Approximating the Minimum k-way Cut in a Graph via Minimum 3-way Cuts

For an edge weighted undirected graph G and an integer k > 2, a k-way cut is a set of edges whose removal leaves G with at least k components. We propose a simple approximation algorithm to the minimum k-way cut problem. It computes a nearly optimal k-way cut by using a set of minimum 3-way cuts. We show that the performance ratio of our algorithm is 2 – 3/k for an odd k and 2 – (3k – 4)/(k ² – k) for an even k. The running time is O(kmn ³ log(n ²/m)) where n and m are the numbers of vertices and edges respectively.     

Computing maximum upward planar subgraphs of single-source embedded digraphs

We show how to compute a maximum upward planar single-source subgraph of a single-source embedded DAG G _φ . We first show that finding a maximum upward planar subgraph of a single-source embedded digraph is NP-complete. We then give a new characterization of upward planar single-source digraphs. We use this characterization to present an algorithm that computes a maximum upward planar single-source subgraph of a single-source embedded DAG. This algorithm takes O(n ⁴) time in the worst case and O(n ³) time on average.     

Efficient algorithms for finding a longest common increasing subsequence

We study the problem of finding a longest common increasing subsequence (LCIS) of multiple sequences of numbers. The LCIS problem is a fundamental issue in various application areas, including the whole genome alignment. In this paper we give an efficient algorithm to find the LCIS of two sequences in time where n is the length of each sequence andr is the number of ordered pairs of positions at which the two sequences match, ℓ is the length of the LCIS, and Sort(n) is the time to sort n numbers. For m sequences wherem ≥ 3, we find the LCIS in Sort(n)) time where r is the total number of m-tuples of positions at which the m sequences match. The previous results find the LCIS of two sequences in O(n ²) and Sort(n)) time. Our algorithm is faster when r is relatively small, e.g., for .     

The hamiltonian numbers in digraphs

In the paper, we study the hamiltonian numbers in digraphs. A hamiltonian walk of a digraph D is a closed spanning directed walk with minimum length in D. The length of a hamiltonian walk of a digraph D is called the hamiltonian number of D, denoted h(D). We prove that if a digraph D of order n is strongly connected, then $n\leq h(D)\leq\lfloor\frac{(n+1)^{2}}{4} \rfloor$ , and hence characterize the strongly connected digraphs of order n with hamiltonian number $\lfloor\frac{(n+1)^{2}}{4} \rfloor$ . In addition, we show that for each k with $4\leq n\leq k\leq\lfloor \frac{(n+1)^{2}}{4} \rfloor$ , there exists a digraph with order n and hamiltonian number k. Furthermore, we also study the hamiltonian spectra of graphs.     

Maximum size of digraphs with some parameters

In this paper, we determine the maximum sizes of strong digraphs under the constraint that their some parameters are fixed, such as vertex connectivity, edge-connectivity, the number of cut vertices. The corresponding extremal digraphs are also characterized. In addition, we establish Nordhaus–Gaddum type theorem for the diameter when \(\overrightarrow{K_n}\) decomposing into many parts. We also pose a related conjecture for Wiener index of digraphs.     

On some geometric problems of color-spanning sets

In this paper we study several geometric problems of color-spanning sets: given n points with m colors in the plane, selecting m points with m distinct colors such that some geometric properties of the m selected points are minimized or maximized. The geometric properties studied in this paper are the maximum diameter, the largest closest pair, the planar smallest minimum spanning tree, the planar largest minimum spanning tree and the planar smallest perimeter convex hull. We propose an O(n ^1+ε ) time algorithm for the maximum diameter color-spanning set problem where ε could be an arbitrarily small positive constant. Then, we present hardness proofs for the other problems and propose two efficient constant factor approximation algorithms for the planar smallest perimeter color-spanning convex hull problem.     

Hamiltonian numbers in oriented graphs

A hamiltonian walk of a digraph is a closed spanning directed walk with minimum length in the digraph. The length of a hamiltonian walk in a digraph D is called the hamiltonian number of D, denoted by h(D). In Chang and Tong (J Comb Optim 25:694–701, 2013), Chang and Tong proved that for a strongly connected digraph D of order n, \(n\le h(D)\le \lfloor \frac{(n+1)^2}{4} \rfloor \), and characterized the strongly connected digraphs of order n with hamiltonian number \(\lfloor \frac{(n+1)^2}{4} \rfloor \). In the paper, we characterized the strongly connected digraphs of order n with hamiltonian number \(\lfloor \frac{(n+1)^2}{4} \rfloor -1\) and show that for any triple of integers n, k and t with \(n\ge 5\), \(n\ge k\ge 3\) and \(t\ge 0\), there is a class of nonisomorphic digraphs with order n and hamiltonian number \(n(n-k+1)-t\).     

