On the complexity of binary samples

Consider a class of binary functions h: X→{ − 1, + 1} on an interval . Define the sample width of h on a finite subset (a sample) S ⊂ X as ω _S (h) =  min _x ∈ S |ω _h (x)| where ω _h (x) = h(x) max {a ≥ 0: h(z) = h(x), x − a ≤ z ≤ x + a}. Let be the space of all samples in X of cardinality ℓ and consider sets of wide samples, i.e., hypersets which are defined as Through an application of the Sauer-Shelah result on the density of sets an upper estimate is obtained on the growth function (or trace) of the class , β > 0, i.e., on the number of possible dichotomies obtained by intersecting all hypersets with a fixed collection of samples of cardinality m. The estimate is .      

Exponential Lower Bounds for the Running Time of DPLL Algorithms on Satisfiable Formulas

DPLL (for Davis, Putnam, Logemann, and Loveland) algorithms form the largest family of contemporary algorithms for SAT (the propositional satisfiability problem) and are widely used in applications. The recursion trees of DPLL algorithm executions on unsatisfiable formulas are equivalent to treelike resolution proofs. Therefore, lower bounds for treelike resolution (known since the 1960s) apply to them. However, these lower bounds say nothing about the behavior of such algorithms on satisfiable formulas. Proving exponential lower bounds for them in the most general setting is impossible without proving P ≠ NP; therefore, to prove lower bounds, one has to restrict the power of branching heuristics. In this paper, we give exponential lower bounds for two families of DPLL algorithms: generalized myopic algorithms, which read up to n ^1−ε of clauses at each step and see the remaining part of the formula without negations, and drunk algorithms, which choose a variable using any complicated rule and then pick its value at random. ^★Extended abstract of this paper appeared in Proceedings of ICALP 2004, LNCS 3142, Springer, 2004, pp. 84–96. ^†Supported by CCR grant CCR-0324906. ^‡Supported in part by Russian Science Support Foundation, RAS program of fundamental research “Research in principal areas of contemporary mathematics,” and INTAS grant 04-77-7173. ^§Supported in part by INTAS grant 04-77-7173.     

Tuples of Disjoint $\mathsf{NP}$ -Sets

Disjoint -pairs are a well studied complexity-theoretic concept with important applications in cryptography and propositional proof complexity. In this paper we introduce a natural generalization of the notion of disjoint -pairs to disjoint k-tuples of -sets for k≥2. We define subclasses of the class of all disjoint k-tuples of -sets. These subclasses are associated with a propositional proof system and possess complete tuples which are defined from the proof system. In our main result we show that complete disjoint -pairs exist if and only if complete disjoint k-tuples of -sets exist for all k≥2. Further, this is equivalent to the existence of a propositional proof system in which the disjointness of all k-tuples is shortly provable. We also show that a strengthening of this conditions characterizes the existence of optimal proof systems. An extended abstract of this paper appeared in the proceedings of the conference CSR 2006 (Lecture Notes in Computer Science 3967, 80–91, 2006). Supported by DFG grant KO 1053/5-1.     

Bent and hyper-bent functions over a field of 2ℓ elements

We study the parameters of bent and hyper-bent (HB) functions in n variables over a field $ P = \mathbb{F}_q We study the parameters of bent and hyper-bent (HB) functions in n variables over a field with q = 2^ℓ elements, ℓ > 1. Any such function is identified with a function F: Q → P, where . The latter has a reduced trace representation F = tr _P ^Q (Φ), where Φ(x) is a uniquely defined polynomial of a special type. It is shown that the most accurate generalization of results on parameters of bent functions from the case ℓ = 1 to the case ℓ > 1 is obtained if instead of the nonlinearity degree of a function one considers its binary nonlinearity index (in the case ℓ = 1 these parameters coincide). We construct a class of HB functions that generalize binary HB functions found in [1]; we indicate a set of parameters q and n for which there are no other HB functions. We introduce the notion of the period of a function and establish a relation between periods of (hyper-)bent functions and their frequency characteristics. Original Russian Text ? A.S. Kuz’min, V.T. Markov, A.A. Nechaev, V.A. Shishkin, A.B. Shishkov, 2008, published in Problemy Peredachi Informatsii, 2008, Vol. 44, No. 1, pp. 15–37. Supported in part by the Russian Foundation for Basic Research, project nos. 05-01-01018 and 05-01-01048, and the President of the Russian Federation Council for State Support of Leading Scientific Schools, project nos. NSh-8564.2006.10 and NSh-5666.2006.1. A part of the results were obtained in the course of research in the Cryptography Academy of the Russian Federation.     

