Standard Complexity Assumptions Research Articles

A Subexponential Parameterized Algorithm for Directed Subset Traveling Salesman Problem on Planar Graphs

There are numerous examples of the so-called ``square root phenomenon'' in the field of parameterized algorithms: many of the most fundamental graph problems, parameterized by some natural parameter $k$, become significantly simpler when restricted to planar graphs and in particular the best possible running time is exponential in $O(\sqrt{k})$ instead of $O(k)$ (modulo standard complexity assumptions). We consider a classic optimization problem Subset Traveling Salesman, where we are asked to visit all the terminals $T$ by a minimum-weight closed walk. We investigate the parameterized complexity of this problem in planar graphs, where the number $k=|T|$ of terminals is regarded as the parameter. We show that Subset TSP can be solved in time $2^{O(\sqrt{k}\log k)}\cdot n^{O(1)}$ even on edge-weighted directed planar graphs. This improves upon the algorithm of Klein and Marx [SODA 2014] with the same running time that worked only on undirected planar graphs with polynomially large integer weights.

Disjunctive Temporal Problems under Structural Restrictions

The disjunctive temporal problem (DTP) is an expressive temporal formalism that extends Dechter et al.'s simple temporal problem. The DTP is well studied in the literature and has many important applications. It is known that deciding satisfiability of DTPs is NP-hard and that, in many cases, single-exponential algorithms (running in O(c^n) time) do not exist under the Exponential-Time Hypothesis. The computational hardness makes it worthwhile to identify restricted problems that are efficiently solvable. One way of doing this is to restrict the interactions of variables and constraints. We show that instances of DTP of any arity with integers bounded by poly(n) can be solved in n^{f(w)} time, where n denotes the problem size, w is the treewidth of the incidence graph and f is a computable function; in other words, this problem is in the complexity class XP and it can be solved in polynomial time whenever w is fixed. We complement this result by showing that binary DTPs that only involve the integers 0 and 1 are not fixed-parameter tractable with respect to treewidth, i.e. they do not admit a f(w)poly(n)$ time algorithm for any computable function f, under standard complexity assumptions. For instances with unbounded integers, we show that even binary DTPs parameterized by treewidth cannot be in XP, unless P = NP.

Parameterized Approximation Algorithms for Bidirected Steiner Network Problems

The D irected S teiner N etwork (DSN) problem takes as input a directed graph G =( V , E ) with non-negative edge-weights and a set D ⊆ V × V of k demand pairs. The aim is to compute the cheapest network N⊆ G for which there is an s\rightarrow t path for each ( s , t )∈ D. It is known that this problem is notoriously hard, as there is no k 1/4− o (1) -approximation algorithm under Gap-ETH, even when parametrizing the runtime by k [Dinur & Manurangsi, ITCS 2018]. In light of this, we systematically study several special cases of DSN and determine their parameterized approximability for the parameter k . For the bi -DSNP lanar problem, the aim is to compute a solution N⊆ G whose cost is at most that of an optimum planar solution in a bidirected graph G , i.e., for every edge uv of G the reverse edge vu exists and has the same weight. This problem is a generalization of several well-studied special cases. Our main result is that this problem admits a parameterized approximation scheme (PAS) for k . We also prove that our result is tight in the sense that (a) the runtime of our PAS cannot be significantly improved, and (b) no PAS exists for any generalization of bi-DSNP lanar , under standard complexity assumptions. The techniques we use also imply a polynomial-sized approximate kernelization scheme (PSAKS). Additionally, we study several generalizations of bi -DSNP lanar and obtain upper and lower bounds on obtainable runtimes parameterized by k . One important special case of DSN is the S trongly C onnected S teiner S ubgraph (SCSS) problem, for which the solution network N⊆ G needs to strongly connect a given set of k terminals. It has been observed before that for SCSS a parameterized 2-approximation exists for parameter k [Chitnis et al., IPEC 2013]. We give a tight inapproximability result by showing that for k no parameterized (2 − ε)-approximation algorithm exists under Gap-ETH. Additionally, we show that when restricting the input of SCSS to bidirected graphs, the problem remains NP-hard but becomes FPT for k .

