SIAM Digital Library
 
 
 

SIAM J. on Computing

All Journal content published prior to 1997 is part of LOCUS.

Search Issue | RSS Feeds RSS
Previous Issue Next Issue

2006

Volume 36, Issue 3, pp. 563-843


Time‐Space Lower Bounds for the Polynomial‐Time Hierarchy on Randomized Machines

Scott Diehl and Dieter van Melkebeek

SIAM J. Comput. 36, pp. 563-594 (32 pages)

Online Publication Date: August 25, 2006

Full Text: | Download PDF

Show Abstract
We establish the first polynomial‐strength time‐space lower bounds for problems in the linear‐time hierarchy on randomized machines with two‐sided error. We show that for any integer $\ell > 1$ and constant $c < \ell$, there exists a positive constant $d$ such that QSAT$_{\ell}$ cannot be computed by such machines in time $n^c$ and space $n^d$, where QSAT$_{\ell}$ denotes the problem of deciding the validity of a quantified Boolean formula with at most $\ell - 1$ quantifier alternations. Moreover, $d$ approaches 1/2 from below as $c$ approaches 1 from above for $\ell = 2$, and $d$ approaches 1 from below as $c$ approaches 1 from above for $\ell \ge 3$. In fact, we establish the stronger result that for any constants $a \le 1$ and $c < 1 + (\ell - 1)a$, there exists a positive constant $d$ such that linear‐time alternating machines using space $n^a$ and $\ell - 1$ alternations cannot be simulated by randomized machines with two‐sided error running in time $n^c$ and space $n^d$, where $d$ approaches $a/2$ from below as $c$ approaches 1 from above for $\ell = 2$, and $d$ approaches $a$ from below as $c$ approaches 1 from above for $\ell \ge 3$. Corresponding to $\ell = 1$, we prove that there exists a positive constant $d$ such that the set of Boolean tautologies cannot be decided by a randomized machine with one‐sided error in time $n^{1.759}$ and space $n^d$. As a corollary, this gives the same lower bound for satisfiability on deterministic machines, improving on the previously best known such result.

Infinitely‐Often Autoreducible Sets

Richard Beigel, Lance Fortnow, and Frank Stephan

SIAM J. Comput. 36, pp. 595-608 (14 pages)

Online Publication Date: August 29, 2006

Full Text: | Download PDF

Show Abstract
A set $A$ is autoreducible if one can compute, for all $x$, the value $A(x)$ by querying $A$ only at places $y \neq x$. Furthermore, $A$ is infinitely‐often autoreducible if, for infinitely many $x$, the value $A(x)$ can be computed by querying $A$ only at places $y \neq x$. For all other $x$, the computation outputs a special symbol to signal that the reduction is undefined. It is shown that for polynomial time Turing and truth‐table autoreducibility there are $A$, $B$, $C$ in the class EXP of all exponential‐time computable sets such that $A$ is not infinitely‐often Turing autoreducible, $B$ is Turing autoreducible but not infinitely‐often truth‐table autoreducible and $C$ is truth‐table autoreducible with $g(n)+1$ queries but not infinitely‐often Turing autoreducible with $g(n)$ queries. Here $n$ is the length of the input, $g$ is nondecreasing, and there exists a polynomial $p$ such that $p(n)$ bounds both the computation time and the value of $g$ at input of length $n$. Furthermore, connections between notions of infinitely‐often autoreducibility and notions of approximability are investigated. The Hausdorff‐dimension of the class of sets which are not infinitely‐often autoreducible is shown to be $1$.

An Extension of the Lovász Local Lemma, and its Applications to Integer Programming

Aravind Srinivasan

SIAM J. Comput. 36, pp. 609-634 (26 pages) | Cited 1 time

Online Publication Date: September 21, 2006

Full Text: | Download PDF

Show Abstract
The Lovász local lemma due to Erdős and Lovász (Infinite and Finite Sets, Colloq. Math. Soc. J. Bolyai 11, 1975, pp. 609–627) is a powerful tool in proving the existence of rare events. We present an extension of this lemma, which works well when the event to be shown to exist is a conjunction of individual events, each of which asserts that a random variable does not deviate much from its mean. As applications, we consider two classes of NP‐hard integer programs: minimax and covering integer programs. A key technique, randomized rounding of linear relaxations, was developed by Raghavan and Thompson (Combinatorica, 7 (1987), pp. 365–374) to derive good approximation algorithms for such problems. We use our extension of the local lemma to prove that randomized rounding produces, with nonzero probability, much better feasible solutions than known before, if the constraint matrices of these integer programs are column‐sparse (e.g., routing using short paths, problems on hypergraphs with small dimension/degree). This complements certain well‐known results from discrepancy theory. We also generalize the method of pessimistic estimators due to Raghavan (J. Comput. System Sci., 37 (1988), pp. 130–143), to obtain constructive (algorithmic) versions of our results for covering integer programs.

