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Algorithmica ◽  
2021 ◽  
Author(s):  
Julian Dörfler ◽  
Marc Roth ◽  
Johannes Schmitt ◽  
Philip Wellnitz

AbstractWe study the problem $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) of counting all induced subgraphs of size k in a graph G that satisfy the property $$\varPhi $$ Φ . It is shown that, given any graph property $$\varPhi $$ Φ that distinguishes independent sets from bicliques, $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) is hard for the class $$\#\mathsf {W[1]}$$ # W [ 1 ] , i.e., the parameterized counting equivalent of $${{\mathsf {N}}}{{\mathsf {P}}}$$ N P . Under additional suitable density conditions on $$\varPhi $$ Φ , satisfied e.g. by non-trivial monotone properties on bipartite graphs, we strengthen $$\#\mathsf {W[1]}$$ # W [ 1 ] -hardness by establishing that $$\#\textsc {IndSub}(\varPhi )$$ # I N D S U B ( Φ ) cannot be solved in time $$f(k)\cdot n^{o(k)}$$ f ( k ) · n o ( k ) for any computable function f, unless the Exponential Time Hypothesis fails. Finally, we observe that our results remain true even if the input graph G is restricted to be bipartite and counting is done modulo a fixed prime.


Author(s):  
Nikolay Bazhenov ◽  
Manat Mustafa

In computability theory, the standard tool to classify preorders is provided by the computable reducibility. If [Formula: see text] and [Formula: see text] are preorders with domain [Formula: see text], then [Formula: see text] is computably reducible to [Formula: see text] if and only if there is a computable function [Formula: see text] such that for all [Formula: see text] and [Formula: see text], [Formula: see text] [Formula: see text][Formula: see text]. We study the complexity of preorders which arise in a natural way in computable structure theory. We prove that the relation of computable isomorphic embeddability among computable torsion abelian groups is a [Formula: see text] complete preorder. A similar result is obtained for computable distributive lattices. We show that the relation of primitive recursive embeddability among punctual structures (in the setting of Kalimullin et al.) is a [Formula: see text] complete preorder.


Author(s):  
Johannes Blum

AbstractWe study the k-Center problem, where the input is a graph $$G=(V,E)$$ G = ( V , E ) with positive edge weights and an integer k, and the goal is to select k center vertices $$C \subseteq V$$ C ⊆ V such that the maximum distance from any vertex to the closest center vertex is minimized. In general, this problem is $$\mathsf {NP}$$ NP -hard and cannot be approximated within a factor less than 2. Typical applications of the k-Center problem can be found in logistics or urban planning and hence, it is natural to study the problem on transportation networks. Common characterizations of such networks are graphs that are (almost) planar or have low doubling dimension, highway dimension or skeleton dimension. It was shown by Feldmann and Marx that k-Center is $$\mathsf {W[1]}$$ W [ 1 ] -hard on planar graphs of constant doubling dimension when parameterized by the number of centers k, the highway dimension $$hd$$ hd and the pathwidth $$pw$$ pw (Feldmann and Marx 2020). We extend their result and show that even if we additionally parameterize by the skeleton dimension $$\kappa $$ κ , the k-Center problem remains $$\mathsf {W[1]}$$ W [ 1 ] -hard. Moreover, we prove that under the Exponential Time Hypothesis there is no exact algorithm for k-Center that has runtime $$f(k,hd,pw,\kappa ) \cdot \vert V \vert ^{o(pw+ \kappa + \sqrt{k+hd})}$$ f ( k , h d , p w , κ ) · | V | o ( p w + κ + k + h d ) for any computable function f.


Algorithmica ◽  
2021 ◽  
Author(s):  
Édouard Bonnet ◽  
Nidhi Purohit

AbstractA resolving set S of a graph G is a subset of its vertices such that no two vertices of G have the same distance vector to S. The Metric Dimension problem asks for a resolving set of minimum size, and in its decision form, a resolving set of size at most some specified integer. This problem is NP-complete, and remains so in very restricted classes of graphs. It is also W[2]-complete with respect to the size of the solution. Metric Dimension has proven elusive on graphs of bounded treewidth. On the algorithmic side, a polynomial time algorithm is known for trees, and even for outerplanar graphs, but the general case of treewidth at most two is open. On the complexity side, no parameterized hardness is known. This has led several papers on the topic to ask for the parameterized complexity of Metric Dimension with respect to treewidth. We provide a first answer to the question. We show that Metric Dimension parameterized by the treewidth of the input graph is W[1]-hard. More refinedly we prove that, unless the Exponential Time Hypothesis fails, there is no algorithm solving Metric Dimension in time $$f(\text {pw})n^{o(\text {pw})}$$ f ( pw ) n o ( pw ) on n-vertex graphs of constant degree, with $$\text {pw}$$ pw the pathwidth of the input graph, and f any computable function. This is in stark contrast with an FPT algorithm of Belmonte et al. (SIAM J Discrete Math 31(2):1217–1243, 2017) with respect to the combined parameter $$\text {tl}+\Delta$$ tl + Δ , where $$\text {tl}$$ tl is the tree-length and $$\Delta$$ Δ the maximum-degree of the input graph.


