SIAM Digital Library
 
 
 

SIAM J. on Computing

SECTION LISTING

BROWSE VOLUMES

Year Range: 

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

Search Issue | RSS Feeds RSS
Next Issue

2007

Volume 37, Issue 1, pp. 1-357

† Special Section on Foundations of Computer Science

On the Computational Complexity of the Forcing Chromatic Number

Frank Harary, Wolfgang Slany, and Oleg Verbitsky

SIAM J. Comput. 37, pp. 1-19 (19 pages)

Online Publication Date: March 30, 2007

Full Text: | Download PDF

Show Abstract
We consider vertex colorings of graphs in which adjacent vertices have distinct colors. A graph is $s$‐chromatic if it is colorable in $s$ colors and any coloring of it uses at least $s$ colors. The forcing chromatic number $F_{\chi}(G)$ of an $s$‐chromatic graph $G$ is the smallest number of vertices which must be colored so that, with the restriction that $s$ colors are used, every remaining vertex has its color determined uniquely. We estimate the computational complexity of $\force G$ relating it to the complexity class US introduced by Blass and Gurevich [Inform. Control, 55 (1982), pp. 80–88]. We prove that recognizing whether $F_{\chi}(G)\le2$ is US‐hard with respect to polynomial‐time many‐one reductions. Moreover, this problem is coNP‐hard even under the promises that $F_{\chi}(G)\le3$ and $G$ is 3‐chromatic. On the other hand, recognizing whether $F_{\chi}(G)\le k$, for each constant $k$, is reducible to a problem in US via a disjunctive truth‐table reduction. Similar results are obtained also for forcing variants of the clique and the domination numbers of a graph.

Lower Bounds for Quantum Communication Complexity

Hartmut Klauck

SIAM J. Comput. 37, pp. 20-46 (27 pages) | Cited 5 times

Online Publication Date: March 30, 2007

Full Text: | Download PDF

Show Abstract
We prove lower bounds on the bounded error quantum communication complexity. Our methods are based on the Fourier transform of the considered functions. First we generalize a method for proving classical communication complexity lower bounds developed by Raz [Comput. Complexity, 5 (1995), pp. 205–221] to the quantum case. Applying this method, we give an exponential separation between bounded error quantum communication complexity and nondeterministic quantum communication complexity. We develop several other lower bound methods based on the Fourier transform, notably showing that $\sqrt{\bar{s}(f)/\log n}$, for the average sensitivity $\bar{s}(f)$ of a function $f$, yields a lower bound on the bounded error quantum communication complexity of $f((x \wedge y)\oplus z)$, where $x$ is a Boolean word held by Alice and $y,z$ are Boolean words held by Bob. We then prove the first large lower bounds on the bounded error quantum communication complexity of functions, for which a polynomial quantum speedup is possible. For all the functions we investigate, the only previously applied general lower bound method based on discrepancy yields bounds that are $O(\log n)$.

Adiabatic Quantum State Generation

Dorit Aharonov and Amnon Ta‐Shma

SIAM J. Comput. 37, pp. 47-82 (36 pages) | Cited 6 times

Online Publication Date: March 30, 2007

Full Text: | Download PDF

Show Abstract
The design of new quantum algorithms has proven to be an extremely difficult task. This paper considers a different approach to this task by studying the problem of quantum state generation. We motivate this problem by showing that the entire class of statistical zero knowledge, which contains natural candidates for efficient quantum algorithms such as graph isomorphism and lattice problems, can be reduced to the problem of quantum state generation. To study quantum state generation, we define a paradigm which we call adiabatic state generation (ASG) and which is based on adiabatic quantum computation. The ASG paradigm is not meant to replace the standard quantum circuit model or to improve on it in terms of computational complexity. Rather, our goal is to provide a natural theoretical framework, in which quantum state generation algorithms could be designed. The new paradigm seems interesting due to its intriguing links to a variety of different areas: the analysis of spectral gaps and ground‐states of Hamiltonians in physics, rapidly mixing Markov chains, adiabatic computation, and approximate counting. To initiate the study of ASG, we prove several general lemmas that can serve as tools when using this paradigm. We demonstrate the application of the paradigm by using it to turn a variety of (classical) approximate counting algorithms into efficient quantum state generators of nontrivial quantum states, including, for example, the uniform superposition over all perfect matchings in a bipartite graph.

