe selection using d-disjunct matrix

In this paper, we studied the MINimum-d-Disjunct Submatrix (MIN-d-DS), which can be used to select the minimum number of non-unique probes for viruses identification. We prove that MIN-d-DS is NP-hard for any fixed d. Using d-disjunct matrix, we present an O(log k)-approximation algorithm where k is an upper bound on the maximum number of targets hybridized to a probe. We also present a (1+(d+1)log n)-approximation algorithm to identify at most d targets in the presence of experimental errors. Our approximation algorithms also yield a linear time complexity for the decoding algorithms. The research of T. Znati was supported in part by National Science Foundation under grant CCF-0548895.     

A dynamic programming approach of finding an optimal broadcast schedule in minimizing total flow time

We study the problem of (off-line) broadcast scheduling in minimizing total flow time and propose a dynamic programming approach to compute an optimal broadcast schedule. Suppose the broadcast server has k pages and the last page request arrives at time n. The optimal schedule can be computed in O(k³(n+k)^k−1) time for the case that the server has a single broadcast channel. For m channels case, i.e., the server can broadcast m different pages at a time where m < k, the optimal schedule can be computed in O(n^k−m) time when k and m are constants. Note that this broadcast scheduling problem is NP-hard when k is a variable and will take O(n^k−m+1) time when k is fixed and m ≥ 1 with the straightforward implementation of the dynamic programming approach. The preliminary version of this paper appeared in Proceedings of the 11th Annual International Computing and Combinatorics Conference as “Off-line Algorithms for Minimizing the Total Flow Time in Broadcast Scheduling”.     

Minimal Tetrahedralizations of a Class of Polyhedra

Given a simple polyhedron P in the three dimensional Euclidean space, different tetrahedralizations of P may contain different numbers of tetrahedra. The minimal tetrahedralization is a tetrahedralization with the minimum number of tetrahedra. In this paper, we present some properties of the graph of polyhedra. Then we identify a class of polyhedra and show that this kind of polyhedra can be minimally tetrahedralized in O(n ²) time.     

Efficient Algorithms and Implementations for Optimizing the Sum of Linear Fractional Functions,with Applications

This paper presents an improved algorithm for solving the sum of linear fractional functions (SOLF) problem in 1-D and 2-D. A key subproblem to our solution is the off-line ratio query (OLRQ) problem, which asks to find the optimal values of a sequence of m linear fractional functions (called ratios), each ratio subject to a feasible domain defined by O(n) linear constraints. Based on some geometric properties and the parametric linear programming technique, we develop an algorithm that solves the OLRQ problem in O((m+n)log (m+n)) time. The OLRQ algorithm can be used to speed up every iteration of a known iterative SOLF algorithm, from O(m(m+n)) time to O((m+n)log (m+n)), in 1-D and 2-D. Implementation results of our improved 1-D and 2-D SOLF algorithm have shown that in most cases it outperforms the commonly-used approaches for the SOLF problem. We also apply our techniques to some problems in computational geometry and other areas, improving the previous results.This research was supported in part by the National Science Foundation under Grant CCR-9623585.The research of this author was supported in part by National Science Foundation under grant CCF-0430366.Grant-in-Aid of Ministry of Science, Culture and Education of Japan, No. 10780274.The research of this author was supported in part by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Scientific Researchon Priority Areas     

Finding an anti-risk path between two nodes in undirected graphs

Given a weighted graph G=(V,E) with a source s and a destination t, a traveler has to go from s to t. However, some of the edges may be blocked at certain times, and the traveler only observes that upon reaching an adjacent site of the blocked edge. Let ℘={P _G (s,t)} be the set of all paths from s to t. The risk of a path is defined as the longest travel under the assumption that any edge of the path may be blocked. The paper will propose the Anti-risk Path Problem of finding a path P _G (s,t) in ℘ such that it has minimum risk. We will show that this problem can be solved in O(mn+n ²log n) time suppose that at most one edge may be blocked, where n and m denote the number of vertices and edges in G, respectively. This research is supported by NSF of China under Grants 70525004, 60736027, 70121001 and Postdoctoral Science Foundation of China under Grant 20060401003.     

