On balanced versus unbalanced computation trees

Restricted relativizations of probabilistic polynomial time

We examine the power of nondeterministic finite automata with acceptance of an input word defined by a so called leaf language, i. e., a condition on the sequence of leaves in the automaton's computation tree. We study leaf languages either taken from one of the classes of the Chomsky hierarchy, or taken from a time- or space-bounded complexity class. We contrast the obtained results with those known for leaf languages for Turing machines and Boolean circuits.

We study computational complexity theory and define a class of optimization problems called OptP (Optimization Polynomial Time), and we show that TRAVELLING SALESPERSON, KNAPSACK and 0-1 INTEGER LINEAR PROGRAMMING are complete for OptP. OptP is a natural generalization of NP (Nondeterministic Polynomial Time), but while NP only considers problems at the level of their yes/no question, the value of an OptP function is the optimal value of the problem. This approach enables us to show a deeper level of structure in these problems than is possible in NP. OptP is a subset of FPSAT, the class of functions computable in polynomial time with an oracle for NP. Our central result is that any FPSAT function decomposes into an OptP function followed by polynomial-time computation. The significance of this result is that it quantifies how much NP-completeness is in a problem, i.e., the number of NP queries it takes to compute the function. It also allows us to unify the classes NP, DP and DELTA-2 in a natural way. For example, we prove that an OptP-completeness result implies, as corollaries, NP-, DP- and DELTA-2- completeness results. We also prove separation results on subclasses of FPSAT by restricting the number of calls to the NP oracle. For example, TRAVELLING SALESPERSON is complete for O(n) queries, CLIQUE is complete for O(log n) queries and BIN PACKING can be solved in O(log log n) queries. We prove these classes distinct under the assumption that P does not equal NP. Finally, we consider generalizations of OptP to higher levels in the Polynomial-Time Hierarchy. We define the DOUBLE KNAPSACK problem and prove that it is complete for DELTA-3, the first example of a natural complete problem for this class, and the highest level in the Polynomial Hierarchy with a known natural complete problem.

We have characterized the relationship between accurate phylogenetic reconstruction and sequence similarity, testing whether high levels of sequence similarity can consistently produce accurate evolutionary trees. We generated protein families with known phylogenies using a modified version of the PAML/EVOLVER program that produces insertions and deletions as well as substitutions. Protein families were evolved over a range of 100-400 point accepted mutations; at these distances 63% of the families shared significant sequence similarity. Protein families were evolved using balanced and unbalanced trees, with ancient or recent radiations. In families sharing statistically significant similarity, about 60% of multiple sequence alignments were 95% identical to true alignments. To compare recovered topologies with true topologies, we used a score that reflects the fraction of clades that were correctly clustered. As expected, the accuracy of the phylogenies was greatest in the least divergent families. About 88% of phylogenies clustered over 80% of clades in families that shared significant sequence similarity, using Bayesian, parsimony, distance, and maximum likelihood methods. However, for protein families with short ancient branches (ancient radiation), only 30% of the most divergent (but statistically significant) families produced accurate phylogenies, and only about 70% of the second most highly conserved families, with median expectation values better than 10(-60), produced accurate trees. These values represent upper bounds on expected tree accuracy for sequences with a simple divergence history; proteins from 700 Giardia families, with a similar range of sequence similarities but considerably more gaps, produced much less accurate trees. For our simulated insertions and deletions, correct multiple sequence alignments did not perform much better than those produced by T-COFFEE, and including sequences with expressed sequence tag-like sequencing errors did not significantly decrease phylogenetic accuracy. In general, although less-divergent sequence families produce more accurate trees, the likelihood of estimating an accurate tree is most dependent on whether radiation in the family was ancient or recent. Accuracy can be improved by combining genes from the same organism when creating species trees or by selecting protein families with the best bootstrap values in comprehensive studies.

The authors study two different ways to restrict the power of NP. They consider languages accepted by nondeterministic polynomial-time machines with a small number of accepting paths in case of acceptance, and they investigate three subclasses of NP that are low for complexity classes not known to be in the polynomial-time hierarchy. The subclasses, UP, FewP, and Few, are all defined in terms of nondeterministic machines with a bounded number of accepting paths for every input string, but for the last two classes this number is not known beforehand and can range over a space of polynomial size. The authors prove lowness properties of the class Few and some other interesting sets that are low for the class PP. The lowness results are used to obtain positive relativizations of complexity classes. >

We define the counting classes NC/sup 1/, GapNC/sup 1/ PNC/sup 1/ and C/sub =/NC/sup 1/. We prove that Boolean circuits, algebraic circuits, programs over nondeterministic finite automata, and programs over constant integer matrices yield equivalent definitions of the latter three classes. We investigate closure properties. We observe that NC/sup 1//spl sube/L and that C/sub =/NC/sup 1//spl sube/L. Then we exploit our finite automaton model and extend the padding techniques used to investigate leaf languages. Finally, we draw some consequences from the resulting body of leaf language characterizations of complexity classes, including the unconditional separation of ACC/sup 0/ from MOD-PH as well as that of TC/sup 0/ from the counting hierarchy. Moreover we obtain that dlogtime-uniformity and logspace-uniformity for AC/sup 0/ coincide if and only if the polynomial time hierarchy equals PSPACE.

