31,644 research outputs found
Threesomes, Degenerates, and Love Triangles
The 3SUM problem is to decide, given a set of real numbers, whether any
three sum to zero. It is widely conjectured that a trivial -time
algorithm is optimal and over the years the consequences of this conjecture
have been revealed. This 3SUM conjecture implies lower bounds on
numerous problems in computational geometry and a variant of the conjecture
implies strong lower bounds on triangle enumeration, dynamic graph algorithms,
and string matching data structures.
In this paper we refute the 3SUM conjecture. We prove that the decision tree
complexity of 3SUM is and give two subquadratic 3SUM
algorithms, a deterministic one running in
time and a randomized one running in time with
high probability. Our results lead directly to improved bounds for -variate
linear degeneracy testing for all odd . The problem is to decide, given
a linear function and a set , whether . We show the
decision tree complexity of this problem is .
Finally, we give a subcubic algorithm for a generalization of the
-product over real-valued matrices and apply it to the problem of
finding zero-weight triangles in weighted graphs. We give a
depth- decision tree for this problem, as well as an
algorithm running in time
Quantum query complexity of minor-closed graph properties
We study the quantum query complexity of minor-closed graph properties, which
include such problems as determining whether an -vertex graph is planar, is
a forest, or does not contain a path of a given length. We show that most
minor-closed properties---those that cannot be characterized by a finite set of
forbidden subgraphs---have quantum query complexity \Theta(n^{3/2}). To
establish this, we prove an adversary lower bound using a detailed analysis of
the structure of minor-closed properties with respect to forbidden topological
minors and forbidden subgraphs. On the other hand, we show that minor-closed
properties (and more generally, sparse graph properties) that can be
characterized by finitely many forbidden subgraphs can be solved strictly
faster, in o(n^{3/2}) queries. Our algorithms are a novel application of the
quantum walk search framework and give improved upper bounds for several
subgraph-finding problems.Comment: v1: 25 pages, 2 figures. v2: 26 page
Search-to-Decision Reductions for Lattice Problems with Approximation Factors (Slightly) Greater Than One
We show the first dimension-preserving search-to-decision reductions for
approximate SVP and CVP. In particular, for any ,
we obtain an efficient dimension-preserving reduction from -SVP to -GapSVP and an efficient dimension-preserving reduction
from -CVP to -GapCVP. These results generalize the known
equivalences of the search and decision versions of these problems in the exact
case when . For SVP, we actually obtain something slightly stronger
than a search-to-decision reduction---we reduce -SVP to
-unique SVP, a potentially easier problem than -GapSVP.Comment: Updated to acknowledge additional prior wor
Sculpting Quantum Speedups
Given a problem which is intractable for both quantum and classical
algorithms, can we find a sub-problem for which quantum algorithms provide an
exponential advantage? We refer to this problem as the "sculpting problem." In
this work, we give a full characterization of sculptable functions in the query
complexity setting. We show that a total function f can be restricted to a
promise P such that Q(f|_P)=O(polylog(N)) and R(f|_P)=N^{Omega(1)}, if and only
if f has a large number of inputs with large certificate complexity. The proof
uses some interesting techniques: for one direction, we introduce new
relationships between randomized and quantum query complexity in various
settings, and for the other direction, we use a recent result from
communication complexity due to Klartag and Regev. We also characterize
sculpting for other query complexity measures, such as R(f) vs. R_0(f) and
R_0(f) vs. D(f).
Along the way, we prove some new relationships for quantum query complexity:
for example, a nearly quadratic relationship between Q(f) and D(f) whenever the
promise of f is small. This contrasts with the recent super-quadratic query
complexity separations, showing that the maximum gap between classical and
quantum query complexities is indeed quadratic in various settings - just not
for total functions!
Lastly, we investigate sculpting in the Turing machine model. We show that if
there is any BPP-bi-immune language in BQP, then every language outside BPP can
be restricted to a promise which places it in PromiseBQP but not in PromiseBPP.
Under a weaker assumption, that some problem in BQP is hard on average for
P/poly, we show that every paddable language outside BPP is sculptable in this
way.Comment: 30 page
On the Closest Vector Problem with a Distance Guarantee
We present a substantially more efficient variant, both in terms of running
time and size of preprocessing advice, of the algorithm by Liu, Lyubashevsky,
and Micciancio for solving CVPP (the preprocessing version of the Closest
Vector Problem, CVP) with a distance guarantee. For instance, for any , our algorithm finds the (unique) closest lattice point for any target
point whose distance from the lattice is at most times the length of
the shortest nonzero lattice vector, requires as preprocessing advice only vectors, and runs in
time .
As our second main contribution, we present reductions showing that it
suffices to solve CVP, both in its plain and preprocessing versions, when the
input target point is within some bounded distance of the lattice. The
reductions are based on ideas due to Kannan and a recent sparsification
technique due to Dadush and Kun. Combining our reductions with the LLM
algorithm gives an approximation factor of for search
CVPP, improving on the previous best of due to Lagarias, Lenstra,
and Schnorr. When combined with our improved algorithm we obtain, somewhat
surprisingly, that only O(n) vectors of preprocessing advice are sufficient to
solve CVPP with (the only slightly worse) approximation factor of O(n).Comment: An early version of the paper was titled "On Bounded Distance
Decoding and the Closest Vector Problem with Preprocessing". Conference on
Computational Complexity (2014
- …