Subhypergraph counts in extremal and random hypergraphs and the fractional <Emphasis Type="Italic">q</Emphasis>-independence

We study the extremal parameter N(n,m,H) which is the largest number of copies of a hypergraph H that can be formed of at most n vertices and m edges. Generalizing previous work of Alon (Isr. J. Math. 38:116–130, 1981), Friedgut and Kahn (Isr. J. Math. 105:251–256, 1998) and Janson, Oleszkiewicz and the third author (Isr. J. Math. 142:61–92, 2004), we obtain an asymptotic formula for N(n,m,H) which is strongly related to the solution α _q (H) of a linear programming problem, called here the fractional q-independence number of H. We observe that α _q (H) is a piecewise linear function of q and determine it explicitly for some ranges of q and some classes of H. As an application, we derive exponential bounds on the upper tail of the distribution of the number of copies of H in a random hypergraph.     

Finding longest increasing and common subsequences in streaming data

We present algorithms and lower bounds for the Longest Increasing Subsequence (LIS) and Longest Common Subsequence (LCS) problems in the data-streaming model. To decide if the LIS of a given stream of elements drawn from an alphabet αbet has length at least k, we discuss a one-pass algorithm using O(k log αbetsize) space, with update time either O(log k) or O(log log αbetsize); for αbetsize = O(1), we can achieve O(log k) space and constant-time updates. We also prove a lower bound of Ω(k) on the space requirement for this problem for general alphabets αbet, even when the input stream is a permutation of αbet. For finding the actual LIS, we give a ⌈log (1 + 1/ɛ)-pass algorithm using O(k^1+ɛlog αbetsize) space, for any ɛ > 0. For LCS, there is a trivial Θ(1)-approximate O(log n)-space streaming algorithm when αbetsize = O(1). For general alphabets αbet, the problem is much harder. We prove several lower bounds on the LCS problem, of which the strongest is the following: it is necessary to use Ω(n/ρ²) space to approximate the LCS of two n-element streams to within a factor of ρ, even if the streams are permutations of each other. A preliminary version of this paper appears in the Proceedings of the 11th International Computing and Combinatorics Conference (COCOON'05), August 2005, pp. 263–272.     

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     

Tree edge decomposition with an application to minimum ultrametric tree approximation

A k-decomposition of a tree is a process in which the tree is recursively partitioned into k edge-disjoint subtrees until each subtree contains only one edge. We investigated the problem how many levels it is sufficient to decompose the edges of a tree. In this paper, we show that any n-edge tree can be 2-decomposed (and 3-decomposed) within at most ⌈1.44 log n⌉ (and ⌈log n⌉ respectively) levels. Extreme trees are given to show that the bounds are asymptotically tight. Based on the result, we designed an improved approximation algorithm for the minimum ultrametric tree.     

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).     

Constructions of given-depth and optimal multirate rearrangeably nonblocking distributors

Rearrangeable multirate multicast switching networks are customarily called rearrangeable multirate distributors. It has been known for a long time that rearrangeable multirate distributors with cross-point complexity O(nlog?² n) can be constructed, where n is the number of inputs (and outputs) of the switching network. The problem of constructing optimal distributors remains open thus far. This paper gives a general construction of rearrangeable multirate distributors with given depths. One byproduct is a rearrangeable multirate distributor with crosspoint complexity O(nlog?n). We also show that this cross-point complexity is optimal, settling the aforementioned open problem. One of the key ingredients of our new construction is the notion of multirate concentrators. The second ingredient is a multirate version of the Pippenger network. We show how to construct given-depth multirate concentrators and given-depth multirate Pippenger networks with small sizes. When the depth is chosen to optimize the size, we obtain the optimal O(nlog?n) cross-point complexity.     

RNA multiple structural alignment with longest common subsequences

In this paper, we present a new model for RNA multiple sequence structural alignment based on the longest common subsequence. We consider both the off-line and on-line cases. For the off-line case, i.e., when the longest common subsequence is given as a linear graph with n vertices, we first present a polynomial O(n ²) time algorithm to compute its maximum nested loop. We then consider a slightly different problem—the Maximum Loop Chain problem and present an algorithm which runs in O(n ⁵) time. For the on-line case, i.e., given m RNA sequences of lengths n, compute the longest common subsequence of them such that this subsequence either induces a maximum nested loop or the maximum number of matches, we present efficient algorithms using dynamic programming when m is small. This research is partially supported by EPSCOR Visiting Scholar's Program and MSU Short-term Professional Development Program.     

Broadcast Routing with Minimum Wavelength Conversion in WDM Optical Networks

We consider dynamic routing of broadcast connections in WDM optical networks. Given the current network state, we want to find a minimum set of network nodes S such that a broadcast routing using only the nodes in S as wavelength conversion nodes can be found. This ensures that the average conversion delay from the source to all destinations is minimized. We refer to the problem as Broadcast Conversion Node Selection (BCNS) problem. We prove that BCNS has no polynomial-time approximation with performance ratio ln n for < 1 unless NPDTIME(n^{O(log log n)}) where n is the number of vertices in the input graph. We present a greedy approximation algorithm for BCNS that achieves approximation ratio 2+ln n. Simulation results show that the algorithm performs very well in practice, obtaining optimal solutions in most of the cases.     

Computing monotone disjoint paths on polytopes

The Holt-Klee Condition states that there exist at least d vertex-disjoint strictly monotone paths from the source to the sink of a polytopal digraph consisting of the set of vertices and arcs of a polytope P directed by a linear objective function in general position. The study of paths on polytopal digraphs stems from a long standing problem, that of designing a polynomial-time pivot method, or proving none exists. To study disjoint paths it would be useful to have a tool to compute them. Without explicitly computing the digraph we develop an algorithm to compute a maximum cardinality set of source to sink paths in a polytope, even in the presence of degeneracy. The algorithm uses a combination of networks flows, the simplex method, and reverse search. An implementation is available.     

Flows with unit path capacities and related packing and covering problems

Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log?m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an $O(\sqrt{m})Since the seminal work of Ford and Fulkerson in the 1950s, network flow theory is one of the most important and most active areas of research in combinatorial optimization. Coming from the classical maximum flow problem, we introduce and study an apparently basic but new flow problem that features a couple of interesting peculiarities. We derive several results on the complexity and approximability of the new problem. On the way we also discover two closely related basic covering and packing problems that are of independent interest. Starting from an LP formulation of the maximum s-t-flow problem in path variables, we introduce unit upper bounds on the amount of flow being sent along each path. The resulting (fractional) flow problem is NP-hard; its integral version is strongly NP-hard already on very simple classes of graphs. For the fractional problem we present an FPTAS that is based on solving the k shortest paths problem iteratively. We show that the integral problem is hard to approximate and give an interesting O(log m)-approximation algorithm, where m is the number of arcs in the considered graph. For the multicommodity version of the problem there is an O(?m)O(\sqrt{m}) -approximation algorithm. We argue that this performance guarantee is best possible, unless P=NP.     

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.     

Standard directed search strategies and their applications

Given a digraph that contains an intruder hiding on vertices or along edges, a directed searching problem is to find the minimum number of searchers required to capture the intruder. In this paper, we propose the standard directed search strategies, which can simplify arguments in proving search numbers, lower bounds, and relationships between search models. Using these strategies, we characterize k-directed-searchable graphs when k≤3. We also use standard directed search strategies to investigate the searching problems on oriented grids. We give tight lower and upper bounds for the directed search number of general oriented grids. Finally, we show how to compute the directed search number of a Manhattan grid. Yang’s research was supported in part by NSERC and MITACS.     

Faster algorithm to find anti-risk path between two nodes of an undirected graph

For a weighted 2-edge connected graph G=(V,E), we are to find a “minimum risk path” from source s to destination t. This is a shortest s?t path under the assumption that at most one edge on the path may be blocked. The fact that the edge is blocked is known only when we reach a site adjacent to the blocked edge. If n and m are the number of nodes and edges of G, then we show that this problem can be solved in O(n ²) time using only simple data structures. This is an improvement over the previous O(mn+n ²logn) time algorithm. Moreover, with use of more complicated data structures like Fibonacci Heaps and transmuters the time can be further reduced to O(m+nlogn).