42 research outputs found
Hardness of Detecting Abelian and Additive Square Factors in Strings
We prove 3SUM-hardness (no strongly subquadratic-time algorithm, assuming the
3SUM conjecture) of several problems related to finding Abelian square and
additive square factors in a string. In particular, we conclude conditional
optimality of the state-of-the-art algorithms for finding such factors.
Overall, we show 3SUM-hardness of (a) detecting an Abelian square factor of
an odd half-length, (b) computing centers of all Abelian square factors, (c)
detecting an additive square factor in a length- string of integers of
magnitude , and (d) a problem of computing a double 3-term
arithmetic progression (i.e., finding indices such that
) in a sequence of integers of
magnitude .
Problem (d) is essentially a convolution version of the AVERAGE problem that
was proposed in a manuscript of Erickson. We obtain a conditional lower bound
for it with the aid of techniques recently developed by Dudek et al. [STOC
2020]. Problem (d) immediately reduces to problem (c) and is a step in
reductions to problems (a) and (b). In conditional lower bounds for problems
(a) and (b) we apply an encoding of Amir et al. [ICALP 2014] and extend it
using several string gadgets that include arbitrarily long Abelian-square-free
strings.
Our reductions also imply conditional lower bounds for detecting Abelian
squares in strings over a constant-sized alphabet. We also show a subquadratic
upper bound in this case, applying a result of Chan and Lewenstein [STOC 2015].Comment: Accepted to ESA 202
Translating Hausdorff is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation
Computing the similarity of two point sets is a ubiquitous task in medical
imaging, geometric shape comparison, trajectory analysis, and many more
settings. Arguably the most basic distance measure for this task is the
Hausdorff distance, which assigns to each point from one set the closest point
in the other set and then evaluates the maximum distance of any assigned pair.
A drawback is that this distance measure is not translational invariant, that
is, comparing two objects just according to their shape while disregarding
their position in space is impossible.
Fortunately, there is a canonical translational invariant version, the
Hausdorff distance under translation, which minimizes the Hausdorff distance
over all translations of one of the point sets. For point sets of size and
, the Hausdorff distance under translation can be computed in time for the and norm [Chew, Kedem SWAT'92] and for the norm [Huttenlocher, Kedem, Sharir DCG'93].
As these bounds have not been improved for over 25 years, in this paper we
approach the Hausdorff distance under translation from the perspective of
fine-grained complexity theory. We show (i) a matching lower bound of
for and (and all other norms) assuming
the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of
for in the imbalanced case of assuming the 3SUM
Hypothesis
Multiplicative Auction Algorithm for Approximate Maximum Weight Bipartite Matching
We present an auction algorithm using
multiplicative instead of constant weight updates to compute a
(1-\eps)-approximate maximum weight matching (MWM) in a bipartite graph with
vertices and edges in time O(m\eps^{-1}\log(\eps^{-1})), matching the
running time of the linear-time approximation algorithm of Duan and Pettie
[JACM '14]. Our algorithm is very simple and it can be extended to give a
dynamic data structure that maintains a (1-\eps)-approximate maximum weight
matching under (1) edge deletions in amortized O(\eps^{-1}\log(\eps^{-1}))
time and (2) one-sided vertex insertions. If all edges incident to an inserted
vertex are given in sorted weight the amortized time is
O(\eps^{-1}\log(\eps^{-1})) per inserted edge. If the inserted incident edges
are not sorted, the amortized time per inserted edge increases by an additive
term of . The fastest prior dynamic (1-\eps)-approximate algorithm
in weighted graphs took time O(\sqrt{m}\eps^{-1}\log (w_{max})) per updated
edge, where the edge weights lie in the range .Comment: To appear in IPCO 202
Translating Hausdorff Is Hard: Fine-Grained Lower Bounds for Hausdorff Distance Under Translation
Computing the similarity of two point sets is a ubiquitous task in medical imaging, geometric shape comparison, trajectory analysis, and many more settings. Arguably the most basic distance measure for this task is the Hausdorff distance, which assigns to each point from one set the closest point in the other set and then evaluates the maximum distance of any assigned pair. A drawback is that this distance measure is not translational invariant, that is, comparing two objects just according to their shape while disregarding their position in space is impossible.
Fortunately, there is a canonical translational invariant version, the Hausdorff distance under translation, which minimizes the Hausdorff distance over all translations of one of the point sets. For point sets of size n and m, the Hausdorff distance under translation can be computed in time ??(nm) for the L? and L_? norm [Chew, Kedem SWAT\u2792] and ??(nm (n+m)) for the L? norm [Huttenlocher, Kedem, Sharir DCG\u2793].
As these bounds have not been improved for over 25 years, in this paper we approach the Hausdorff distance under translation from the perspective of fine-grained complexity theory. We show (i) a matching lower bound of (nm)^{1-o(1)} for L? and L_? assuming the Orthogonal Vectors Hypothesis and (ii) a matching lower bound of n^{2-o(1)} for L? in the imbalanced case of m = ?(1) assuming the 3SUM Hypothesis
Parameterized Matroid-Constrained Maximum Coverage
In this paper, we introduce the concept of Density-Balanced Subset in a matroid, in which independent sets can be sampled so as to guarantee that (i) each element has the same probability to be sampled, and (ii) those events are negatively correlated. These Density-Balanced Subsets are subsets in the ground set of a matroid in which the traditional notion of uniform random sampling can be extended.
