The Difference between Polynomial-Time Many-One and Truth-Table Reducibilities on Distributional Problems

In this paper we separate many-one reducibility from truth-table reducibility for distributional problems in DistNP under the hypothesis that P ≠ NP . As a first example we consider the 3-Satisfiability problem (3SAT) with two different distributions on 3CNF formulas. We show that 3SAT with a version of the standard distribution is truth-table reducible but not many-one reducible to 3SAT with a less redundant distribution unless P = NP . We extend this separation result and define a distributional complexity class C with the following properties: (1) C is a subclass of DistNP, this relation is proper unless P = NP. (2) C contains DistP, but it is not contained in AveP unless DistNP \subseteq AveZPP. (3) C has a ≤ ^p _m -complete set. (4) C has a ≤ ^p _tt -complete set that is not ≤ ^p _m -complete unless P = NP. This shows that under the assumption that P ≠ NP, the two completeness notions differ on some nontrivial subclass of DistNP.     

Exact 3-satisfiability is decidable in time O(20.16254n)

Let F = C ₁ C _m be a Boolean formula in conjunctive normal form over a set V of n propositional variables, s.t. each clause C _i contains at most three literals l over V. Solving the problem exact 3-satisfiability (X3SAT) for F means to decide whether there is a truth assignment setting exactly one literal in each clause of F to true (1). As is well known X3SAT is NP-complete [6]. By exploiting a perfect matching reduction we prove that X3SAT is deterministically decidable in time O(2^0.18674n ). Thereby we improve a result in [2,3] stating X3SAT O(2^0.2072n ) and a bound of O(2^0.200002n ) for the corresponding enumeration problem #X3SAT stated in a preprint [1]. After that by a more involved deterministic case analysis we are able to show that X3SAT O(2^0.16254n ).An extended abstract of this paper was presented at the Fifth International Symposium on the Theory and Applications of Satisfiability Testing (SAT 2002).     

Solving HPP and SAT by P Systems with Active Membranes and Separation Rules

The P systems (or membrane systems) are a class of distributed parallel computing devices of a biochemical type, where membrane division is the frequently investigated way for obtaining an exponential working space in a linear time, and on this basis solving hard problems, typically NP-complete problems, in polynomial (often, linear) time. In this paper, using another way to obtain exponential working space – membrane separation, it was shown that Satisfiability Problem and Hamiltonian Path Problem can be deterministically solved in linear or polynomial time by a uniform family of P systems with separation rules, where separation rules are not changing labels, but polarizations are used. Some related open problems are mentioned.     

Some structural properties of SAT

The following four conjectures about structural of SAT are studied in this paper.(1) SAT∈P^SPARSE∩NP;(2)SAT∈SRTDtt;(3)SAT∈Ptt^bAPP;(4)FPtt^SAT=FTlog^SAT.It is proved that some pairs of these conjectures imply P=NP ,for example,if SAT∈P^SPARSE∩NP and SAT∈Ptt^bAPP,or if SAT∈SRTDtt and SAT∩PttbAPP,then P=NP.This improves previous results in literature.     

Approximating Satisfiable Satisfiability Problems

We study the approximability of the Maximum Satisfiability Problem (MAX SAT) and of the boolean k -ary Constraint Satisfaction Problem (MAX k CSP) restricted to satisfiable instances. For both problems we improve on the performance ratios of known algorithms for the unrestricted case. Our approximation for satisfiable MAX 3CSP instances is better than any possible approximation for the unrestricted version of the problem (unless P=NP). This result implies that the requirement of perfect completeness weakens the acceptance power of non-adaptive PCP verifiers that read 3 bits. We also present the first non-trivial results about PCP classes defined in terms of free bits that collapse to P. Received August 1997; revised February 1999.     

Sorting,linear time and the satisfiability problem

Let DLIN denote the class of problems that are computable withinO(n/logn) uniform time on a RAM which can read its input (of lengthn) in blocks and which only uses integersO(n/logn). We prove that the sorting problem belongs to DLIN and formulate the Linear Time Thesis: Each problem computable by a linear time bounded algorithm (in an intuitive sense) belongs to DLIN. We also study the subclass DTIME_SORT(n) of problems computable within linear time on a Turing machine using in addition a fixed number of sortings, and we show how the reductions of this class, so-called sort-lin reductions, are useful to classify NP problems: e.g. there is a sort-lin reduction from SAT to 3-SAT (using no sorting) and a sort-lin reduction from the NP-complete graph problem KERNEL to SAT (but we do not know of any similar reduction in DTIME(n)). Similarly, problem 2-SAT is linearly equivalent to the problem of Horn renaming (via sort-lin reductions). Finally, SAT is compared with many other combinatorial problems. A problemA is SAT-easy (resp. SAT-hard) if there is a sort-lin reduction fromA to SAT (resp. SAT toA). A SAT-equivalent problem is a problem both SAT-easy and SAT-hard. It is shown that the class of SAT-easy (resp. SAT-equivalent) problems is very large and that its generalization to so-called special clauses or, more generally, regular clauses, does not enlarge it. Moreover, we justify our opinion that problem SAT is, in some sense, a minimal NP-complete problem.     