On the Complexity of 2-Monotone Restarting Automata

The -automaton is the weakest form of the nondeterministic version of the restarting automaton that was introduced by Jančar et al. to model the so-called analysis by reduction. Here it is shown that the class ℒ(R) of languages that are accepted by -automata is incomparable under set inclusion to the class of Church-Rosser languages and to the class of growing context-sensitive languages. In fact this already holds for the class of languages that are accepted by 2-monotone -automata. In addition, we prove that already the latter class contains -complete languages, showing that already the 2-monotone -automaton has a surprisingly large expressive power. The results of this paper have been announced at DLT 2004 in Auckland, New Zealand. This work was mainly carried out while T. Jurdziński was visiting the University of Kassel, supported by a grant from the Deutsche Forschungsgemeinschaft (DFG). F. Mráz and M. Plátek were partially supported by the Grant Agency of the Czech Republic under Grant-No. 201/04/2102 and by the program ‘Information Society’ under project 1ET100300517. F. Mráz was also supported by the Grant Agency of Charles University in Prague under Grant-No. 358/2006/A-INF/MFF.     

Approximation Algorithms for Multi-Criteria Traveling Salesman Problems

We analyze approximation algorithms for several variants of the traveling salesman problem with multiple objective functions. First, we consider the symmetric TSP (STSP) with γ-triangle inequality. For this problem, we present a deterministic polynomial-time algorithm that achieves an approximation ratio of and a randomized approximation algorithm that achieves a ratio of . In particular, we obtain a 2+ε approximation for multi-criteria metric STSP. Then we show that multi-criteria cycle cover problems admit fully polynomial-time randomized approximation schemes. Based on these schemes, we present randomized approximation algorithms for STSP with γ-triangle inequality (ratio ), asymmetric TSP (ATSP) with γ-triangle inequality (ratio ), STSP with weights one and two (ratio 4/3) and ATSP with weights one and two (ratio 3/2). A preliminary version of this work has been presented at the 4th Workshop on Approximation and Online Algorithms (WAOA 2006) (Lecture Notes in Computer Science, vol. 4368, pp. 302–315, 2007). B. Manthey is supported by the Postdoc-Program of the German Academic Exchange Service (DAAD). He is on leave from Saarland University and has done part of the work at the Institute for Theoretical Computer Science of the University of Lübeck supported by DFG research grant RE 672/3 and at the Department of Computer Science at Saarland University.     

Approximately Fair Cost Allocation in Metric Traveling Salesman Games

A traveling salesman game is a cooperative game . Here N, the set of players, is the set of cities (or the vertices of the complete graph) and c _D is the characteristic function where D is the underlying cost matrix. For all S⊆N, define c _D (S) to be the cost of a minimum cost Hamiltonian tour through the vertices of S∪{0} where is called as the home city. Define Core as the core of a traveling salesman game . Okamoto (Discrete Appl. Math. 138:349–369, [2004]) conjectured that for the traveling salesman game with D satisfying triangle inequality, the problem of testing whether Core is empty or not is -hard. We prove that this conjecture is true. This result directly implies the -hardness for the general case when D is asymmetric. We also study approximately fair cost allocations for these games. For this, we introduce the cycle cover games and show that the core of a cycle cover game is non-empty by finding a fair cost allocation vector in polynomial time. For a traveling salesman game, let and ∀ S⊆N, x(S)≤ε⋅c _D (S)} be an ε-approximate core, for a given ε>1. By viewing an approximate fair cost allocation vector for this game as a sum of exact fair cost allocation vectors of several related cycle cover games, we provide a polynomial time algorithm demonstrating the non-emptiness of the log ₂(|N|−1)-approximate core by exhibiting a vector in this approximate core for the asymmetric traveling salesman game. We improve it further by finding a -approximate core in polynomial time for some constant c. We also show that there exists an ε ₀>1 such that it is -hard to decide whether ε ₀-Core is empty or not. A preliminary version of the paper appeared in the third Workshop on Approximation and Online Algorithms (WAOA), 2005.     

Random Geometric Graph Diameter in the Unit Ball

The unit ball random geometric graph has as its vertices n points distributed independently and uniformly in the unit ball in , with two vertices adjacent if and only if their ℓ_p-distance is at most λ. Like its cousin the Erdos-Renyi random graph, G has a connectivity threshold: an asymptotic value for λ in terms of n, above which G is connected and below which G is disconnected. In the connected zone we determine upper and lower bounds for the graph diameter of G. Specifically, almost always, , where is the ℓ_p-diameter of the unit ball B. We employ a combination of methods from probabilistic combinatorics and stochastic geometry.     

A ( $2-c\frac{1}{\sqrt{N}}$ )-Approximation Algorithm for the Stable Marriage Problem

We consider the problem of finding a stable matching of maximum size when both ties and unacceptable partners are allowed in preference lists. This problem is NP-hard and the current best known approximation algorithm achieves the approximation ratio 2−c(log N)/N, where c is an arbitrary positive constant and N is the number of men in an input. In this paper, we improve the ratio to , where c is an arbitrary constant that satisfies . A preliminary version of this paper was presented at the 16th Annual International Symposium on Algorithms and Computation, ISAAC 2005.     