Approximating Longest Cycles in Graphs with Bounded Degrees

Guantao Chen, Zhicheng Gao, Xingxing Yu, and Wenan Zang

SIAM J. Comput. 36, pp. 635-656 (22 pages) | Cited 1 time

Online Publication Date: October 03, 2006

Full Text: | Download PDF

Show Abstract
Jackson and Wormald conjecture that if $G$ is a 3‐connected $n$‐vertex graph with maximum degree $d\ge 4$, then $G$ has a cycle of length $\Omega(n^{\log_{d-1}2})$. We show that this conjecture holds when $d-1$ is replaced by $\max\{64,4d+1\}$. Our proof implies a cubic algorithm for finding such a cycle.

Fairness Measures for Resource Allocation

Amit Kumar and Jon Kleinberg

SIAM J. Comput. 36, pp. 657-680 (24 pages) | Cited 5 times

Online Publication Date: October 10, 2006

Full Text: | Download PDF

Show Abstract
In many optimization problems, one seeks to allocate a limited set of resources to a set of individuals with demands. Thus, such allocations can naturally be viewed as vectors, with one coordinate representing each individual. Motivated by work in network routing and bandwidth assignment, we consider the problem of producing solutions that simultaneously approximate all feasible allocations in a coordinate‐wise sense. This is a very strong type of “global” approximation guarantee, and we explore its consequences in a wide range of discrete optimization problems, including facility location, scheduling, and bandwidth assignment in networks. A fundamental issue—one not encountered in the traditional design of approximation algorithms—is that good approximations in this global sense need not exist for every problem instance; there is no a priori reason why there should be an allocation that simultaneously approximates all others. As a result, the existential questions concerning such good allocations lead to a new perspective on a number of fundamental problems in resource allocation, and on the structure of their feasible solutions.

Dynamic Subgraph Connectivity with Geometric Applications

Timothy M. Chan

SIAM J. Comput. 36, pp. 681-694 (14 pages) | Cited 4 times

Online Publication Date: October 10, 2006

Full Text: | Download PDF

Show Abstract
Inspired by dynamic connectivity applications in computational geometry, we consider a problem we call dynamic subgraph connectivity: design a data structure for an undirected graph $G=(V,E)$ and a subset of vertices $S \subseteq V$ to support insertions/deletions in $S$ and connectivity queries (are two vertices connected?) in the subgraph induced by $S$. We develop the first sublinear, fully dynamic method for this problem for general sparse graphs, using a combination of several simple ideas. Our method requires $\widetilde O(|E|^{4\omega/(3\omega+3)})=O(|E|^{0.94})$ amortized update time, and $\widetilde O(|E|^{1/3})$ query time, after $\widetilde O(|E|^{(5\omega+1)/(3\omega+3)})$ preprocessing time, where ω is the matrix multiplication exponent and $\widetilde O$ hides polylogarithmic factors.

On the Number of Crossing‐Free Matchings, Cycles, and Partitions

Micha Sharir and Emo Welzl

SIAM J. Comput. 36, pp. 695-720 (26 pages) | Cited 1 time

Online Publication Date: October 12, 2006

Full Text: | Download PDF

Show Abstract
We show that a set of $n$ points in the plane has at most $O(10.05^n)$ perfect matchings with crossing‐free straight‐line embedding. The expected number of perfect crossing‐free matchings of a set of $n$ points drawn independently and identically distributed from an arbitrary distribution in the plane is at most $O(9.24^n)$. Several related bounds are derived: (a) The number of all (not necessarily perfect) crossing‐free matchings is at most $O(10.43^n)$. (b) The number of red‐blue perfect crossing‐free matchings (where the points are colored red or blue and each edge of the matching must connect a red point with a blue point) is at most $O(7.61^n)$. (c) The number of left‐right perfect crossing‐free matchings (where the points are designated as left or right endpoints of the matching edges) is at most $O(5.38^n)$. (d) The number of perfect crossing‐free matchings across a line (where all the matching edges must cross a fixed halving line of the set) is at most $4^n$. These bounds are employed to infer that a set of $n$ points in the plane has at most $O(86.81^n)$ crossing‐free spanning cycles (simple polygonizations) and at most $O(12.24^n)$ crossing‐free partitions (these are partitions of the point set so that the convex hulls of the individual parts are pairwise disjoint). We also derive lower bounds for some of these quantities.

