Zhi-Zhong Chen

The Complexity of Selecting Maximal Solutions

Many important computational problems involve finding a maximal (with respect to set inclusion) solution in some combinatorial context. We study such maximality problems from the complexity point of view, and categorize their complexity precisely in terms of tight upper and lower bounds. Our results give characterizations of coNP, DP, ?P2, FPNP||, FNP//OptP log n] and FP?P||2 in terms of subclasses of maximality problems. An important consequence of our results is that finding an X-minimal satisfying truth assignment for a given CNF boolean formula is complete for FNP//OptPlog n], solving an open question by Papadimitriou Proceedings of the 32nd IEEE Symposium on the Foundations of Computer Science, 1991, pp. 163-169].

Computing Phylogenetic Roots with Bounded Degrees and Errors

Given a set of species and their similarity data, an important problem in evolutionary biology is how to reconstruct a phylogeny (also called evolutionary tree) so that species are close in the phylogeny if and only if they have high similarity. Assume that the similarity data are represented as a graph G = (V, E), where each vertex represents a species and two vertices are adjacent if they represent species of high similarity. The phylogeny reconstruction problem can then be abstracted as the problem of finding a (phylogenetic) tree T from the given graph G such that (1) T has no degree-2 internal nodes, (2) the external nodes (i.e., leaves) of T are exactly the elements of V, and (3)

Map graphs

(u, v) \in E

The longest common subsequence problem for sequences with nested arc annotations

if and only if

Reducing Randomness via Irrational Numbers

d_T(u, v) \le k

Planar map graphs

for some fixed threshold k, where dT(u,v) denotes the distance between u and v in tree T. This is called the phylogenetic kth root problem (PRk), and such a tree T, if it exists, is called a phylogenetic kth root of graph G. The computational complexity of PRk} is open, except...

Improved deterministic approximation algorithms for Max TSP

We consider a modified notion of planarity, in which two nations of a map are considered adjacent when they share any point of their boundaries (not necessarily an edge, as planarity requires). Such adjacencies define a map graph. We give an NP characterization for such graphs, derive some consequences regarding sparsity and coloring, and survey some algorithmic results.

Approximation Algorithms for Independent Sets in Map Graphs

Arc-annotated sequences are useful in representing the structural information of RNA and protein sequences. The LONGEST ARc-PRESERVING COMMON SUBSEQUENCE (LAPCS) Problem has been introduced in Evans (Algorithms and complexity for annotated sequence analysis, Ph.D. Thesis, University of Victoria, 1999) as a framework for studying the similarity of arc-annotated sequences. Several algorithmic and complexity results on the LAPCS problem have been presented in Evans (1999) and Jiang et al. (in: Proceedings of the 11th Annual Symposium on Combinatorial Pattern Matching (CPM 2000), Lecture Note in Computer Science, Vol. 1848, 2000, pp. 154-165). In this paper, we continue this line of research and present new algorithmic and complexity results on the LAPCS problem restricted to two nested arc-annotated sequences, denoted as LAPCS(NESTED, NESTED). The restricted problem is perhaps the most interesting variant of the LAPCS problem and has important applications in the comparison of RNA secondary structures. Particularly, we prove that LAPCS(NESTED, NESTED) is NP-hard, which answers an open question in Evans (1999). We then present a polynomial-time approximation scheme for LAPCS(NESTED, NESTED) with an additional c-diagonal restriction. An interesting special case, UNARY LAPCS(NESTED, NESTEO), is also investigated, for which we show the NP-hardness and present a better approximation algorithm than the one for general LAPCS(NESTED, NESTED).

Reducing randomness via irrational numbers

We propose a general methodology for testing whether a given polynomial with integer coefficients is identically zero. The methodology evaluates the polynomial at efficiently computable approximations of suitable irrational points. In contrast to the classical technique of DeMillo, Lipton, Schwartz, and Zippel, this methodology can decrease the error probability by increasing the precision of the approximations instead of using more random bits. Consequently, randomized algorithms that use the classical technique can generally be improved using the new methodology. To demonstrate the methodology, we discuss two nontrivial applications. The first is to decide whether a graph has a perfect matching in parallel. Our new NC algorithm uses fewer random bits while doing less work than the previously best NC algorithm by Chari, Rohatgi, and Srinivasan. The second application is to test the equality of two multisets of integers. Our new algorithm improves upon the previously best algorithms by Blum and Kannan and can speed up their checking algorithm for sorting programs on a large range of inputs.

Exact algorithms for haplotype assembly from whole-genome sequence data

We introduce and study a modi ed notion of planarity, in which two regions of a map are considered adjacent when they share any point of their boundaries (not an edge, as standard planarity requires). We seek to characterize the abstract graphs realized by such map adjacencies. We prove some preliminary properties of such graphs, and give a polynomial time algorithm for the following restricted problem: given an abstract graph, decide whether it is realized by a map in which at most four regions meet at any point. The general recognition problem remains open.