Explicit List-Decodable Codes with Optimal Rate for Computationally Bounded Channels

A stochastic code is a pair of encoding and decoding procedures (Enc, Dec) where \({{\rm Enc} : \{0, 1\}^{k} \times \{0, 1\}^{d} \rightarrow \{0, 1\}^{n}}\). The code is (p, L)-list decodable against a class \(\mathcal{C}\) of “channel functions” \(C : \{0,1\}^{n} \rightarrow \{0,1\}^{n}\) if for every message \(m \in \{0,1\}^{k}\) and every channel \(C \in \mathcal{C}\) that induces at most pn errors, applying Dec on the “received word” C(Enc(m,S)) produces a list of at most L messages that contain m with high probability over the choice of uniform \(S \leftarrow \{0, 1\}^{d}\). Note that both the channel C and the decoding algorithm Dec do not receive the random variable S, when attempting to decode. The rate of a code is \(R = k/n\), and a code is explicit if Enc, Dec run in time poly(n).Guruswami and Smith (Journal of the ACM, 2016) showed that for every constants \(0 < p < \frac{1}{2}, \epsilon > 0\) and \(c > 1\) there exist a constant L and a Monte Carlo explicit constructions of stochastic codes with rate \(R \geq 1-H(p) - \epsilon\) that are (p, L)-list decodable for size \(n^c\) channels. Here, Monte Carlo means that the encoding and decoding need to share a public uniformly chosen \({\rm poly}(n^c)\) bit string Y, and the constructed stochastic code is (p, L)-list decodable with high probability over the choice of Y.Guruswami and Smith pose an open problem to give fully explicit (that is not Monte Carlo) explicit codes with the same parameters, under hardness assumptions. In this paper, we resolve this open problem, using a minimal assumption: the existence of poly-time computable pseudorandom generators for small circuits, which follows from standard complexity assumptions by Impagliazzo and Wigderson (STOC 97).Guruswami and Smith also asked to give a fully explicit unconditional constructions with the same parameters against \(O(\log n)\)-space online channels. (These are channels that have space \(O(\log n)\) and are allowed to read the input codeword in one pass.) We also resolve this open problem.Finally, we consider a tighter notion of explicitness, in which the running time of encoding and list-decoding algorithms does not increase, when increasing the complexity of the channel. We give explicit constructions (with rate approaching \(1 - H(p)\) for every \(p \leq p_{0}\) for some \(p_{0} >0\) ) for channels that are circuits of size \(2^{n^{\Omega(1/d)}}\) and depth d. Here, the running time of encoding and decoding is a polynomial that does not depend on the depth of the circuit.Our approach builds on the machinery developed by Guruswami and Smith, replacing some probabilistic arguments with explicit constructions. We also present a simplified and general approach that makes the reductions in the proof more efficient, so that we can handle weak classes of channels.

On the Complexity of Learning a Class Ratio from Unlabeled Data

&#x0D; &#x0D; &#x0D; In the problem of learning a class ratio from unlabeled data, which we call CR learning, the training data is unlabeled, and only the ratios, or proportions, of examples receiving each label are given. The goal is to learn a hypothesis that predicts the proportions of labels on the distribution underlying the sample. This model of learning is applicable to a wide variety of settings, including predicting the number of votes for candidates in political elections from polls.&#x0D; In this paper, we formally define this class and resolve foundational questions regarding the computational complexity of CR learning and characterize its relationship to PAC learning. Among our results, we show, perhaps surprisingly, that for finite VC classes what can be efficiently CR learned is a strict subset of what can be learned efficiently in PAC, under standard complexity assumptions. We also show that there exist classes of functions whose CR learnability is independent of ZFC, the standard set theoretic axioms. This implies that CR learning cannot be easily characterized (like PAC by VC dimension).&#x0D; &#x0D; &#x0D;

Election with Bribed Voter Uncertainty: Hardness and Approximation Algorithm

Bribery in election (or computational social choice in general) is an important problem that has received a considerable amount of attention. In the classic bribery problem, the briber (or attacker) bribes some voters in attempting to make the briber’s designated candidate win an election. In this paper, we introduce a novel variant of the bribery problem, “Election with Bribed Voter Uncertainty” or BVU for short, accommodating the uncertainty that the vote of a bribed voter may or may not be counted. This uncertainty occurs either because a bribed voter may not cast its vote in fear of being caught, or because a bribed voter is indeed caught and therefore its vote is discarded. As a first step towards ultimately understanding and addressing this important problem, we show that it does not admit any multiplicative O(1)-approximation algorithm modulo standard complexity assumptions. We further show that there is an approximation algorithm that returns a solution with an additive-ε error in FPT time for any fixed ε.

