2 research outputs found
Minimum Common String Partition: Exact Algorithms
In the minimum common string partition problem (MCSP), one gets two strings and is asked to find the minimum number of cuts in the first string such that the second string can be obtained by rearranging the resulting pieces. It is a difficult algorithmic problem having applications in computational biology, text processing, and data compression. MCSP has been studied extensively from various algorithmic angles: there are many papers studying approximation, heuristic, and parameterized algorithms. At the same time, almost nothing is known about its exact complexity. In this paper, we present new results in this direction. We improve the known 2? upper bound (where n is the length of input strings) to ?? where ? ? 1.618... is the golden ratio. The algorithm uses Fibonacci numbers to encode subsets as monomials of a certain implicit polynomial and extracts one of its coefficients using the fast Fourier transform. Then, we show that the case of constant size alphabet can be solved in subexponential time 2^{O(nlog log n/log n)} by a hybrid strategy: enumerate all long pieces and use dynamic programming over histograms of short pieces. Finally, we prove almost matching lower bounds assuming the Exponential Time Hypothesis
Computations with polynomial evaluation oracle: ruling out superlinear SETH-based lower bounds
The field of fine-grained complexity aims at proving conditional lower bounds
on the time complexity of computational problems. One of the most popular
assumptions, Strong Exponential Time Hypothesis (SETH), implies that SAT cannot
be solved in time. In recent years, it has been proved that
known algorithms for many problems are optimal under SETH. Despite the wide
applicability of SETH, for many problems, there are no known SETH-based lower
bounds, so the quest for new reductions continues.
Two barriers for proving SETH-based lower bounds are known. Carmosino et al.
(ITCS 2016) introduced the Nondeterministic Strong Exponential Time Hypothesis
(NSETH) stating that TAUT cannot be solved in time even if
one allows nondeterminism. They used this hypothesis to show that some natural
fine-grained reductions would be difficult to obtain: proving that, say, 3-SUM
requires time under SETH, breaks NSETH and this, in turn,
implies strong circuit lower bounds. Recently, Belova et al. (SODA 2023)
introduced the so-called polynomial formulations to show that for many NP-hard
problems, proving any explicit exponential lower bound under SETH also implies
strong circuit lower bounds.
We prove that for a range of problems from P, including -SUM and triangle
detection, proving superlinear lower bounds under SETH is challenging as it
implies new circuit lower bounds. To this end, we show that these problems can
be solved in nearly linear time with oracle calls to evaluating a polynomial of
constant degree. Then, we introduce a strengthening of SETH stating that
solving SAT in time is difficult even if one has
constant degree polynomial evaluation oracle calls. This hypothesis is stronger
and less believable than SETH, but refuting it is still challenging: we show
that this implies circuit lower bounds