Modulo Constraints and the Complexity of Typechecking XML Views

The typechecking problem for transformations of relational data into tree data is the following: given a relational-to-XML transformation P, and an XML type d, decide whether for every database instance the result of the transformation P on satisfies d. TreeQL programs with projection-free conjunctive queries (see Alon et al. in ACM Trans. Comput. Log. 4(3):315–354, 2003) are considered as transformations and DTDs with arbitrary regular expressions as XML types. A non-elementary upper bound for the typechecking problem was already given by Alon et al. (ACM Trans. Comput. Log. 4(3):315–354, 2003) (although in a more general setting, where equality and negation in projection-free conjunctive queries and additional universal integrity constraints are allowed). In this paper we show that the typechecking problem is coNEXPTIME-complete. As an intermediate step we consider the following problem, which can be formulated independently of XML notions. Given a set of triples of the form (φ,k,j), where φ is a projection-free conjunctive query and k,j are natural numbers, decide whether there exists a database such that, for each triple (φ,k,j) in the set, there exists a natural number α, such that there are exactly k+j*α tuples satisfying the query φ in . Our main technical contribution consists of a NEXPTIME algorithm for the last problem. Partially supported by Polish Ministry of Science and Higher Education research project N206 022 31/3660, 2006/2009. This paper is an extended version of 20, where the coNEXPTIME upper bound was shown.     

New Linear-Time Algorithms for Edge-Coloring Planar Graphs

We show efficient algorithms for edge-coloring planar graphs. Our main result is a linear-time algorithm for coloring planar graphs with maximum degree Δ with max {Δ,9} colors. Thus the coloring is optimal for graphs with maximum degree Δ≥9. Moreover for Δ=4,5,6 we give linear-time algorithms that use Δ+2 colors. These results improve over the algorithms of Chrobak and Yung (J. Algorithms 10:35–51, 1989) and of Chrobak and Nishizeki (J. Algorithms 11:102–116, 1990) which color planar graphs using max {Δ,19} colors in linear time or using max {Δ,9} colors in time. R. Cole supported in part by NSF grants CCR0105678 and CCF0515127 and IDM0414763. Ł. Kowalik supported in part by KBN grant 4T11C04425. Part of this work was done while Ł. Kowalik was staying at the Max Planck Institute in Saarbruecken, Germany.     

Characterizing Programming Systems Allowing Program Self-Reference

The interest is in characterizing insightfully the power of program self-reference in effective programming systems ( ), the computability-theoretic analogs of programming languages (for the partial computable functions). In an in which the constructive form of Kleene’s Recursion Theorem (KRT) holds, it is possible to construct, algorithmically, from an arbitrary algorithmic task, a self-referential program that, in a sense, creates a self-copy and then performs that task on the self-copy. In an in which the not-necessarily-constructive form of Kleene’s Recursion Theorem (krt) holds, such self-referential programs exist, but cannot, in general, be found algorithmically. In an earlier effort, Royer proved that there is no collection of recursive denotational control structures whose implementability characterizes the in which KRT holds. One main result herein, proven by a finite injury priority argument, is that the in which krt holds are, similarly, not characterized by the implementability of some collection of recursive denotational control structures. On the positive side, however, a characterization of such of a rather different sort is shown herein. Though, perhaps not the insightful characterization sought after, this surprising result reveals that a hidden and inherent constructivity is always present in krt. This paper is an expanded version of [6]. This paper received support from NSF Grant CCR-0208616. “Know thyself.” Greek proverb     

On “Exploiting” Node-Heterogeneous Clusters Optimally

It is proved that “FIFO” worksharing protocols provide asymptotically optimal solutions to two problems related to sharing large collections of independent tasks in a heterogeneous network of workstations (HNOW) . In the , one seeks to accomplish as much work as possible on during a prespecified fixed period of L time units. In the , one seeks to complete W units of work by “renting” for as short a time as necessary. The worksharing protocols we study are crafted within an architectural model that characterizes via parameters that measure ’s workstations’ computational and communicational powers. All valid protocols are self-scheduling, in the sense that they determine completely both an amount of work to allocate to each of ’s workstations and a schedule for all related interworkstation communications. The schedules provide either a value for W given L, or a value for L given W, hence solve both of the motivating problems. A protocol observes a FIFO regimen if it has ’s workstations finish their assigned work, and return their results, in the same order in which they are supplied with their workloads. The proven optimality of FIFO protocols resides in the fact that they accomplish at least as much work as any other protocol during all sufficiently long worksharing episodes, and that they complete sufficiently large given collections of tasks at least as fast as any other protocol. Simulation experiments illustrate that the superiority of FIFO protocols is often observed during worksharing episodes of only a few minutes’ duration. A portion of this research was presented at the 15th ACM Symp. on Parallelism in Algorithms and Architectures (2003).     