Two Variations of the Minimum Steiner Problem

Given a set S of starting vertices and a set T of terminating vertices in a graph G = (V,E) with non-negative weights on edges, the minimum Steiner network problem is to find a subgraph of G with the minimum total edge weight. In such a subgraph, we require that for each vertex s S and t T, there is a path from s to a terminating vertex as well as a path from a starting vertex to t. This problem can easily be proven NP-hard. For solving the minimum Steiner network problem, we first present an algorithm that runs in time and space that both are polynomial in n with constant degrees, but exponential in |S|+|T|, where n is the number of vertices in G. Then we present an algorithm that uses space that is quadratic in n and runs in time that is polynomial in n with a degree O(max {max {|S|,|T|}–2,min {|S|,|T|}–1}). In spite of this degree, we prove that the number of Steiner vertices in our solution can be as large as |S|+|T|–2. Our algorithm can enumerate all possible optimal solutions. The input graph G can either be undirected or directed acyclic. We also give a linear time algorithm for the special case when min {|S|,|T|} = 1 and max {|S|,|T|} = 2.The minimum union paths problem is similar to the minimum Steiner network problem except that we are given a set H of hitting vertices in G in addition to the sets of starting and terminating vertices. We want to find a subgraph of G with the minimum total edge weight such that the conditions required by the minimum Steiner network problem are satisfied as well as the condition that every hitting vertex is on a path from a starting vertex to a terminating vertex. Furthermore, G must be directed acyclic. For solving the minimum union paths problem, we also present algorithms that have a time and space tradeoff similar to algorithms for the minimum Steiner network problem. We also give a linear time algorithm for the special case when |S| = 1, |T| = 1 and |H| = 2.An extended abstract of part of this paper appears in Hsu et al. (1996).Supported in part by the National Science Foundation under Grants CCR-9309743 and INT-9207212, and by the Office of Naval Research under Grant No. N00014-93-1-0272.Supported in part by the National Science Council, Taiwan, ROC, under Grant No. NSC-83-0408-E-001-021.     

Opportunistic data structures for range queries

In this paper, we study the problem of supporting range sum queries on a compressed sequence of values. For a sequence of n k-bit integers, k ≤ O(log n), our data structures require asymptotically the same amount of storage as the compressed sequence if compressed using the Lempel-Ziv algorithm. The basic structure supports range sum queries in O(log n) time. With an increase by a constant factor in the storage complexity, the query time can be improved to O(log log n + k). The work described in this paper is fully supported by a grant from the Research Grant Council of the Hong Kong Special Administrative Region, China (CityU 1071/02E). A preliminary version has appeared in 11th International Conference in Computing and Combinatorics (COCOON'05).     

Perfect Circular Arc Coloring

The circular arc coloring problem is to find a minimum coloring of a set of arcs of a circle so that no two overlapping arcs share a color. This NP-hard problem arises in a rich variety of applications and has been studied extensively. In this paper we present an O(n² m) combinatorial algorithm for optimally coloring any set of arcs that corresponds to a perfect graph, and propose a new approach to the general circular arc coloring problem.Partially supported by Project 02139 of Education Ministry of China.Supported in part by the Research Grants Council of Hong Kong (Project No. HKU7054/03P) and a seed funding for basic research of HKU.     

On 3-Stage Clos Networks with Different Nonblocking Requirements on Two Types of Calls

Hwang and Lin introduced a new nonblocking requirement for 2-cast traffic which imposes different requirements on different types of coexisting calls. The requirement is strictly nonblocking for point-to-point calls among the 2-cast traffic, and is rearrangeable for genuine 2-cast calls. They conjectured that the 3-stage Clos network C(n,n,r,r,2n) satisfies the above requirement. We prove that C(n,n,4,r,2n) satisfies the above requirement.Supported in part by NSC91-2115-M009-002.Supported in part by the National Science Council under grant NSC91-2115-M009-010 and by the Li-Li-Tai-Yang Network Research Center of National Chiao Tung University.     

Hardness of <Emphasis Type="Italic">k</Emphasis>-Vertex-Connected Subgraph Augmentation Problem

Given a k-connected graph G=(V,E) and V ^′⊂V, k-Vertex-Connected Subgraph Augmentation Problem (k-VCSAP) is to find S⊂V∖V ^′ with minimum cardinality such that the subgraph induced by V ^′∪S is k-connected. In this paper, we study the hardness of k-VCSAP in undirect graphs. We first prove k-VCSAP is APX-hard. Then, we improve the lower bound in two ways by relying on different assumptions. That is, we prove no algorithm for k-VCSAP has a PR better than O(log (log n)) unless P=NP and O(log n) unless NP⊆DTIME(n ^{O(log log n)}), where n is the size of an input graph.     

Restricted domination parameters in graphs

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊂eqV(G) is an m-tuple dominating set if S dominates every vertex of G at least m times, and an m-dominating set if S dominates every vertex of G−S at least m times. The minimum cardinality of a dominating set is γ, of an m-dominating set is γ _m , and of an m-tuple dominating set is mtupledom. For a property π of subsets of V(G), with associated parameter f_π, the k-restricted π-number r _k (G,f_π) is the smallest integer r such that given any subset K of (at most) k vertices of G, there exists a π set containing K of (at most) cardinality r. We show that for 1< k < n where n is the order of G: (a) if G has minimum degree m, then r _k (G,γ _m ) < (mn+k)/(m+1); (b) if G has minimum degree 3, then r _k (G,γ) < (3n+5k)/8; and (c) if G is connected with minimum degree at least 2, then r _k (G,ddom) < 3n/4 + 2k/7. These bounds are sharp. Research supported in part by the South African National Research Foundation and the University of KwaZulu-Natal.     