Counting and Enumerating Pointed Pseudotriangulations with the Greedy Flip Algorithm

Hervé Brönnimann, Lutz Kettner, Michel Pocchiola, and Jack Snoeyink

SIAM J. Comput. 36, pp. 721-739 (19 pages) | Cited 3 times

Online Publication Date: October 16, 2006

Full Text: | Download PDF

Show Abstract
We present an algorithm to enumerate the pointed pseudotriangulations of a given point set, based on the greedy flip algorithm of Pocchiola and Vegter [Discrete Comput. Geom. 16 (1996), pp. 419–453]. Our two independent implementations agree and allow us to experimentally verify or disprove conjectures on the numbers of pointed pseudotriangulations and triangulations of a given point set. (For example, we establish that the number of triangulations is bounded by the number of pointed pseudotriangulations for all sets of up to 10 points.)

Random $k$‐SAT: Two Moments Suffice to Cross a Sharp Threshold

Dimitris Achlioptas and Cristopher Moore

SIAM J. Comput. 36, pp. 740-762 (23 pages) | Cited 12 times

Online Publication Date: October 24, 2006

Full Text: | Download PDF

Show Abstract
Many NP‐complete constraint satisfaction problems appear to undergo a “phase transition” from solubility to insolubility when the constraint density passes through a critical threshold. In all such cases it is easy to derive upper bounds on the location of the threshold by showing that above a certain density the first moment (expectation) of the number of solutions tends to zero. We show that in the case of certain symmetric constraints, considering the second moment of the number of solutions yields nearly matching lower bounds for the location of the threshold. Specifically, we prove that the threshold for both random hypergraph 2‐colorability (Property B) and random Not‐All‐Equal $k$‐SAT is $2^{k-1}\ln 2 -O(1)$. As a corollary, we establish that the threshold for random $k$‐SAT is of order $\Theta(2^k)$, resolving a long‐standing open problem.

Quantum Algorithms for Some Hidden Shift Problems

Wim van Dam, Sean Hallgren, and Lawrence Ip

SIAM J. Comput. 36, pp. 763-778 (16 pages) | Cited 2 times

Online Publication Date: October 24, 2006

Full Text: | Download PDF

Show Abstract
Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of “unknown shift” problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure.

Testing Polynomials over General Fields

Tali Kaufman and Dana Ron

SIAM J. Comput. 36, pp. 779-802 (24 pages) | Cited 1 time

Online Publication Date: October 30, 2006

Full Text: | Download PDF

Show Abstract
In this work we fill the knowledge gap concerning testing polynomials over finite fields. As previous works show, when the cardinality of the field, $q$, is sufficiently larger than the degree bound, $d$, then the number of queries sufficient for testing is polynomial or even linear in $d$. On the other hand, when $q=2$ then the number of queries, both sufficient and necessary, grows exponentially with $d$. Here we study the intermediate case where $2 < q = O(d)$ and show a smooth transition between the two extremes. Specifically, let $p$ be the characteristic of the field (so that $p$ is prime and $q = p^s$ for some integer $s \geq 1$). Then the number of queries performed by the test grows like $\ell\cdot q^{2\ell+1}$, where $\ell = \big\lceil \frac{d+1}{q-q/p}\big\rceil $. Furthermore, $q^{\Omega(\ell)}$ queries are necessary when $q = O(d)$. The test itself provides a unifying view of the tests for these two extremes: it considers random affine subspaces of dimension $\ell$ and verifies that the function restricted to the selected subspaces is a polynomial of degree at most $d$. Viewed in the context of coding theory, our result shows that Reed–Muller codes over general fields (usually referred to as generalized Reed–Muller (GRM) codes) are locally testable. In the course of our analysis we provide a characterization of small‐weight words that span the code. Such a characterization was previously known only when the field size is a prime or is sufficiently large, in which case the minimum‐weight words span the code.

