1,754 research outputs found
Why is it hard to beat for Longest Common Weakly Increasing Subsequence?
The Longest Common Weakly Increasing Subsequence problem (LCWIS) is a variant
of the classic Longest Common Subsequence problem (LCS). Both problems can be
solved with simple quadratic time algorithms. A recent line of research led to
a number of matching conditional lower bounds for LCS and other related
problems. However, the status of LCWIS remained open.
In this paper we show that LCWIS cannot be solved in strongly subquadratic
time unless the Strong Exponential Time Hypothesis (SETH) is false.
The ideas which we developed can also be used to obtain a lower bound based
on a safer assumption of NC-SETH, i.e. a version of SETH which talks about NC
circuits instead of less expressive CNF formulas
Combined super-/substring and super-/subsequence problems
Super-/substring problems and super-/subsequence problems are well-known problems in stringology that have applications in a variety of areas, such as manufacturing systems design and molecular biology. Here we investigate the complexity of a new type of such problem that forms a combination of a super-/substring and a super-/subsequence problem. Moreover we introduce different types of minimal superstring and maximal substring problems. In particular, we consider the following problems: given a set L of strings and a string S, (i) find a minimal superstring (or maximal substring) of L that is also a supersequence (or a subsequence) of S, (ii) find a minimal supersequence (or maximal subsequence) of L that is also a superstring (or a substring) of S. In addition some non-super-/non-substring and non-super-/non-subsequence variants are studied. We obtain several NP-hardness or even MAX SNP-hardness results and also identify types of "weak minimal" superstrings and "weak maximal" substrings for which (i) is polynomial-time solvable
Tree Contractions and Evolutionary Trees
An evolutionary tree is a rooted tree where each internal vertex has at least
two children and where the leaves are labeled with distinct symbols
representing species. Evolutionary trees are useful for modeling the
evolutionary history of species. An agreement subtree of two evolutionary trees
is an evolutionary tree which is also a topological subtree of the two given
trees. We give an algorithm to determine the largest possible number of leaves
in any agreement subtree of two trees T_1 and T_2 with n leaves each. If the
maximum degree d of these trees is bounded by a constant, the time complexity
is O(n log^2(n)) and is within a log(n) factor of optimal. For general d, this
algorithm runs in O(n d^2 log(d) log^2(n)) time or alternatively in O(n d
sqrt(d) log^3(n)) time
Distributed PCP Theorems for Hardness of Approximation in P
We present a new distributed model of probabilistically checkable proofs
(PCP). A satisfying assignment to a CNF formula is
shared between two parties, where Alice knows , Bob knows
, and both parties know . The goal is to have
Alice and Bob jointly write a PCP that satisfies , while
exchanging little or no information. Unfortunately, this model as-is does not
allow for nontrivial query complexity. Instead, we focus on a non-deterministic
variant, where the players are helped by Merlin, a third party who knows all of
.
Using our framework, we obtain, for the first time, PCP-like reductions from
the Strong Exponential Time Hypothesis (SETH) to approximation problems in P.
In particular, under SETH we show that there are no truly-subquadratic
approximation algorithms for Bichromatic Maximum Inner Product over
{0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate
Regular Expression Matching, and Diameter in Product Metric. All our
inapproximability factors are nearly-tight. In particular, for the first two
problems we obtain nearly-polynomial factors of ; only
-factor lower bounds (under SETH) were known before
Completeness Results for Parameterized Space Classes
The parameterized complexity of a problem is considered "settled" once it has
been shown to lie in FPT or to be complete for a class in the W-hierarchy or a
similar parameterized hierarchy. Several natural parameterized problems have,
however, resisted such a classification. At least in some cases, the reason is
that upper and lower bounds for their parameterized space complexity have
recently been obtained that rule out completeness results for parameterized
time classes. In this paper, we make progress in this direction by proving that
the associative generability problem and the longest common subsequence problem
are complete for parameterized space classes. These classes are defined in
terms of different forms of bounded nondeterminism and in terms of simultaneous
time--space bounds. As a technical tool we introduce a "union operation" that
translates between problems complete for classical complexity classes and for
W-classes.Comment: IPEC 201
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