On the power of parity polynomial time

This paper proves that the complexity class P, parity polynomial time [PZ], contains the class of languages accepted byNP machines with few accepting paths. Indeed, P contains a broad class of languages accepted by path-restricted nondeterministic machines. In particular, P contains the polynomial accepting path versions ofNP, of the counting hierarchy, and of Mod _m NP form>1. We further prove that the class of nondeterministic path-restricted languages is closed under bounded truth-table reductions.These results were announced at the 6th Symposium on Theoretical Aspects of Computer Science [CH3]. Jin-yi Cai was supported in part by NSF Grant CCR-8709818 and the work was done while he was at Yale University. Lane A. Hemachandra was supported in part by a Hewlett-Packard Corporation equipment grant and the National Science Foundation under Grant CCR-8809174/CCR-8996198 and a Presidential Young Investigator Award. His work was done in part while at Columbia University.     

Inapproximability Results for Set Splitting and Satisfiability Problems with No Mixed Clauses

We prove hardness results for approximating set splitting problems and also instances of satisfiability problems which have no mixed clauses, i.e., every clause has either all its literals unnegated or all of them negated. Results of Håstad imply tight hardness results for set splitting when all sets have size exactly $k \ge 4$ elements and also for non-mixed satisfiability problems with exactly $k$ literals in each clause for $k \ge 4$. We consider the case $k=3$. For the MAX E3-SET SPLITTING, problem in which all sets have size exactly 3, we prove an NP-hardness result for approximating within any factor better than ${\frac{19}{20}}$. This result holds even for satisfiable instances of MAX E3-SET SPLITTING, and is based on a PCP construction due to Håstad. For non-mixed MAX 3SAT, we give a PCP construction which is a slight variant of the one given by Håstad for MAX 3SAT, and use it to prove the NP-hardness of approximating within a factor better than ${\frac{11}{12}}$.     

Power indices and easier hard problems

The concepts of power_index, satisfiability hypothesis (SH), and structure tree are introduced and used to make sharper hypotheses about a problem's complexity than the problem isNP-complete. These concepts are used to characterize the complexities of a number of basicNP-complete problems, including both CLIQUE and PARTITION which are shown to have power-indices at most 1/2. Also, the problem 3SAT is shown to be solvable deterministically in time exponential only in thesquare root ofv+c, wherev is the number of variables andc is the number of crossovers needed to layout the formula in the plane.The research of R. E. Stearns was supported by NSF Grants DCR 83-03932 and CCR 89-03319, and that of H. B. Hunt was supported by NSF Grants DCR 86-03184 and CCR 89-03319.     

An Efficient Approach for Solving Optimization over Linear Arithmetic Constraints

Satisfiability Modulo Theories (SMT) have been widely investigated over the last decade. Recently researchers have extended SMT to the optimization problem over linear arithmetic constraints. To the best of our knowledge, Symba and OPT-MathSAT are two most efficient solvers available for this problem. The key algorithms used by Symba and OPT-MathSAT consist of the loop of two procedures: 1) critical finding for detecting a critical point, which is very likely to be globally optimal, and 2) global checking for confirming the critical point is really globally optimal. In this paper, we propose a new approach based on the Simplex method widely used in operation research. Our fundamental idea is to find several critical points by constructing and solving a series of linear problems with the Simplex method. Our approach replaces the algorithms of critical finding in Symba and OPT-MathSAT, and reduces the runtime of critical finding and decreases the number of executions of global checking. The correctness of our approach is proved. The experiment evaluates our implementation against Symba and OPT-MathSAT on a critical class of problems in real-time systems. Our approach outperforms Symba on 99.6% of benchmarks and is superior to OPT-MathSAT in large-scale cases where the number of tasks is more than 24. The experimental results demonstrate that our approach has great potential and competitiveness for the optimization problem.     

Reconstructing (h,v)-convex 2-dimensional patterns of objects from approximate horizontal and vertical projections

The problem of reconstructing a pattern of an object from its approximate discrete orthogonal projections in a 2-dimensional grid, may have no solution because the inaccuracy in the measurements of the projections may generate an inconsistent problem. To attempt to overcome this difficulty, one seeks to reconstruct a pattern with projection values having possibly some bounded differences with the given projection values and minimizing the sum of the absolute differences.