A Strong Designated Verifier Proxy Re-Signature Scheme for IoT Environments

With the rapid popularization of the Internet of Things (IoT) in our daily lives, the communication security and identity privacy of IoT devices must be ensured. However, traditional authentication mechanisms utilized in IoT cannot completely ensure a user’s privacy when his/her messages are routed via an untrusted intermediate device. Strong designated-verifier proxy re-signature (SDVPRS) is a new cryptographic technology that combines the advantages of strong designated verifier signature and proxy re-signature. Therefore, SDVPRS is considered to be a better approach to maintain data integrity and protect the identity privacy of the signer in a resource-limited IoT device. Nevertheless, designing a secure SDVPRS scheme without random oracles is still a challenging task. In this paper, we mainly focus on such a construction by providing a new method. We first provide the formal definition of SDVPRS and its security model. Then, we present the first SDVPRS scheme, which is bidirectional, multi-use and non-transferable, and we prove its security under the standard complexity assumptions in the standard model. The analysis results show that our SDVPRS scheme can not only protect the privacy of the signer’s identity, but also provide non-delegatability for signature verification. We present an example of potential application to environmental monitoring systems using our SDVPRS scheme.

Parameterized complexity of a coupled-task scheduling problem

In this article, we investigate the parameterized complexity of coupled-task scheduling in the presence of compatibility constraint given by a compatibility graph. In this model, each task contains two sub-tasks delayed by an idle time, and a sub- task of a task can be performed during the idle time of another one if, and only if, the two tasks are compatible. We consider a parameterized version of the scheduling problem: is there a schedule in which at least k coupled-tasks possess a completion time before a fixed due date? It is known that this problem is NP-complete ([23] and [24]), and we prove that it is fixed-parameter tractable (FPT ) if the total duration time of each task is bounded by a constant, whereas the problem becomes W[1]-hard if it is not the case. We also show that in the former case the problem does not admit a polynomial kernel under some standard complexity assumptions. Moreover, we obtain also an FPT algorithm when the problem is parameterized by the size of a vertex cover of the compatibility graph.

Fixing improper colorings of graphs

In this paper we consider a variation of a recoloring problem, called the Color-Fixing. Let us have some non-proper r-coloring φ of a graph G. We investigate the problem of finding a proper r-coloring of G, which is “the most similar” to φ, i.e., the number k of vertices that have to be recolored is minimum possible. We observe that the problem is NP-complete for any fixed r≥3, even for bipartite planar graphs. Moreover, it is W[1]-hard even for bipartite graphs, when parameterized by the number k of allowed recoloring transformations. On the other hand, the problem is fixed-parameter tractable, when parameterized by k and the number r of colors.We provide a 2n⋅nO(1) algorithm for the problem and a linear algorithm for graphs with bounded treewidth. We also show several lower complexity bounds, using standard complexity assumptions. Finally, we investigate the fixing number of a graph G. It is the minimum k such that k recoloring transformations are sufficient to transform any coloring of G into a proper one.

SignORKE: improving pairing‐based one‐round key exchange without random oracles

The study presents a new efficient way to construct the one-round key exchange (ORKE) without random oracles based on standard hard complexity assumptions. The authors propose a (PKI-based) ORKE protocol which is more computational efficient than existing pairing-based ORKE protocols without random oracles in the post-specified peer setting. The core idea of this construction is to integrate the consistency check of the ephemeral public key and the verification of the signature into the session key generation. This enables us to roughly save two pairing operations. The authors just call this kind of scheme that is deeply composed by signature and one-round key exchange as SignORKE. The authors’ protocol is shown to be secure in a variant of the Canetti–Krawczyk security model which covers the majority of state-of-the-art active attacks.