Single-machine batch delivery scheduling with an assignable common due window

This paper addresses a batch delivery single-machine scheduling problem in which jobs have an assignable common due window. Each job will incur an early (tardy) penalty if it is early (tardy) with respect to the common due window under a given schedule. There is no capacity limit on each delivery batch, and the cost per batch delivery is fixed and independent of the number of jobs in the batch. The objective is to find the optimal size and location of the window, the optimal dispatch date for each job, as well as an optimal job sequence to minimize a cost function based on earliness, tardiness, holding time, window location, window size, and batch delivery. We show that the problem can be optimally solved in O(n⁸)$O(n^8)$ time by a dynamic programming algorithm under a reasonable assumption on the relationships among the cost parameters. A computational experiment is also conducted to evaluate the performance of the proposed algorithm. We also show that some special cases of the problem can be optimally solved by lower order algorithms.     

The web proxy location problem in general tree of rings networks

The Web proxy location problem in general networks is an NP-hard problem. In this paper, we study the problem in networks showing a general tree of rings topology. We improve the results of the tree case in literature and get an exact algorithm with time complexity O(nhk), where n is the number of nodes in the tree, h is the height of the tree (the server is in the root of the tree), and k is the number of web proxies to be placed in the net. For the case of networks with a general tree of rings topology we present an exact algorithm with O(kn ²) time complexity.This research has been supported by NSF of China (No. 10371028) and the Educational Department grant of Zhejiang Province (No. 20030622).     

Incrementing Bipartite Digraph Edge-Connectivity

This paper solves the problem of increasing the edge-connectivity of a bipartite digraph by adding the smallest number of new edges that preserve bipartiteness. A natural application arises when we wish to reinforce a 2-dimensional square grid framework with cables. We actually solve the more general problem of covering a crossing family of sets with the smallest number of directed edges, where each new edge must join the blocks of a given bipartition of the elements. The smallest number of new edges is given by a min-max formula that has six infinite families of exceptional cases. We discuss a problem on network flows whose solution has a similar formula with three infinite families of exceptional cases. We also discuss a problem on arborescences whose solution has five infinite families of exceptions. We give an algorithm that increases the edge-connectivity of a bipartite digraph in the same time as the best-known algorithm for the problem without the bipartite constraint: O(km log n) for unweighted digraphs and O(nm log (n ²/m)) for weighted digraphs, where n, m and k are the number of vertices and edges of the given graph and the target connectivity, respectively.     

Computational Comparison Studies of Quadratic Assignment Like Formulations for the In Silico Sequence Selection Problem in De Novo Protein Design

In this paper an O(n²) mathematical formulation for in silico sequence selection in de novo protein design proposed by Klepeis et al. (2003, 2004), in which the number of additional variables and linear constraints scales with the square of the number of binary variables, is compared to three O(n) formulations. It is found that the O(n²) formulation is superior to the O(n) formulations on most sequence search spaces. The superiority of the O(n²) formulation is due to the reformulation linearization techniques (RLTs), since the O(n²) formulation without RLTs is found to be computationally less efficient than the O(n) formulations. In addition, new algorithmic enhancing components of RLTs with inequality constraints, triangle inequalities, and Dead-End Elimination (DEE) type preprocessing are added to the O(n²) formulation. The current best O(n²) formulation, which is the original formulation from Klepeis et al. (2003, 2004) plus DEE type preprocessing, is proposed for in silico sequence search. For a test problem with a search space of 3.4×10⁴⁵ sequences, this new improved model is able to reduce the required CPU time by 67%.     

Node-weighted Steiner tree approximation in unit disk graphs

Given a graph G=(V,E) with node weight w:V→R ⁺ and a subset S⊆V, find a minimum total weight tree interconnecting all nodes in S. This is the node-weighted Steiner tree problem which will be studied in this paper. In general, this problem is NP-hard and cannot be approximated by a polynomial time algorithm with performance ratio aln n for any 0<a<1 unless NP⊆DTIME(n ^O(log n)), where n is the number of nodes in s. In this paper, we are the first to show that even though for unit disk graphs, the problem is still NP-hard and it has a polynomial time constant approximation. We present a 2.5ρ-approximation where ρ is the best known performance ratio for polynomial time approximation of classical Steiner minimum tree problem in graphs. As a corollary, we obtain that there is a polynomial time (9.875+ε)-approximation algorithm for minimum weight connected dominating set in unit disk graphs, and also there is a polynomial time (4.875+ε)-approximation algorithm for minimum weight connected vertex cover in unit disk graphs.     

Separating sublinear time computations by approximate diameter

We study the problem of separating sublinear time computations via approximating the diameter for a sequence S=p ₁ p ₂ ⋅⋅⋅ p _n of points in a metric space, in which any two consecutive points have the same distance. The computation is considered respectively under deterministic, zero error randomized, and bounded error randomized models. We obtain a class of separations using various versions of the approximate diameter problem based on restrictions on input data. We derive tight sublinear time separations for each of the three computation models via proving that computation with O(n ^r ) time is strictly more powerful than that with O(n ^r−ε ) time, where r and ε are arbitrary parameters in (0,1) and (0,r) respectively. We show that, for any parameter r∈(0,1), the bounded error randomized sublinear time computation in time O(n ^r ) cannot be simulated by any zero error randomized sublinear time algorithm in o(n) time or queries; and the same is true for zero error randomized computation versus deterministic computation.