This paper addresses the problem of reconstructing a pattern with a difference at most equal to +1 or −1 between each of its projection values and the corresponding given projection value. We deal with the case of patterns which have to be horizontally and vertically convex and the case of patterns which have to be moreover connected, the so-called convex polyominoes. We show that in both cases, the problem of reconstructing a pattern can be transformed into a Satisfiability (SAT) Problem. This is done in order to take advantage of the recent advances in the design of solvers for the SAT Problem. We show, experimentally, that by adding two important features to CSAT (an efficient SAT solver), optimal patterns can be found if there exist feasible ones. These two features are: first, a method that extracts in linear time an optimal pattern from a set of feasible patterns grouped in a generic pattern (obtaining a generic pattern may be exponential in the worst case) and second, a method that computes actively a lower bound of the sum of absolute differences that can be obtained from a partially defined pattern. This allows to prune the search tree if this lower bound exceeds the best sum of absolute differences found so far.     

An observation on probability versus randomness with applications to complexity classes

Every class C of languages satisfying a simple topological condition is shown to have probability one if and only if it contains some language that is algorithmically random in the sense of Martin-Löf. This result is used to derive separation properties of algorithmically random oracles and to give characterizations of the complexity classesP, BPP, AM, andPH in terms of reducibility to such oracles. These characterizations lead to results like:P =NP if and only if an algorithmically random set exists that is _btt ^P -hard forNP.The work of the first author was supported in part by the Alexander-von-Humboldt-Stiftung and by the National Science Foundation under Grant CCR-8913584 while he visited the Lehrstuhl für Theoretische Informatik, Institut für Informatik, Universität Würzburg, Germany. The work of the second author was supported in part by the National Science Foundation under Grant CCR-8809238 and in part by DIMACS, where he was a visitor while a portion of his work was done.     

Onk-positive satisfiability problem

An algorithm for solving the satisfiability problem is presented. It is proved that this algorithm solves 2-SAT and Horn-SAT in linear time andk-positive SAT (in which every clause contains at mostk positive literals) in timeO(|F|·ξ _n ^k ), where |F| is the length of inputF, n is the number of atoms occurring inF, and ξ _k is the greatest real number satisfying the equation . Compared with previous results, this nontrivial upper bound on time complexity could only be obtained fork-SAT, which is a subproblem ofk-positive SAT. Research partially supported by NSFC grant 221-4-1439. HUANG Xiong received his B.S. and M.S. degrees in computer science from Peking University in 1992 and 1995 respectively. Now he is a Ph.D. candidate in Beijing University of Aeronautics and Astronautics. His major research interests are design and analysis of algorithms, computational complexity, and satisfiability problem. LI Wei received his B.S. degree in mathematics from Peking University in 1966 and his Ph.D. degree in computer science from The University of Edinburgh in 1983. Since 1986, he has been a Professor in computer science at Beijing University of Aeronautics and Astronautics. He has published over 100 papers in the areas of concurrent programming languages, operational semantics, type theory, and logical foundation of artificial intelligence.     

On a Refined Analysis of Some Problems in Interval Arithmetic Using Real Number Complexity Theory

We study some problems in interval arithmetic treated in Kreinovich et al. [13]. First, we consider the best linear approximation of a quadratic interval function. Whereas this problem (as decision problem) is known to be NP-hard in the Turing model, we analyze its complexity in the real number model and the analogous class NP . Our results substantiate that most likely it does not any longer capture the difficulty of NP in such a real number setting. More precisely, we give upper complexity bounds for the approximation problem for interval functions by locating it in (a real analogue of). This result allows several conclusions: the problem is not (any more) NP -hard under so called weak polynomial time reductions and likely not to be NP -hard under (full) polynomial time reductions; for fixed dimension the problem is polynomial time solvable; this extends the results in Koshelev et al. [12] and answers a question left open in [13].We also study several versions of interval linear systems and show similar results as for the approximation problem.Our methods combine structural complexity theory with issues from semi-infinite optimization theory.     