2020 ◽  
Vol 177 (1) ◽  
pp. 69-93
Author(s):  
Purnata Ghosal ◽  
B.V. Raghavendra Rao

We consider the problem of obtaining parameterized lower bounds for the size of arithmetic circuits computing polynomials with the degree of the polynomial as the parameter. We consider the following special classes of multilinear algebraic branching programs: 1) Read Once Oblivious Branching Programs (ROABPs), 2) Strict interval branching programs, 3) Sum of read once formulas with restricted ordering. We obtain parameterized lower bounds (i.e., nΩ(t(k)) lower bound for some function t of k) on the size of the above models computing a multilinear polynomial that can be computed by a depth four circuit of size g(k)nO(1) for some computable function g. Further, we obtain a parameterized separation between ROABPs and read-2 ABPs. This is obtained by constructing a degree k polynomial that can be computed by a read-2 ABP of small size such that the rank of the partial derivative matrix under any partition of the variables is large.


Information ◽  
2020 ◽  
Vol 11 (10) ◽  
pp. 457
Author(s):  
Kee Sung Kim

As data outsourcing services have been becoming common recently, developing skills to search over encrypted data has received a lot of attention. Order-revealing encryption (OREnc) enables performing a range of queries on encrypted data through a publicly computable function that outputs the ordering information of the underlying plaintexts. In 2016, Lewi et al. proposed an OREnc scheme that is more secure than the existing practical (stateless and non-interactive) schemes by constructing an ideally-secure OREnc scheme for small domains and a “domain-extension” scheme for obtaining the final OREnc scheme for large domains. They encoded a large message into small message blocks of equal size to apply them to their small-domain scheme, thus their resulting OREnc scheme reveals the index of the first differing message block. In this work, we introduce a new ideally-secure OREnc scheme for small domains with shorter ciphertexts. We also present an alternative message-block encoding technique. Combining the proposed constructions with the domain-extension scheme of Lewi et al., we can obtain a new large-domain OREnc scheme with shorter ciphertexts or with different leakage information, but longer ciphertexts.


Author(s):  
Alexsander Andrade de Melo ◽  
Mateus De Oliveira Oliveira

A fundamental drawback that arises when one is faced with the task of deterministically certifying solutions to computational problems in PSPACE is the fact that witnesses may have superpolynomial size, assuming that NP is not equal to PSPACE. Therefore, the complexity of such a deterministic verifier may already be super-polynomially lower-bounded by the size of a witness. In this work, we introduce a new symbolic framework to address this drawback. More precisely, we introduce a PSPACE-hard notion of symbolic constraint satisfaction problem where both instances and solutions for these instances are implicitly represented by ordered decision diagrams (i.e. read-once, oblivious, branching programs). Our main result states that given an ordered decision diagram D of length k and width w specifying a CSP instance, one can determine in time f(w,w')*k whether there is an ODD of width at most w' encoding a solution for this instance. Intuitively, while the parameter w quantifies the complexity of the instance, the parameter w' quantifies the complexity of a prospective solution. We show that CSPs of constant width can be used to formalize natural PSPACE hard problems, such as reachability of configurations for Turing machines working in nondeterministic linear space. For such problems, our main result immediately yields an algorithm that determines the existence of solutions of width w in time g(w)*n, where g:N->N is a suitable computable function, and n is the size of the input.


Author(s):  
Rod Downey ◽  
Noam Greenberg

This chapter discusses the notion of α‎-c.a. functions. The main issue is to properly define what is meant by a computable function o from N to α‎, which is required for the definition of α‎-computable approximations. Naturally, to deal with an ordinal α‎ computably, one needs a notation for this ordinal, or more generally, a computable well-ordering of order-type α‎. To form the basis of a solid hierarchy, the notion of α‎-c.a. should not depend on which well-ordering one takes, rather it should only depend on its order-type. Thus, one cannot consider all computable copies of α‎. Rather, one restricts one's self to a class of particularly well-behaved well-orderings, in a way that ensures that they are all computably isomorphic. Having defined α‎-c.a. functions, the chapter relates these functions to iterations of the bounded jump (the jump inside the weak truth-table degrees).


Author(s):  
Giuseppe Primiero

This chapter illustrates the basic tools of computability theory, essential to the formulation of the decision problem and the definition of the notion of computable function.


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