185,590 research outputs found
On the Complexity of #CSP^d
Counting CSP^d is the counting constraint satisfaction problem (#CSP in short) restricted to the instances where every variable occurs a multiple of d times. This paper revisits tractable structures in #CSP and gives a complexity classification theorem for #CSP^d with algebraic complex weights. The result unifies affine functions (stabilizer states in quantum information theory) and related variants such as the local affine functions, the discovery of which leads to all the recent progress on the complexity of Holant problems.
The Holant is a framework that generalizes counting CSP. In the literature on Holant problems, weighted constraints are often expressed as tensors (vectors) such that projections and linear transformations help analyze the structure. This paper gives an example showing that different classes of tensors distinguished by these algebraic operations may share the same closure property under tensor product and contraction
Towards a complexity theory for the congested clique
The congested clique model of distributed computing has been receiving
attention as a model for densely connected distributed systems. While there has
been significant progress on the side of upper bounds, we have very little in
terms of lower bounds for the congested clique; indeed, it is now know that
proving explicit congested clique lower bounds is as difficult as proving
circuit lower bounds.
In this work, we use various more traditional complexity-theoretic tools to
build a clearer picture of the complexity landscape of the congested clique:
-- Nondeterminism and beyond: We introduce the nondeterministic congested
clique model (analogous to NP) and show that there is a natural canonical
problem family that captures all problems solvable in constant time with
nondeterministic algorithms. We further generalise these notions by introducing
the constant-round decision hierarchy (analogous to the polynomial hierarchy).
-- Non-constructive lower bounds: We lift the prior non-uniform counting
arguments to a general technique for proving non-constructive uniform lower
bounds for the congested clique. In particular, we prove a time hierarchy
theorem for the congested clique, showing that there are decision problems of
essentially all complexities, both in the deterministic and nondeterministic
settings.
-- Fine-grained complexity: We map out relationships between various natural
problems in the congested clique model, arguing that a reduction-based
complexity theory currently gives us a fairly good picture of the complexity
landscape of the congested clique
String Synchronizing Sets: Sublinear-Time BWT Construction and Optimal LCE Data Structure
Burrows-Wheeler transform (BWT) is an invertible text transformation that,
given a text of length , permutes its symbols according to the
lexicographic order of suffixes of . BWT is one of the most heavily studied
algorithms in data compression with numerous applications in indexing, sequence
analysis, and bioinformatics. Its construction is a bottleneck in many
scenarios, and settling the complexity of this task is one of the most
important unsolved problems in sequence analysis that has remained open for 25
years. Given a binary string of length , occupying machine
words, the BWT construction algorithm due to Hon et al. (SIAM J. Comput., 2009)
runs in time and space. Recent advancements (Belazzougui,
STOC 2014, and Munro et al., SODA 2017) focus on removing the alphabet-size
dependency in the time complexity, but they still require time.
In this paper, we propose the first algorithm that breaks the -time
barrier for BWT construction. Given a binary string of length , our
procedure builds the Burrows-Wheeler transform in time and
space. We complement this result with a conditional lower bound
proving that any further progress in the time complexity of BWT construction
would yield faster algorithms for the very well studied problem of counting
inversions: it would improve the state-of-the-art -time
solution by Chan and P\v{a}tra\c{s}cu (SODA 2010). Our algorithm is based on a
novel concept of string synchronizing sets, which is of independent interest.
As one of the applications, we show that this technique lets us design a data
structure of the optimal size that answers Longest Common
Extension queries (LCE queries) in time and, furthermore, can be
deterministically constructed in the optimal time.Comment: Full version of a paper accepted to STOC 201
A closer look at the sanitation ladder: issues of monitoring the sector
The WHO/UNICEF Joint Monitoring Programme (JMP) reports progress in sanitation by looking at the use of a set of pre-defined sanitation technologies. A technology-based approach offers several advantages, but it does not effectively deal with such problems as hygienic conditions of infrastructure and equity. In this study the monitoring strategy employed by the JMP is reviewed with reference to experience in Ethiopia and Tanzania. Although findings may not accurately represent the complexity of the sanitation status worldwide, a comparison of the two cases shows that many of the issues to effective monitoring are generic. On the basis of this analysis, two recommendations are identified for further study: (i) to extend list of criteria when assessing improved sanitation by considering aspects related with hygienic condition of the latrine; and (ii) to review and validate those aspects that prevent shared latrines from counting towards improved sanitation
The parameterised complexity of counting even and odd induced subgraphs
We consider the problem of counting, in a given graph, the number of induced k-vertex subgraphs which have an even number of edges, and also the complementary problem of counting the k-vertex induced subgraphs having an odd number of edges. We demonstrate that both problems are #W[1]-hard when parameterised by k, in fact proving a somewhat stronger result about counting subgraphs with a property that only holds for some subset of k-vertex subgraphs which have an even (respectively odd) number of edges. On the other hand, we show that each of the problems admits an FPTRAS. These approximation schemes are based on a surprising structural result, which exploits ideas from Ramsey theory
Obstructions to combinatorial formulas for plethysm
Motivated by questions of Mulmuley and Stanley we investigate
quasi-polynomials arising in formulas for plethysm. We demonstrate, on the
examples of and , that these need not be counting
functions of inhomogeneous polytopes of dimension equal to the degree of the
quasi-polynomial. It follows that these functions are not, in general, counting
functions of lattice points in any scaled convex bodies, even when restricted
to single rays. Our results also apply to special rectangular Kronecker
coefficients.Comment: 7 pages; v2: Improved version with further reaching counterexamples;
v3: final version as in Electronic Journal of Combinatoric
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