122 research outputs found
The Quantum Query Complexity of Algebraic Properties
We present quantum query complexity bounds for testing algebraic properties.
For a set S and a binary operation on S, we consider the decision problem
whether is a semigroup or has an identity element. If S is a monoid, we
want to decide whether S is a group.
We present quantum algorithms for these problems that improve the best known
classical complexity bounds. In particular, we give the first application of
the new quantum random walk technique by Magniez, Nayak, Roland, and Santha
that improves the previous bounds by Ambainis and Szegedy. We also present
several lower bounds for testing algebraic properties.Comment: 13 pages, 0 figure
Deterministic Identity Testing for Sum of Read-Once Oblivious Arithmetic Branching Programs
A read-once oblivious arithmetic branching program (ROABP) is an arithmetic
branching program (ABP) where each variable occurs in at most one layer. We
give the first polynomial time whitebox identity test for a polynomial computed
by a sum of constantly many ROABPs. We also give a corresponding blackbox
algorithm with quasi-polynomial time complexity . In both the
cases, our time complexity is double exponential in the number of ROABPs.
ROABPs are a generalization of set-multilinear depth- circuits. The prior
results for the sum of constantly many set-multilinear depth- circuits were
only slightly better than brute-force, i.e. exponential-time.
Our techniques are a new interplay of three concepts for ROABP: low
evaluation dimension, basis isolating weight assignment and low-support rank
concentration. We relate basis isolation to rank concentration and extend it to
a sum of two ROABPs using evaluation dimension (or partial derivatives).Comment: 22 pages, Computational Complexity Conference, 201
The Isomorphism Problem for Planar 3-Connected Graphs is in Unambiguous Logspace
The isomorphism problem for planar graphs is known to be efficiently
solvable. For planar 3-connected graphs, the isomorphism problem can be solved
by efficient parallel algorithms, it is in the class . In this paper we
improve the upper bound for planar 3-connected graphs to unambiguous logspace,
in fact to . As a consequence of our method we get that the
isomorphism problem for oriented graphs is in . We also show that the
problems are hard for
Restricted Information from Nonadaptive Queries to NP
AbstractWe investigate classes of sets that can be decided by bounded truth-table reductions to an NP set in which evaluators donothave full access to the answers to the queries but get only restricted information such as the number of queries that are in the oracle set or even just this number modulom, for somem⩾2. We also investigate the case in which evaluators are nondeterministic. We show that when we vary the information that the evaluators get, this can change the resulting power of the evaluators. We locate all these classes within levels of the Boolean hierarchy which allows us to compare the complexity of such classes
The complexity of the characteristic and the minimal polynomial
AbstractWe investigate the complexity of (1) computing the characteristic polynomial, the minimal polynomial, and all the invariant factors of an integer matrix, and of (2) verifying them, when the coefficients are given as input.It is known that each coefficient of the characteristic polynomial of a matrix A is computable in GapL, and the constant term, the determinant of A, is complete for GapL. We show that the verification of the characteristic polynomial is complete for complexity class C=L (exact counting logspace).We show that each coefficient of the minimal polynomial of a matrix A can be computed in AC0(GapL), the AC0-closure of GapL, and there is a coefficient which is hard for GapL. Furthermore, the verification of the minimal polynomial is in AC0(C=L) and is hard for C=L. The hardness result extends to (computing and verifying) the system of all invariant factors of a matrix
On sets bounded truth-table reducible to P-selective sets
We show that if every NP set is ≤
P
btt
-reducible to some P-selective set, then NP is contained in DTIME(2nO(1/logn√)). The result is extended for some unbounded reducibilities such as ≤
P
polylog-tt
-reducibility
04421 Abstracts Collection -- Algebraic Methods in Computational Complexity
From 10.10.04 to 15.10.04, the Dagstuhl Seminar 04421
``Algebraic Methods in Computational Complexity\u27\u27
was held in the International Conference and Research Center (IBFI),
Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Pinpointing computation with modular queries in the Boolean hierarchy
A modular query consists of asking how many (modulo m) of k strings belong to a fixed NP language. Modular queries provide a form of restricted access to an NP oracle. For each k and m, we consider the class of languages accepted by NP machines that ask a single modular query. Han and Thierauf [HT95] showed that these classes coincide with levels of the Boolean hierarchy when m is even or k≤2m, and they determined the exact levels. Until now, the remaining case-odd m and large k - looked quite difficult. We pinpoint the level in the Boolean hierarchy for the remaining case; thus, these classes coincide with levels of the Boolean hierarchy for every k and m. In addition we characterize the classes obtained by using an NP(l) acceptor in place of an NP acceptor (NP(l) is the lth level of the Boolean hierarchy). As before, these all coincide with levels in the Boolean hierarchy
Algebraic and Combinatorial Methods in Computational Complexity
At its core, much of Computational Complexity is concerned with combinatorial objects and structures. But it has often proven true that the best way to prove things about these combinatorial objects is by establishing a connection (perhaps approximate) to a more well-behaved algebraic setting. Indeed, many of the deepest and most powerful results in Computational Complexity rely on algebraic proof techniques. The PCP characterization of NP and the Agrawal-Kayal-Saxena polynomial-time primality test are two prominent examples. Recently, there have been some works going in the opposite direction, giving alternative combinatorial proofs for results that were originally proved algebraically. These alternative proofs can yield important improvements because they are closer to the underlying problems and avoid the losses in passing to the algebraic setting. A prominent example is Dinur's proof of the PCP Theorem via gap amplification which yielded short PCPs with only a polylogarithmic length blowup (which had been the focus of significant research effort up to that point). We see here (and in a number of recent works) an exciting interplay between algebraic and combinatorial techniques. This seminar aims to capitalize on recent progress and bring together researchers who are using a diverse array of algebraic and combinatorial methods in a variety of settings
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