A Probabilistic Study on Combinatorial Expanders and Hashing

Phillip G. Bradford and Michael N. Katehakis

SIAM J. Comput. 37, pp. 83-111 (29 pages)

Online Publication Date: March 30, 2007

Full Text: | Download PDF

Show Abstract
This paper gives a new way of showing that certain constant degree graphs are graph expanders. This is done by giving new proofs of expansion for three permutations of the Gabber–Galil expander. Our results give an expansion factor of $\frac{3}{16}$ for subgraphs of these three‐regular graphs with $(p-1)^2$ inputs for $p$ prime. The proofs are not based on eigenvalue methods or higher algebra. The same methods show the expected number of probes for unsuccessful search in double hashing is bounded by $\frac{1}{1-\alpha}$, where α is the load factor. This assumes a double hashing scheme in which two hash functions are randomly and independently chosen from a specified uniform distribution. The result is valid regardless of the distribution of the inputs. This is analogous to Carter and Wegman’s result for hashing with chaining. This paper concludes by elaborating on how any sufficiently sized subset of inputs in any distribution expands in the subgraph of the Gabber–Galil graph expander of focus. This is related to any key distribution having expected $\frac{1}{1 - \alpha}$ probes for unsuccessful search for double hashing given the initial random, independent, and uniform choice of two universal hash functions.

Hardness of the Undirected Congestion Minimization Problem

Matthew Andrews and Lisa Zhang

SIAM J. Comput. 37, pp. 112-131 (20 pages) | Cited 2 times

Online Publication Date: April 10, 2007

Full Text: | Download PDF

Show Abstract
We show that there is no $\gamma\log\log M/\log\log\log M$‐approximation for the undirected congestion minimization problem unless $NP \subseteq ZPTIME(n^{{\rm polylog} n})$, where $M$ is the size of the graph and γ is some positive constant.

Constraint Satisfaction, Logic and Forbidden Patterns

Florent Madelaine and Iain A. Stewart

SIAM J. Comput. 37, pp. 132-163 (32 pages) | Cited 2 times

Online Publication Date: April 10, 2007

Full Text: | Download PDF

Show Abstract
In the 1990s, Feder and Vardi attempted to find a large subclass of NP which exhibits a dichotomy, that is, where every problem in the subclass is either solvable in polynomial‐time or NP‐complete. Their studies resulted in a candidate class of problems, namely, those definable in the logic MMSNP. While it remains open as to whether MMSNP exhibits a dichotomy, for various reasons it remains a strong candidate. Feder and Vardi added to the significance of MMSNP by proving that, although MMSNP strictly contains CSP, the class of constraint satisfaction problems, MMSNP and CSP are computationally equivalent. We introduce here a new class of combinatorial problems, the class of forbidden patterns problems FPP, and characterize MMSNP as the finite unions of problems from FPP. We use our characterization to detail exactly those problems that are in MMSNP but not in CSP. Furthermore, given a problem in MMSNP, we are able to decide whether the problem is in CSP or not (this whole process is effective). If the problem is in CSP, then we can construct a template for this problem; otherwise, for any given candidate for the role of template, we can build a counterexample (again, this process is effective).
back to top
RSS Feeds
FREE

Special Section on Foundations of Computer Science

Dimitris Achlioptas, Guest Editors and Vladlen Koltun, Guest Editors

SIAM J. Comput. 37, pp. 165-165 (1 page)

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
This volume comprises the polished and fully refereed versions of a selection of papers presented at the Forty‐Fifth Annual IEEE Symposium on Foundations of Computer Science (FOCS 2004), held in Rome, Italy, October 17–19, 2004. Unrefereed preliminary versions of the papers presented at the symposium appeared in the proceedings of the meeting, published by IEEE.
The FOCS 2004 Program Committee consisted of Dimitris Achlioptas, Micah Adler, Eli Ben‐Sasson, Faith Fich, Oded Goldreich, Martin Grohe, Sean Hallgren, Johan Håstad, Giuseppe F. Italiano, Vladlen Koltun, Yuval Rabani, Miklos Santha, Leonard Schulman, Rocco Servedio, D. Sivakumar, Eli Upfal, and David Williamson.
Out of 272 “Extended Abstracts” submitted to the FOCS 2004 Program Committee, 64 were selected for presentation at the symposium. Eight of those 64 papers are included in this volume. This collection encompasses a wide variety of questions and methods in theoretical computer science, often shedding new light on entire areas with a fresh approach. The topics include fundamental questions of complexity theory and algorithms as well as foundational mathematical problems. All papers were refereed in accordance with SICOMP’s stringent standards, and most were substantially updated in the process.
We take this opportunity to thank all the referees whose anonymous work has significantly contributed to the value of this volume. It was an honor to edit this special section in SIAM Journal on Computing.

Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation

Dorit Aharonov, Wim van Dam, Julia Kempe, Zeph Landau, Seth Lloyd, and Oded Regev

SIAM J. Comput. 37, pp. 166-194 (29 pages) | Cited 37 times

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two‐dimensional grid with nearest neighbor interactions. The equivalence between the models allows stating the main open problems in quantum computation using well‐studied mathematical objects such as eigenvectors and spectral gaps of sparse matrices.

On the List and Bounded Distance Decodability of Reed–Solomon Codes

Qi Cheng and Daqing Wan

SIAM J. Comput. 37, pp. 195-209 (15 pages) | Cited 7 times

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
For an error‐correcting code and a distance bound, the list decoding problem is to compute all the codewords within a given distance to a received message. The bounded distance decoding problem is to find one codeword if there is at least one codeword within the given distance, or to output the empty set if there is not. Obviously the bounded distance decoding problem is not as hard as the list decoding problem. For a Reed–Solomon code $[n,k]_q$, a simple counting argument shows that for any integer $0<g < n $, there exists at least one Hamming ball of radius $n-g$, which contains at least ${n \choose g}/q^{g-k} $ many codewords. Let $\hat{g}(n,k,q) $ be the smallest positive integer $g$ such that ${{n \choose g}/ q^{g-k}} \leq 1 $. One knows that $$k-1\leq \hat{g}(n,k,q) \leq \sqrt{n(k-1)}\leq n.$$ For the distance bound up to $n-\sqrt{n(k-1)}$, it is known that both the list and bounded distance decoding can be solved efficiently. For the distance bound between $n - \sqrt{n(k-1)}$ and $n- \hat{g}(n,k,q) $, we do not know whether the Reed–Solomon code is list or bounded distance decodable; nor do we know whether there are polynomially many codewords in all balls of the radius. It is generally believed that the answer to both questions is no. In this paper, we prove the following: (1) List decoding cannot be done for radius $n - \hat{g}(n,k,q) $ or larger, unless the discrete logarithm over ${\bf F}_{q^{\hat{g}(n,k,q) -k}}$ is easy. (2) Let $h$ and $g$ be positive integers satisfying $ q\geq \max(g^2, (h-1)^{2+\epsilon})$ and $g \geq ({4 \over \epsilon } + 2) (h+1)$ for a constant $\epsilon > 0 $. We show that the discrete logarithm problem over ${\bf F}_{q^{h}}$ can be efficiently reduced by a randomized algorithm to the bounded distance decoding problem of the Reed–Solomon code $[q, g-h]_q$ with radius $q - g$. These results show that the decoding problems for the Reed–Solomon code are at least as hard as the discrete logarithm problem over certain finite fields. For the list decoding problem of Reed–Solomon codes, although the infeasible radius that we obtain is much larger than the radius, which is known to be feasible, it is the first nontrivial bound. Our result on the bounded distance decodability of Reed–Solomon codes is also the first of its kind. The main tools for obtaining these results are an interesting connection between the problem of list decoding of Reed–Solomon code, the problem of a discrete logarithm over finite fields, and a generalization of Katz’s theorem on representations of elements in an extension finite field by products of distinct linear factors.

Quantum Walk Algorithm for Element Distinctness

Andris Ambainis

SIAM J. Comput. 37, pp. 210-239 (30 pages) | Cited 21 times

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
We use quantum walks to construct a new quantum algorithm for element distinctness and its generalization. For element distinctness (the problem of finding two equal items among $N$ given items), we get an $O(N^{2/3})$ query quantum algorithm. This improves the previous $O(N^{3/4})$ quantum algorithm of Buhrman et al. [SIAM J. Comput., 34 (2005), pp. 1324–1330] and matches the lower bound of Aaronson and Shi [J. ACM, 51 (2004), pp. 595–605]. We also give an $O(N^{k/(k+1)})$ query quantum algorithm for the generalization of element distinctness in which we have to find $k$ equal items among $N$ items.