We then provide an application of this concept to the Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid ? = (V, ?) of rank k on a ground set V and a coverage function f on V, the goal is to find an independent set S ? ? maximizing f(S). This problem is an important special case of the much-studied submodular function maximization problem subject to a matroid constraint; this is also a generalization of the maximum k-cover problem in a graph. In this paper, assuming that the coverage function has a bounded frequency ? (i.e., any element of the underlying universe of the coverage function appears in at most ? sets), we design a procedure, parameterized by some integer ?, to extract in polynomial time an approximate kernel of size ? ? k that is guaranteed to contain a 1 - (? - 1)/? approximation of the optimal solution. This procedure can then be used to get a Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a 1 - ? approximation in time (?/?)^O(k) ? |V|^O(1). This generalizes and improves the results of [Manurangsi, 2019] and [Huang and Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid. Moreover, as the kernel has a very simple characterization, it can be constructed in the streaming setting
Parameterized Matroid-Constrained Maximum Coverage
In this paper, we introduce the concept of Density-Balanced Subset in a
matroid, in which independent sets can be sampled so as to guarantee that (i)
each element has the same probability to be sampled, and (ii) those events are
negatively correlated. These Density-Balanced Subsets are subsets in the ground
set of a matroid in which the traditional notion of uniform random sampling can
be extended. We then provide an application of this concept to the
Matroid-Constrained Maximum Coverage problem. In this problem, given a matroid
of rank on a ground set and a coverage
function on , the goal is to find an independent set
maximizing . This problem is an important special case of the
much-studied submodular function maximization problem subject to a matroid
constraint; this is also a generalization of the maximum -cover problem in a
graph. In this paper, assuming that the coverage function has a bounded
frequency (i.e., any element of the underlying universe of the coverage
function appears in at most sets), we design a procedure, parameterized
by some integer , to extract in polynomial time an approximate kernel of
size that is guaranteed to contain a
approximation of the optimal solution. This procedure can then be used to get a
Fixed-Parameter Tractable Approximation Scheme (FPT-AS) providing a approximation in time .
This generalizes and improves the results of [Manurangsi, 2019] and [Huang and
Sellier, 2022], providing the first FPT-AS working on an arbitrary matroid.
Moreover, because of its simplicity, the kernel construction can be performed
in the streaming setting
Stronger 3-SUM Lower Bounds for Approximate Distance Oracles via Additive Combinatorics
The "short cycle removal" technique was recently introduced by Abboud,
Bringmann, Khoury and Zamir (STOC '22) to prove fine-grained hardness of
approximation. Its main technical result is that listing all triangles in an
-regular graph is -hard under the 3-SUM conjecture even
when the number of short cycles is small; namely, when the number of -cycles
is for .
Abboud et al. achieve by applying structure vs. randomness
arguments on graphs. In this paper, we take a step back and apply conceptually
similar arguments on the numbers of the 3-SUM problem. Consequently, we achieve
the best possible and the following lower bounds under the 3-SUM
conjecture:
* Approximate distance oracles: The seminal Thorup-Zwick distance oracles
achieve stretch after preprocessing a graph in
time. For the same stretch, and assuming the query time is Abboud et
al. proved an lower bound on the
preprocessing time; we improve it to which is only a
factor 2 away from the upper bound. We also obtain tight bounds for stretch
and and higher lower bounds for dynamic shortest paths.
* Listing 4-cycles: Abboud et al. proved the first super-linear lower bound
for listing all 4-cycles in a graph, ruling out time
algorithms where is the number of 4-cycles. We settle the complexity of
this basic problem by showing that the
upper bound is tight up to factors.
Our results exploit a rich tool set from additive combinatorics, most notably
the Balog-Szemer\'edi-Gowers theorem and Rusza's covering lemma. A key
ingredient that may be of independent interest is a subquadratic algorithm for
3-SUM if one of the sets has small doubling.Comment: Abstract shortened to fit arXiv requirement
Deterministic Fully Dynamic SSSP and More
We present the first non-trivial fully dynamic algorithm maintaining exact
single-source distances in unweighted graphs. This resolves an open problem
stated by Sankowski [COCOON 2005] and van den Brand and Nanongkai [FOCS 2019].
Previous fully dynamic single-source distances data structures were all
approximate, but so far, non-trivial dynamic algorithms for the exact setting
could only be ruled out for polynomially weighted graphs (Abboud and
Vassilevska Williams, [FOCS 2014]). The exact unweighted case remained the main
case for which neither a subquadratic dynamic algorithm nor a quadratic lower
bound was known.
Our dynamic algorithm works on directed graphs, is deterministic, and can
report a single-source shortest paths tree in subquadratic time as well. Thus
we also obtain the first deterministic fully dynamic data structure for
reachability (transitive closure) with subquadratic update and query time. This
answers an open problem of van den Brand, Nanongkai, and Saranurak [FOCS 2019].
Finally, using the same framework we obtain the first fully dynamic data
structure maintaining all-pairs -approximate distances within
non-trivial sub- worst-case update time while supporting optimal-time
approximate shortest path reporting at the same time. This data structure is
also deterministic and therefore implies the first known non-trivial
deterministic worst-case bound for recomputing the transitive closure of a
digraph.Comment: Extended abstract to appear in FOCS 202