Absorbing random walks and the NAE2SAT problem

In this paper, we propose a simple randomized algorithm for the NAE2SAT problem. The analysis of the algorithm uses the theory of symmetric, absorbing random walks. Inasmuch as the NAE2SAT problem is in the complexity class L (Deterministic Logarithmic Space) and L ? P ? RP, the existence of such an algorithm is not surprising. However, the simplicity of our approach and the insights revealed by its analysis make the current study worthwhile. Not-all-equal SAT (NAESAT) is the variant of the satisfiability problem (SAT), in which we are interested in an assignment that satisfies all the clauses, but falsifies at least one literal in each clause. NAESAT is an interesting SAT variant in its own right and has been studied by SAT theoreticians, on account of its connections to the graph colouring problem. It is well-known that the variant of NAESAT with exactly three literals per clause (NAE3SAT) is NP-complete [M.R. Garey and D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-Completeness, W. H. Freeman Company, San Francisco, 1979]. Likewise, there seem to be a number of polynomial time algorithms for NAE2SAT as part of algorithm design folklore [D.S. Johnson, Personal Communication]. Here we show that the NAE2SAT problem admits an extremely simple, literal-flipping algorithm, in precisely the same way that 2SAT does. On a satisfiable instance involving n variables, the probability that our algorithm does not find a satisfying assignment is at most 1/24. The randomized algorithm takes O(1) extra space, in the presence of a verifier and provides an interesting insight into checking whether a graph is bipartite. It must be noted that space-parsimonious randomized algorithms for a problem, such as the one proposed in this paper, invariably lead to error–bounded, online algorithms for the same. As part of our analysis, we argue that a restricted variant of NAE2SAT is L-complete. We note that the bounds derived in this paper are sharper than the ones in Papadimitriou [C.H. Papadimitriou, On selecting a satisfying truth assignment, in Proceedings of the 32nd Annual Symposium on Foundations of Computer Science, San Juan, Puerto Rico, 1–4 October 1991, IEEE, ed., IEEE, Washington, DC, 1991, pp. 163–169].     

The SAT2002 competition

SAT Competition 2002 held in March–May 2002 in conjunction with SAT 2002 (the Fifth International Symposium on the Theory and Applications of Satisfiability Testing). About 30 solvers and 2300 benchmarks took part in the competition, which required more than 2 CPU years to complete the evaluation. In this report, we give the results of the competition, try to interpret them, and give suggestions for future competitions.     

Infeasibility of instance compression and succinct PCPs for NP

The OR-SAT problem asks, given Boolean formulae ${?}_1,\dots ,{?}_m$ each of size at most n, whether at least one of the ${?}_i$'s is satisfiable. We show that there is no reduction from OR-SAT to any set A where the length of the output is bounded by a polynomial in n, unless $\mathsf{NP}?\mathsf{coNP}/\mathrm{poly}$, and the Polynomial-Time Hierarchy collapses. This result settles an open problem proposed by Bodlaender et al. (2008) [6] and Harnik and Naor (2006) [20] and has a number of implications. (i) A number of parametric $\mathsf{NP}$ problems, including Satisfiability, Clique, Dominating Set and Integer Programming, are not instance compressible or polynomially kernelizable unless $\mathsf{NP}?\mathsf{coNP}/\mathrm{poly}$. (ii) Satisfiability does not have PCPs of size polynomial in the number of variables unless $\mathsf{NP}?\mathsf{coNP}/\mathrm{poly}$. (iii) An approach of Harnik and Naor to constructing collision-resistant hash functions from one-way functions is unlikely to be viable in its present form. (iv) (Buhrman–Hitchcock) There are no subexponential-size hard sets for NP unless NP is in co-NP/poly. We also study probabilistic variants of compression, and show various results about and connections between these variants. To this end, we introduce a new strong derandomization hypothesis, the Oracle Derandomization Hypothesis, and discuss how it relates to traditional derandomization assumptions.     

Positive Versions of Polynomial Time

We show that restricting a number of characterizations of the complexity classPto be positive (in natural ways) results in the same class of (monotone) problems, which we denote byposP. By a well-known result of Razborov,posPis a proper subclass of the class of monotone problems inP. We exhibit complete problems forposPvia weak logical reductions, as we do for other logically defined classes of problems. Our work is a continuation of research undertaken by Grigni and Sipser, and subsequently Stewart; indeed, we introduce the notion of a positive deterministic Turing machine and consequently solve a problem posed by Grigni and Sipser.     

The Data Broadcast Problem with Non-Uniform Transmission Times

Abstract. The Data Broadcast Problem consists of finding an infinite schedule to broadcast a given set of messages so as to minimize a linear combination of the average service time to clients requesting messages, and of the cost of the broadcast. This problem also models the Maintenance Scheduling Problem and the Multi-Item Replenishment Problem. Previous work concentrated on a discrete-time restriction where all messages have transmission time equal to 1. Here, we study a generalization of the model to a setting of continuous time and messages of non-uniform transmission times. We prove that the Data Broadcast Problem is strongly NP -hard, even if the broadcast costs are all zero, and give 3 -approximation algorithms.