Naor–Yung paradigm with shared randomness and applications

The Naor–Yung paradigm [63] allows to generically boost security under chosen-plaintext attacks (CPA) to security against chosen-ciphertext attacks (CCA) for public-key encryption (PKE) schemes. The main idea is to encrypt the plaintext twice (under independent public keys), and to append a non-interactive zero-knowledge (NIZK) proof that the two ciphertexts indeed encrypt the same message. Later work by Camenisch, Chandran, and Shoup [32] and Naor and Segev [28,30] established that the very same technique can also be used in the settings of key-dependent message (KDM) and key-leakage attacks (respectively).In this paper we study the conditions under which the two ciphertexts in the Naor–Yung construction can share the same random coins. We find that this is possible, provided that the underlying PKE scheme meets an additional simple property. The motivation for re-using the same random coins is that this allows to design much more efficient NIZK proofs. We showcase such an improvement in the random oracle model, under standard complexity assumptions including Decisional Diffie–Hellman, Quadratic Residuosity, and Subset Sum. The length of the resulting ciphertexts is reduced by 50%, yielding truly efficient PKE schemes achieving CCA security under KDM and key-leakage attacks.As an additional contribution, we design the first PKE scheme whose CPA security under KDM attacks can be directly reduced to (low-density instances of) the Subset Sum assumption. Our PKE scheme supports key-dependent messages computed via any affine function of the secret key.

Optimizing Tree Patterns for Querying Graph- and Tree-Structured Data

Many of today's graph query languages are based on graph pattern matching. We investigate optimization for treeshaped patterns with transitive closure. Such patterns are quite expressive, yet can be evaluated efficiently. The minimization problem aims at reducing the number of nodes in patterns and goes back to the early 2000's. We provide an example showing that, in contrast to earlier claims, tree patterns cannot be minimized by deleting nodes only. The example resolves the M ?/= NR problem, which asks if a tree pattern is minimal if and only if it is nonredundant. The example can be adapted to also understand the complexity of minimization, which was another question that was open since the early research on the problem. Interestingly, the latter result also shows that, unless standard complexity assumptions are false, more general approaches for minimizing tree patterns are also bound to fail in some cases.

Sparse Fault-Tolerant BFS Structures

A fault-tolerant structure for a network is required for continued functioning following the failure of some of the network’s edges or vertices. This article considers breadth-first search (BFS) spanning trees and addresses the problem of designing a sparse fault-tolerant BFS structure (FT-BFS structure), namely, a sparse subgraph T of the given network G such that subsequent to the failure of a single edge or vertex, the surviving part T ′ of T still contains a BFS spanning tree for (the surviving part of) G . For a source node s , a target node t , and an edge e ∈ G , the shortest s − t path P s , t , e that does not go through e is known as a replacement path . Thus, our FT-BFS structure contains the collection of all replacement paths P s , t , e for every t ∈ V ( G ) and every failed edge e ∈ E ( G ). Our main results are as follows. We present an algorithm that for every n -vertex graph G and source node s constructs a (single edge failure) FT-BFS structure rooted at s with O ( n ċ min {Depth( s ), √n{) edges, where Depth( s ) is the depth of the BFS tree rooted at s . This result is complemented by a matching lower bound, showing that there exist n -vertex graphs with a source node s for which any edge (or vertex) FT-BFS structure rooted at s has Ω( n 3/2 ) edges. We then consider fault-tolerant multi-source BFS structures (FT-MBFS structures), aiming to provide (following a failure) a BFS tree rooted at each source s ∈ S for some subset of sources S ⊆ V . Again, tight bounds are provided, showing that there exists a poly-time algorithm that for every n -vertex graph and source set S ⊆ V of size σ constructs a (single failure) FT-MBFS structure T *( S ) from each source s i ∈ S , with O (√σ ċ n &lt;sup;&gt;3/2&lt;/sup;&gt;) edges, and, on the other hand, there exist n -vertex graphs with source sets S ⊆ V of cardinality σ, on which any FT-MBFS structure from S has Ω(√σ ċ n 3/2 ) edges. Finally, we propose an O (log n ) approximation algorithm for constructing FT-BFS and FT-MBFS structures. The latter is complemented by a hardness result stating that there exists no Ω(log n ) approximation algorithm for these problems under standard complexity assumptions. In comparison with previous constructions, our algorithm is deterministic and may improve the number of edges by a factor of up to √ n for some instances. All our algorithms can be extended to deal with one vertex failure as well, with the same performance.

