2,835 research outputs found
Lower Bounds for Monotone Counting Circuits
A {+,x}-circuit counts a given multivariate polynomial f, if its values on
0-1 inputs are the same as those of f; on other inputs the circuit may output
arbitrary values. Such a circuit counts the number of monomials of f evaluated
to 1 by a given 0-1 input vector (with multiplicities given by their
coefficients). A circuit decides if it has the same 0-1 roots as f. We
first show that some multilinear polynomials can be exponentially easier to
count than to compute them, and can be exponentially easier to decide than to
count them. Then we give general lower bounds on the size of counting circuits.Comment: 20 page
Fast and Deterministic Constant Factor Approximation Algorithms for LCS Imply New Circuit Lower Bounds
The Longest Common Subsequence (LCS) is one of the most basic similarity measures and it captures important applications in bioinformatics and text analysis. Following the SETH-based nearly-quadratic time lower bounds for LCS from recent years, it is a major open problem to understand the complexity of approximate LCS.
In the last ITCS [AB17] drew an interesting connection between this problem and the area of circuit complexity:
they proved that approximation algorithms for LCS in deterministic truly-subquadratic time imply new circuit lower bounds (E^NP does not have non-uniform linear-size Valiant Series Parallel circuits).
In this work, we strengthen this connection between approximate LCS and circuit complexity by applying the Distributed PCP framework of [ARW17].
We obtain a reduction that holds against much larger approximation factors (super-constant versus 1+o(1)), yields a lower bound for a larger class of circuits (linear-size NC^1), and is also easier to analyze
Lower bounds on dynamic programming for maximum weight independent set
Publisher Copyright: © 2021 Tuukka Korhonen.We prove lower bounds on pure dynamic programming algorithms for maximum weight independent set (MWIS). We model such algorithms as tropical circuits, i.e., circuits that compute with max and + operations. For a graph G, an MWIS-circuit of G is a tropical circuit whose inputs correspond to vertices of G and which computes the weight of a maximum weight independent set of G for any assignment of weights to the inputs. We show that if G has treewidth w and maximum degree d, then any MWIS-circuit of G has 2Ω(w/d) gates and that if G is planar, or more generally H-minor-free for any fixed graph H, then any MWIS-circuit of G has 2Ω(w) gates. An MWIS-formula is an MWIScircuit where each gate has fan-out at most one. We show that if G has treedepth t and maximum degree d, then any MWIS-formula of G has 2Ω(t/d) gates. It follows that treewidth characterizes optimal MWIS-circuits up to polynomials for all bounded degree graphs and H-minor-free graphs, and treedepth characterizes optimal MWIS-formulas up to polynomials for all bounded degree graphs.Peer reviewe
Computing the Maximum using (min, +) Formulas
We study computation by formulas over (min,+). We consider the
computation of max{x_1,...,x_n} over N as a difference of
(min,+) formulas, and show that size n + n log n is sufficient
and necessary. Our proof also shows that any (min,+) formula
computing the minimum of all sums of n-1 out of n variables must
have n log n leaves; this too is tight. Our proofs use a
complexity measure for (min,+) functions based on minterm-like
behaviour and on the entropy of an associated graph
Fine-grained Complexity Meets IP = PSPACE
In this paper we study the fine-grained complexity of finding exact and
approximate solutions to problems in P. Our main contribution is showing
reductions from exact to approximate solution for a host of such problems.
As one (notable) example, we show that the Closest-LCS-Pair problem (Given
two sets of strings and , compute exactly the maximum with ) is equivalent to its approximation version
(under near-linear time reductions, and with a constant approximation factor).
More generally, we identify a class of problems, which we call BP-Pair-Class,
comprising both exact and approximate solutions, and show that they are all
equivalent under near-linear time reductions.
Exploring this class and its properties, we also show:
Under the NC-SETH assumption (a significantly more relaxed
assumption than SETH), solving any of the problems in this class requires
essentially quadratic time.
Modest improvements on the running time of known algorithms
(shaving log factors) would imply that NEXP is not in non-uniform
.
Finally, we leverage our techniques to show new barriers for
deterministic approximation algorithms for LCS.
At the heart of these new results is a deep connection between interactive
proof systems for bounded-space computations and the fine-grained complexity of
exact and approximate solutions to problems in P. In particular, our results
build on the proof techniques from the classical IP = PSPACE result
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