Speeding Up HMM Decoding and Training by Exploiting Sequence Repetitions

We present a method to speed up the dynamic program algorithms used for solving the HMM decoding and training problems for discrete time-independent HMMs. We discuss the application of our method to Viterbi’s decoding and training algorithms (IEEE Trans. Inform. Theory IT-13:260–269, 1967), as well as to the forward-backward and Baum-Welch (Inequalities 3:1–8, 1972) algorithms. Our approach is based on identifying repeated substrings in the observed input sequence. Initially, we show how to exploit repetitions of all sufficiently small substrings (this is similar to the Four Russians method). Then, we describe four algorithms based alternatively on run length encoding (RLE), Lempel-Ziv (LZ78) parsing, grammar-based compression (SLP), and byte pair encoding (BPE). Compared to Viterbi’s algorithm, we achieve speedups of Θ(log n) using the Four Russians method, using RLE, using LZ78, using SLP, and Ω(r) using BPE, where k is the number of hidden states, n is the length of the observed sequence and r is its compression ratio (under each compression scheme). Our experimental results demonstrate that our new algorithms are indeed faster in practice. We also discuss a parallel implementation of our algorithms. A preliminary version of this paper appeared in Proc. 18th Annual Symposium on Combinatorial Pattern Matching (CPM), pp. 4–15, 2007. Y. Lifshits’ research was supported by the Center for the Mathematics of Information and the Lee Center for Advanced Networking. S. Mozes’ work conducted while visiting MIT.     

Quantum Algorithms for Matching Problems

We present quantum algorithms for the following matching problems in unweighted and weighted graphs with n vertices and m edges:

Many-to-Many Communication in Radio Networks

Radio networks model wireless data communication when the bandwidth is limited to one wave frequency. The key restriction of such networks is mutual interference of packets arriving simultaneously at a node. The many-to-many (m2m) communication primitive involves p participant nodes from among n nodes in the network, where the distance between any pair of participants is at most d. The task is to have all the participants get to know all the input messages. We consider three cases of the m2m communication problem. In the ad-hoc case, each participant knows only its name and the values of n, p and d. In the partially centralized case, each participant knows the topology of the network and the values of p and d, but does not know the names of the other participants. In the centralized case, each participant knows the topology of the network and the names of all the participants. For the centralized m2m problem, we give deterministic protocols, for both undirected and directed networks, working in time, which is provably optimal. For the partially centralized m2m problem, we give a randomized protocol for undirected networks working in time with high probability (whp), and we show that any deterministic protocol requires time. For the ad-hoc m2m problem, we develop a randomized protocol for undirected networks that works in time whp. We show two lower bounds for the ad-hoc m2m problem. One lower bound states that any randomized protocol for the m2m ad hoc problem requires expected time. Another lower bound states that for any deterministic protocol for the m2m ad hoc problem, there is a network on which the protocol requires time when n−p(n)=Ω(n) and d>1, and that it requires Ω(n) time when n−p(n)=o(n). The results of this paper appeared in a preliminary form in “On many-to-many communication in packet radio networks” in Proceedings of the 10th Conference on Principles of Distributed Systems (OPODIS), Bordeaux, France, 2006, Lecture Notes in Computer Science 4305, Springer, Heidelberg, pp. 258–272. The work of B.S. Chlebus was supported by NSF Grant 0310503.     

Algorithms for Modular Counting of Roots of Multivariate Polynomials

Given a multivariate polynomial P(X ₁,…,X _n ) over a finite field , let N(P) denote the number of roots over . The modular root counting problem is given a modulus r, to determine N _r (P)=N(P)mod r. We study the complexity of computing N _r (P), when the polynomial is given as a sum of monomials. We give an efficient algorithm to compute N _r (P) when the modulus r is a power of the characteristic of the field. We show that for all other moduli, the problem of computing N _r (P) is -hard. We present some hardness results which imply that our algorithm is essentially optimal for prime fields. We show an equivalence between maximum-likelihood decoding for Reed-Solomon codes and a root-finding problem for symmetric polynomials. P. Gopalan’s and R.J Lipton’s research was supported by NSF grant CCR-3606B64. V. Guruswami’s research was supported in part by NSF grant CCF-0343672 and a Sloan Research Fellowship.     

Planar and Grid Graph Reachability Problems

We study the complexity of restricted versions of s-t-connectivity, which is the standard complete problem for . In particular, we focus on different classes of planar graphs, of which grid graphs are an important special case. Our main results are: