Weitian Tong

The Steiner Traveling Salesman Problem with online edge blockages

We consider the online Steiner Traveling Salesman Problem. In this problem, we are given an edge-weighted graph G = (V, E) and a subset D⊆V of destination vertices, with the optimization goal to find a minimum weight closed tour that traverses every destination vertex of D at least once. During the traversal, the salesman could encounter at most k non-recoverable blocked edges. The edge blockages are real-time, meaning that the salesman knows about a blocked edge whenever it occurs. We first show a lower bound on the competitive ratio and present an online optimal algorithm for the problem. While this optimal algorithm has non-polynomial running time, we present another online polynomial-time near optimal algorithm for the problem. Experimental results show that our online polynomial-time algorithm produces solutions very close to the offline optimal solutions.

An FPTAS for the parallel two-stage flowshop problem

We consider the NP-hard m-parallel two-stage flowshop problem, abbreviated as the ( m , 2 ) -PFS problem, where we need to schedule n jobs to m parallel identical two-stage flowshops in order to minimize the makespan, i.e. the maximum completion time of all the jobs on the m flowshops. The ( m , 2 ) -PFS problem can be decomposed into two subproblems: to assign the n jobs to the m parallel flowshops, and for each flowshop to schedule the jobs assigned to the flowshop. We first present a pseudo-polynomial time dynamic programming algorithm to solve the ( m , 2 ) -PFS problem optimally, for any fixed m, based on an earlier idea for solving the ( 2 , 2 ) -PFS problem. Using the dynamic programming algorithm as a subroutine, we design a fully polynomial-time approximation scheme (FPTAS) for the ( m , 2 ) -PFS problem.

The Steiner traveling salesman problem with online advanced edge blockages

The package delivery in an urban road network is formulated as an online Steiner traveling salesman problem, where the driver (i.e. the salesman) receives road (i.e. edge) blockage messages when he is at a certain distance to the respective blocked edges. Such road blockages are referred to as advanced information. With these online advanced road blockages, the driver wishes to deliver all the packages to their respective customers and returns back to the service depot through a shortest route. During the entire delivery process, there will be at most k road blockages, and they are non-recoverable. When the driver knows about road blockages at a distance α OPT , where α ? 0 , 1 is referred to as the forecasting ratio and OPT denotes the length of the offline shortest route, we first prove that max { ( 1 - 2 α ) k + 1 , 1 } is a lower bound on the competitive ratio. We then present a polynomial time online algorithm with a competitive ratio very close to this lower bound. Computational results show that our algorithm is efficient and produces near optimal solutions. Similar results for a variation, in which the driver does not need to return to the service depot, are also achieved. HighlightsThe sTSP with online advanced edge blockages is formulated from an application.Lower bounds on the competitive ratio are proved.An efficient routing algorithm with proven performance is proposed.Extensive computational experiments show both the efficiency and the effectiveness.

Set Cover, Set Packing and Hitting Set for Tree Convex and Tree-Like Set Systems

A set system is a collection of subsets of a given finite universe. A tree convex set system has a tree defined on the universe, such that each subset in the system induces a subtree. A circular convex set system has a circular ordering defined on the universe, such that each subset in the system induces a circular arc. A tree-like set system has a tree defined on the system, such that for each element in the universe, all subsets in the system containing this element induce a subtree. A circular-like set system has a circular ordering defined on the system, such that for each element in the universe, all subsets in the system containing this element induce a circular arc. In this paper, we restrict the trees to be stars, combs, triads, respectively, and restrict the set system to be unweighted. We show tractability of Triad Convex Set Cover, Circular-like Set Packing, and Triad-like Hitting Set, intractability of Comb Convex Set Cover and Comb-like Hitting Set. Our results not only complement the known results in literatures, but also rise interesting questions such as which other kind of trees will lead to tractability or intractability results of Set Cover, Set Packing and Hitting Set for tree convex and tree-like set systems.

Approximation Algorithms for the Maximum Multiple RNA Interaction Problem

RNA interactions are fundamental in many cellular processes, which can involve two or more RNA molecules. Multiple RNA interactions are also believed to be much more complex than pairwise interactions. Recently, multiple RNA interaction prediction has been formulated as a maximization problem. Here we extensively examine this optimization problem under several biologically meaningful interaction models. We present a polynomial time algorithm for the problem when the order of interacting RNAs is known and pseudoknot interactions are allowed; for the general problem without an assumed RNA order, we prove the NP-hardness for both variants (allowing and disallowing pseudoknot interactions), and present a constant ratio approximation algorithm for each of them.

On the approximability of the exemplar adjacency number problem for genomes with gene repetitions

In this paper, we apply a measure, exemplar adjacency number, which complements and extends the well-studied breakpoint distance between two permutations, to measure the similarity between two genomes (or in general, between any two sequences drawn from the same alphabet). For two genomes G and H drawn from the same set of n gene families and containing gene repetitions, we consider the corresponding Exemplar Adjacency Number problem (EAN), in which we delete duplicated genes from G and H such that the resultant exemplar genomes (permutations) G and H have the maximum adjacency number. We obtain the following results. First, we prove that the one-sided 2-repetitive EAN problem, i.e., when one of G and H is given exemplar and each gene occurs in the other genome at most twice, can be linearly reduced from the Maximum Independent Set problem. This implies that EAN does not admit any O(n^0^.^5^-^@e)-approximation algorithm, for any @e>0, unless P = NP. This hardness result also implies that EAN, parameterized by the optimal solution value, is W[1]-hard. Secondly, we show that the two-sided 2-repetitive EAN problem has an O(n^0^.^5)-approximation algorithm, which is tight up to a constant factor.

An improved approximation algorithm for the bandpass problem

The general Bandpass-B problem is NP-hard and can be approximated by a reduction into the B -set packing problem, with a worst case performance ratio of O (B 2). When B =2, a maximum weight matching gives a 2-approximation to the problem. The Bandpass-2 problem, or simply the Bandpass problem, can be viewed as a variation of the maximum traveling salesman problem, in which the edge weights are dynamic rather than given at the front. We present in this paper a

Isomorphism and similarity for 2-generation pedigrees

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An Improved Approximation Algorithm for the Minimum Common Integer Partition Problem

-approximation algorithm for the Bandpass problem, which is the first improvement over the simple maximum weight matching based 2-approximation algorithm.

Algorithms for Cut Problems on Trees

We consider the emerging problem of comparing the similarity between (unlabeled) pedigrees. More specifically, we focus on the simplest pedigrees, namely, the 2-generation pedigrees. We show that the isomorphism testing for two 2-generation pedigrees is GI-hard. If the 2-generation pedigrees are monogamous (i.e., each individual at level-1 can mate with exactly one partner) then the isomorphism testing problem can be solved in polynomial time. We then consider the problem by relaxing it into an NP-complete decomposition problem which can be formulated as the Minimum Common Integer Pair Partition (MCIPP) problem, which we show to be FPT by exploiting a property of the optimal solution. While there is still some difficulty to overcome, this lays down a solid foundation for this research.