Conflicting relativizations, also known as “Baker-Gill-Solovay phenomena,” are common place in complexity theory. However, Book, Long, and Selman (SIAM J. Comput.,13 (1984), 461–487) have shown thatP(A) = NP B (A) for all oraclesA if and only ifP = NP, whereNP B (A) denotes the class of languages accepted by nondeterministic machines which possess only a polynomial number of query strings in the computation tree. It is shown in this paper that any superpolynomial bound on the number of queries already yields the BGS phenomenon. Similar results hold for theNP = co-NP andNPC = NP questions. A second objective of this paper is to point out a technique of Hartmanis with which to achieve oracle constructions by using the Kolmogorov complexity. This relatively new technique seems to be adequate for obtaining separation results for complexity classes which are not enumerable.

Languages polylog-time reducible to dot-depth 1/2

We construct an oracleA such that\(\Sigma _2^{P,A} \nsubseteq PP^A \). So the polynomial time hierarchy is separated from the polynomial time probabilistic complexity class in relativization.

Classifies control problems by exhibiting their alternating quantifier structure. This classification allows the authors to relate these control problems to the computational complexity classes of the polynomial time hierarchy. A specific synthesis problem for uncertain systems is shown to be hard in the class /spl Pi//sub 2//sup p/.

We give resource bounded versions of the Completeness Theorem for propositional and predicate logic. For example, it is well known that every computable consistent propositional theory has a computable complete consistent extension. We show that, when length is measured relative to the binary representation of natural numbers and formulas, every polynomial time decidable propositional theory has an exponential time (EXPTIME) complete consistent extension whereas there is a nondeterministic polynomial time (NP) decidable theory which has no polynomial time complete consistent extension when length is measured relative to the binary representation of natural numbers and formulas. It is well known that a propositional theory is axiomatizable (respectively decidable) if and only if it may be represented as the set of infinite paths through a computable tree (respectively a computable tree with no dead ends). We show that any polynomial time decidable theory may be represented as the set of paths through a polynomial time decidable tree. On the other hand, the statement that every polynomial time decidable tree represents the set of complete consistent extensions of some theory which is polynomial time decidable, relative to the tally representation of natural numbers and formulas, is equivalent to P = NP. For predicate logic, we develop a complexity theoretic version of the Henkin construction to prove a complexity theoretic version of the Completeness Theorem. Our results imply that that any polynomial space decidable theory Δ possesses a polynomial space computable model which is exponential space decidable and thus Δ has an exponential space complete consistent extension. Similar results are obtained for other notions of complexity.

The domination and Hamilton circuit problems are of interest both in algorithm design and complexity theory. The domination problem has applications in facility location and the Hamilton circuit problem has applications in routing problems in communications and operations research.The problem of deciding if G has a dominating set of cardinality at most k, and the problem of determining if G has a Hamilton circuit are NP-Complete. Polynomial time algorithms are, however, available for a large number of restricted classes. A motivation for the study of these algorithms is that they not only give insight into the characterization of these classes but also require a variety of algorithmic techniques and data structures. So the search for efficient algorithms, for these problems in many classes still continues.A class of perfect graphs which is practically important and mathematically interesting is the class of permutation graphs. The domination problem is polynomial time solvable on permutation graphs. Algorithms that are already available are of time complexity O(n2) or more, and space complexity O(n2) on these graphs. The Hamilton circuit problem is open for this class.We present a simple O(n) time and O(n) space algorithm for the domination problem on permutation graphs. Unlike the existing algorithms, we use the concept of geometric representation of permutation graphs. Further, exploiting this geometric notion, we develop an O(n2) time and O(n) space algorithm for the Hamilton circuit problem.KeywordsFacility LocationAdjacency RegionPolynomial Time AlgorithmGeometric RepresentationMinimum AngleThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

We investigate the search cost in terms of number of messages generated for routing queries in tree-based P2P structured systems including binary and k-ary tree structures with different arities and different degrees of imbalance in the tree shape. This work is motivated by the fact that k-ary balanced tree access structures can greatly reduce the number of hops for searching compared to the binary trees. We study to what extent the same fact is true when the tree-like structures for access in P2P environments are unbalanced. Another important issue related to P2P environments is how to build these structures in a self-organizing way. We propose a mechanism for constructing k-ary tree based decentralized access structure in a self-organizing way and based on local interactions only. The ability to search efficiently also on unbalanced k-ary trees opens interesting opportunities for load balancing as has been shown in earlier work on P-Grid, our approach to structured P2P systems.KeywordsCommunication CostAccess StructureSearch CostDistribute Hash TableRout Table EntryThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

The label tree-based classification is one of the most popular approaches for reducing the testing complexity to sublinear with the large number of classes. One of the popular approaches to generate the label tree is to apply recursively a spectral clustering algorithm to an affinity matrix for partition set of class labels into subsets, each subset corresponds to a child node of the tree. To obtain the affinity matrix from confusion matrix, a set of N binary one-versus-all classifiers is trained and applied on validation set, where N is number of classes. However, these approaches are not reliable when there are a large number of classes because it is too costly to train these classifiers. Furthermore, the affinity matrix could not reflect the real similarity among classes due to the classification accuracy can be low. In addition, the resulting label tree may not be balanced due to the objective function of spectral clustering penalizes unbalanced partitions. In this paper, to achieve better similarity measurement between classes and without using one-versus-all classifiers, we adopt the sum match kernel to get similarity matrix. Moreover, we propose a heuristic for learning balanced tree structure by adjusting the number of class labels in clusters after the spectral clustering is done. The experimental results on benchmark datasets SUN-397 and Caltech-256 show that the performance of the proposed approach outperforms significantly the other approaches.

Robust simulations and significant separations