Dynamic Optimality—Almost

Erik D. Demaine, Dion Harmon, John Iacono, and Mihai Pǎtraşcu

SIAM J. Comput. 37, pp. 240-251 (12 pages)

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
We present an $O(\lg \lg n)$‐competitive online binary search tree, improving upon the best previous (trivial) competitive ratio of $O(\lg n)$. This is the first major progress on Sleator and Tarjan’s dynamic optimality conjecture of 1985 that $O(1)$‐competitive binary search trees exist.

Algebras with Polynomial Identities and Computing the Determinant

Steve Chien and Alistair Sinclair

SIAM J. Comput. 37, pp. 252-266 (15 pages)

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
In 1991, Nisan proved an exponential lower bound on the size of an algebraic branching program (ABP) that computes the determinant of a matrix in the noncommutative “free algebra” setting, in which there are no nontrivial relationships between the matrix entries. By contrast, when the matrix entries commute there are polynomial size ABPs for the determinant. This paper extends Nisan’s result to a much wider class of noncommutative algebras, including all nontrivial matrix algebras over any field of characteristic 0, group algebras of all nonabelian finite groups over algebraically closed fields of characteristic 0, the quaternion algebra, and the Clifford algebras. As a result, we obtain more compelling evidence for the essential role played by commutativity in the efficient computation of the determinant. The key to our approach is a characterization of noncommutative algebras by means of the polynomial identities that they satisfy. Extending Nisan’s lower bound framework, we find that any reduction in complexity compared to the free algebra must arise from the ability of the identities to reduce the rank of certain naturally associated matrices. Using results from the theory of algebras with polynomial identities, we are able to show that none of the identities of the above classes of algebras is able to achieve such a rank reduction.

Worst‐Case to Average‐Case Reductions Based on Gaussian Measures

Daniele Micciancio and Oded Regev

SIAM J. Comput. 37, pp. 267-302 (36 pages) | Cited 5 times

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
We show that finding small solutions to random modular linear equations is at least as hard as approximating several lattice problems in the worst case within a factor almost linear in the dimension of the lattice. The lattice problems we consider are the shortest vector problem, the shortest independent vectors problem, the covering radius problem, and the guaranteed distance decoding problem (a variant of the well‐known closest vector problem). The approximation factor we obtain is $n \log^{O(1)} n$ for all four problems. This greatly improves on all previous work on the subject starting from Ajtai’s seminal paper [Generating hard instances of lattice problems, in Complexity of Computations and Proofs, Quad. Mat. 13, Dept. Math., Seconda Univ. Napoli, Caserta, Italy, 2004, pp. 1–32] up to the strongest previously known results by Micciancio [SIAM J. Comput., 34 (2004), pp. 118–169]. Our results also bring us closer to the limit where the problems are no longer known to be in NP intersect coNP. Our main tools are Gaussian measures on lattices and the high‐dimensional Fourier transform. We start by defining a new lattice parameter which determines the amount of Gaussian noise that one has to add to a lattice in order to get close to a uniform distribution. In addition to yielding quantitatively much stronger results, the use of this parameter allows us to simplify many of the complications in previous work. Our technical contributions are twofold. First, we show tight connections between this new parameter and existing lattice parameters. One such important connection is between this parameter and the length of the shortest set of linearly independent vectors. Second, we prove that the distribution that one obtains after adding Gaussian noise to the lattice has the following interesting property: the distribution of the noise vector when conditioning on the final value behaves in many respects like the original Gaussian noise vector. In particular, its moments remain essentially unchanged.

A Polynomial Time Algorithm for Computing an Arrow–Debreu Market Equilibrium for Linear Utilities

Kamal Jain

SIAM J. Comput. 37, pp. 303-318 (16 pages) | Cited 1 time

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
We provide the first polynomial time exact algorithm for computing an Arrow–Debreu market equilibrium for the case of linear utilities. Our algorithm is based on solving a convex program using the ellipsoid algorithm and simultaneous diophantine approximation. As a side result, we prove that the set of assignments at equilibrium is convex and the equilibrium prices themselves are log‐convex. Our convex program is explicit and intuitive, which allows maximizing a concave function over the set of equilibria. On the practical side, Ye developed an interior point algorithm [Lecture Notes in Comput. Sci. 3521, Springer, New York, 2005, pp. 3–5] to find an equilibrium based on our convex program. We also derive separate combinatorial characterizations of equilibrium for Arrow–Debreu and Fisher cases. Our convex program can be extended for many nonlinear utilities and production models. Our paper also makes a powerful theorem (Theorem 6.4.1 in [M. Grotschel, L. Lovasz, and A. Schrijver, Geometric Algorithms and Combinatorial Optimization, 2nd ed., Springer‐Verlag, Berlin, Heidelberg, 1993]) even more powerful (in Theorems 12 and 13) in the area of geometric algorithms and combinatorial optimization. The main idea in this generalization is to allow ellipsoids to contain not the whole convex region but a part of it. This theorem is of independent interest.