Exponential Determinization for ω‐Automata with a Strong Fairness Acceptance Condition

Shmuel Safra

SIAM J. Comput. 36, pp. 803-814 (12 pages)

Online Publication Date: November 03, 2006

Full Text: | Download PDF

Show Abstract
In [S. Safra, Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988, pp. 319–327] an exponential determinization procedure for Buchi automata was shown, yielding tight bounds for decision procedures of some logics (see [A. E. Emerson and C. Jutla, Proceedings of the 29th IEEE Symposium on Foundations of Computer Science, 1988, pp. 328–337; Safra (1988); S. Safra and M. Y. Vardi, Proceedings of the 21st ACM Symposium on Theory of Computing, 1989, pp. 127–137; and D. Kozen and J. Tiuryn, Logics of program, in Handbook of Theoretical Computer Science, Elsevier, Amsterdam, 1990, pp. 789–840]). In Safra and Vardi (1989) the complexity of determinization and complementation of ω‐automata was further investigated, leaving as an open question the complexity of the determinization of a single class of ω‐automata. For this class of ω‐automata with strong fairness as an acceptance condition (Streett automata), Safra and Vardi (1989) managed to show an exponential complementation procedure; however, the blow‐up of translating these automata—to any of the classes known to admit exponential determinization—is inherently exponential. This might suggest that the blow‐up of the determinization of Streett automata is inherently doubly exponential. This paper shows an exponential determinization construction for Streett automata. In fact, the complexity of our construction is roughly the same as the complexity achieved in Safra (1988) for Buchi automata. Moreover, a simple observation extends this upper bound to the complementation problem. Since any ω‐automaton that admits exponential determinization can be easily converted into a Streett automaton, we have obtained a single procedure that can be used for all of these conversions. Furthermore, this construction is optimal (up to a constant factor in the exponent) for all of these conversions. Our results imply that Streett automata (with strong fairness as an acceptance condition) can be used instead of Buchi automata (with the weaker acceptance condition) without any loss of efficiency.

Computing Maximally Separated Sets in the Plane

Pankaj K. Agarwal, Mark Overmars, and Micha Sharir

SIAM J. Comput. 36, pp. 815-834 (20 pages)

Online Publication Date: November 03, 2006

Full Text: | Download PDF

Show Abstract
Let $S$ be a set of $n$ points in $\reals^2$. Given an integer $1 \le k \le n$, we wish to find a maximally separated subset $I \subseteq S$ of size $k$; this is a subset for which the minimum among the ${k\choose 2}$ pairwise distances between its points is as large as possible. The decision problem associated with this problem is to determine whether there exists $I\subseteq S$, $|I|=k$, so that all ${k\choose 2}$ pairwise distances in $I$ are at least 2. This problem can also be formulated in terms of disk‐intersection graphs: Let $D$ be the set of unit disks centered at the points of $S$. The disk‐intersection graph $G$ of $D$ has as edges all pairs of disks with nonempty intersection. Any set $I$ with the above properties is then the set of centers of disks that form an independent set in the graph $G$. This problem is known to be NP‐complete if $k$ is part of the input. In this paper we first present a linear‐time $\eps$‐approximation algorithm for any constant $k$. Next we give exact algorithms for the cases $k=3$ and $k=4$ that run in time $O(n^{4/3}\polylog(n))$. We also present a simpler $n^{O(\sqrt{k})}$‐time exact algorithm (as compared with the recent algorithm in [J. Alber and J. Fiala, J. Algorithms, 52 (2004), pp. 134–151]) for arbitrary values of $k$.

A Probabilistic Approach to the Dichotomy Problem

Tomasz Łuczak and Jaroslav Nešetřil

SIAM J. Comput. 36, pp. 835-843 (9 pages)

Online Publication Date: November 14, 2006

Full Text: | Download PDF

Show Abstract
Let ${\mathcal R}(n,k)$ denote the random $k$‐ary relation defined on the set $[n]=\{1,2,\dots,n\}$. We show that the probability that $([n], {\mathcal R}(n,k))$ is projective tends to one, as either $n$ or $k$ tends to infinity. This result implies that for most relational systems $(B,{{\underline{R}}})$ the ${{\textrm{CSP}}}(B,{{\underline{R}}})$ problem is NP‐complete (and thus that the dichotomy conjecture holds with probability 1), and confirms a conjecture of Rosenberg [I. G. Rosenberg, Rocky Mountain J. Math., 3 (1973), pp. 631–639].
Close

Your Recent History Recent History Help

Recently Viewed

  1. The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals
  2. Online Scheduling of Precedence Constrained Tasks
  3. Classifying the Complexity of Constraints Using Finite Algebras
  4. Counting Complexity of Solvable Black-Box Group Problems
  5. On Multidimensional Packing Problems
© SIAM By using SIAM Publications Online you agree to abide by the Terms and Conditions of Use. Banner art adapted from a figure by Hinke M. Osinga and Bernd Krauskopf (University of Bristol, UK).
close