4 research outputs found
A full complexity dichotomy for immanant families
Given an integer and an irreducible character of
for some partition of , the immanant
maps matrices
to . Important special cases
include the determinant and permanent, which are the immanants associated with
the sign and trivial character, respectively.
It is known that immanants can be evaluated in polynomial time for characters
that are close to the sign character: Given a partition of with
parts, let count the boxes to the right of the first
column in the Young diagram of . For a family of partitions ,
let and write Imm
for the problem of evaluating on input and
. If , then Imm is known to be
polynomial-time computable. This subsumes the case of the determinant. On the
other hand, if , then previously known hardness results
suggest that Imm cannot be solved in polynomial time. However, these
results only address certain restricted classes of families .
In this paper, we show that the parameterized complexity assumption FPT
#W[1] rules out polynomial-time algorithms for Imm for any
computationally reasonable family of partitions with
. We give an analogous result in algebraic complexity under
the assumption VFPT VW[1]. Furthermore, if even grows
polynomially in , we show that Imm is hard for #P and VNP.
This concludes a series of partial results on the complexity of immanants
obtained over the last 35 years.Comment: 28 pages, to appear at STOC'2
Modular Counting of Subgraphs: Matchings, Matching-Splittable Graphs, and Paths
We systematically investigate the complexity of counting subgraph patterns
modulo fixed integers. For example, it is known that the parity of the number
of -matchings can be determined in polynomial time by a simple reduction to
the determinant. We generalize this to an -time algorithm to
compute modulo the number of subgraph occurrences of patterns that are
vertices away from being matchings. This shows that the known
polynomial-time cases of subgraph detection (Jansen and Marx, SODA 2015) carry
over into the setting of counting modulo .
Complementing our algorithm, we also give a simple and self-contained proof
that counting -matchings modulo odd integers is Mod_q-W[1]-complete and
prove that counting -paths modulo is Parity-W[1]-complete, answering an
open question by Bj\"orklund, Dell, and Husfeldt (ICALP 2015).Comment: 23 pages, to appear at ESA 202
Parameterized Streaming Algorithms for Min-Ones d-SAT
In this work, we initiate the study of the Min-Ones d-SAT problem in the parameterized streaming model. An instance of the problem consists of a d-CNF formula F and an integer k, and the objective is to determine if F has a satisfying assignment which sets at most k variables to 1. In the parameterized streaming model, input is provided as a stream, just as in the usual streaming model. A key difference is that the bound on the read-write memory available to the algorithm is O(f(k) log n) (f: N -> N, a computable function) as opposed to the O(log n) bound of the usual streaming model. The other important difference is that the number of passes the algorithm makes over its input must be a (preferably small) function of k.
We design a (k + 1)-pass parameterized streaming algorithm that solves Min-Ones d-SAT (d >= 2) using space O((kd^(ck) + k^d)log n) (c > 0, a constant) and a (d + 1)^k-pass algorithm that uses space O(k log n). We also design a streaming kernelization for Min-Ones 2-SAT that makes (k + 2) passes and uses space O(k^6 log n) to produce a kernel with O(k^6) clauses.
To complement these positive results, we show that any k-pass algorithm for or Min-Ones d-SAT (d >= 2) requires space Omega(max{n^(1/k) / 2^k, log(n / k)}) on instances (F, k). This is achieved via a reduction from the streaming problem POT Pointer Chasing (Guha and McGregor [ICALP 2008]), which might be of independent interest. Given this, our (k + 1)-pass parameterized streaming algorithm is the best possible, inasmuch as the number of passes is concerned.
In contrast to the results of Fafianie and Kratsch [MFCS 2014] and Chitnis et al. [SODA 2015], who independently showed that there are 1-pass parameterized streaming algorithms for Vertex Cover (a restriction of Min-Ones 2-SAT), we show using lower bounds from Communication Complexity that for any d >= 1, a 1-pass streaming algorithm for Min-Ones d-SAT requires space Omega(n). This excludes the possibility of a 1-pass parameterized streaming algorithm for the problem. Additionally, we show that any p-pass algorithm for the problem requires space Omega(n/p)