Optimal Inapproximability Results for MAX‐CUT and Other 2‐Variable CSPs?

Subhash Khot, Guy Kindler, Elchanan Mossel, and Ryan O’Donnell

SIAM J. Comput. 37, pp. 319-357 (39 pages) | Cited 15 times

Online Publication Date: May 14, 2007

Full Text: | Download PDF

Show Abstract
In this paper we show a reduction from the Unique Games problem to the problem of approximating MAX‐CUT to within a factor of $\alpha_{\text{\tiny{GW}}} + \epsilon$ for all $\epsilon > 0$; here $\alpha_{\text{\tiny{GW}}} \approx .878567$ denotes the approximation ratio achieved by the algorithm of Goemans and Williamson in [J. Assoc. Comput. Mach., 42 (1995), pp. 1115–1145]. This implies that if the Unique Games Conjecture of Khot in [Proceedings of the 34th Annual ACM Symposium on Theory of Computing, 2002, pp. 767–775] holds, then the Goemans–Williamson approximation algorithm is optimal. Our result indicates that the geometric nature of the Goemans–Williamson algorithm might be intrinsic to the MAX‐CUT problem. Our reduction relies on a theorem we call Majority Is Stablest. This was introduced as a conjecture in the original version of this paper, and was subsequently confirmed in [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21–30]. A stronger version of this conjecture called Plurality Is Stablest is still open, although [E. Mossel, R. O’Donnell, and K. Oleszkiewicz, Proceedings of the 46th Annual IEEE Symposium on Foundations of Computer Science, 2005, pp. 21–30] contains a proof of an asymptotic version of it. Our techniques extend to several other two‐variable constraint satisfaction problems. In particular, subject to the Unique Games Conjecture, we show tight or nearly tight hardness results for MAX‐2SAT, MAX‐$q$‐CUT, and MAX‐2LIN($q$). For MAX‐2SAT we show approximation hardness up to a factor of roughly $.943$. This nearly matches the $.940$ approximation algorithm of Lewin, Livnat, and Zwick in [Proceedings of the 9th Annual Conference on Integer Programming and Combinatorial Optimization, Springer‐Verlag, Berlin, 2002, pp. 67–82]. Furthermore, we show that our .943... factor is actually tight for a slightly restricted version of MAX‐2SAT. For MAX‐$q$‐CUT we show a hardness factor which asymptotically (for large $q$) matches the approximation factor achieved by Frieze and Jerrum [Improved approximation algorithms for MAX k‐CUT and MAX BISECTION, in Integer Programming and Combinatorial Optimization, Springer‐Verlag, Berlin, pp. 1–13], namely $1 - 1/q + 2({\rm ln}\,q)/q^2$. For MAX‐2LIN($q$) we show hardness of distinguishing between instances which are $(1-\epsilon)$‐satisfiable and those which are not even, roughly, $(q^{-\epsilon/2})$‐satisfiable. These parameters almost match those achieved by the recent algorithm of Charikar, Makarychev, and Makarychev [Proceedings of the 38th Annual ACM Symposium on Theory of Computing, 2006, pp. 205–214]. The hardness result holds even for instances in which all equations are of the form $x_i - x_j = c$. At a more qualitative level, this result also implies that $1-\epsilon$ vs. ϵ hardness for MAX‐2LIN($q$) is equivalent to the Unique Games Conjecture.
Close

Your Recent History Recent History Help

Recently Viewed

  1. Complexity of Self‐Assembled Shapes
  2. Testing Polynomials over General Fields
  3. Quantum Algorithms for Some Hidden Shift Problems
  4. The Directed Steiner Network Problem is Tractable for a Constant Number of Terminals
  5. Online Scheduling of Precedence Constrained Tasks
© 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