User Collusion Avoidance Scheme for Privacy-Preserving Decentralized Key-Policy Attribute-Based Encryption

Decentralized attribute-based encryption (ABE) is a variant of multi-authority based ABE whereby any attribute authority (AA) can independently join and leave the system without collaborating with the existing AAs. In this paper, we propose a user collusion avoidance scheme which preserves the user's privacy when they interact with multiple authorities to obtain decryption credentials. The proposed scheme mitigates the well-known user collusion security vulnerability found in previous schemes. We show that our scheme relies on the standard complexity assumption (decisional bilienar Deffie-Hellman assumption). This is contrast to previous schemes which relies on non-standard assumption (q-decisional Diffie-Hellman inversion).

Primal–Dual Algorithms for Precedence Constrained Covering Problems

A covering problem is an integer linear program of type $$\min \{c^Tx\mid Ax\ge D,\ 0\le x\le d,\ x \in \mathbb {Z}\}$$min{cTxźAxźD,0≤x≤d,xźZ} where $$A\in \mathbb {Z}^{m\times n}_+$$AźZ+m×n, $$D\in \mathbb {Z}_+^m$$DźZ+m, and $$c,d\in \mathbb {Z}_+^n$$c,dźZ+n. In this paper, we study covering problems with additional precedence constraints $$\{x_i\le x_j \ \forall j\preceq i \in \mathcal {P}\}$${xi≤xjźjźiźP}, where $$\mathcal {P}=([n], \preceq )$$P=([n],ź) is some arbitrary, but fixed partial order on the items represented by the column-indices of A. Such precedence constrained covering problems (PCCPs) are of high theoretical and practical importance even in the special case of the precedence constrained knapsack problem, that is, where $$m=1$$m=1 and $$d\equiv 1$$dź1. Our main result is a strongly-polynomial primal---dual approximation algorithm for PCCP with $$d\equiv 1$$dź1. Our approach generalizes the well-known knapsack cover inequalities to obtain an IP formulation which renders any explicit precedence constraints redundant. The approximation ratio of this algorithm is upper bounded by the width of $$\mathcal {P}$$P, that is, by the size of a maximum antichain in $$\mathcal {P}$$P. Interestingly, this bound is independent of the number of constraints. We are not aware of any other results on approximation algorithms for PCCP on arbitrary posets $$\mathcal {P}$$P. For the general case with $$d\not \equiv 1$$dź1, we present pseudo-polynomial algorithms. Finally, we show that the problem does not admit a PTAS under standard complexity assumptions.

Efficient Cryptosystems From $$\mathbf{2}^{{\varvec{k}}}$$ 2 k -th Power Residue Symbols

Goldwasser and Micali (J Comput Syst Sci 28(2):270---299, 1984) highlighted the importance of randomizing the plaintext for public-key encryption and introduced the notion of semantic security. They also realized a cryptosystem meeting this security notion under the standard complexity assumption of deciding quadratic residuosity modulo a composite number. The Goldwasser---Micali cryptosystem is simple and elegant but is quite wasteful in bandwidth when encrypting large messages. A number of works followed to address this issue and proposed various modifications. This paper revisits the original Goldwasser---Micali cryptosystem using $$2^k$$2k-th power residue symbols. The so-obtained cryptosystems appear as a very natural generalization for $$k \ge 2$$kź2 (the case $$k=1$$k=1 corresponds exactly to the Goldwasser---Micali cryptosystem). Advantageously, they are efficient in both bandwidth and speed; in particular, they allow for fast decryption. Further, the cryptosystems described in this paper inherit the useful features of the original cryptosystem (like its homomorphic property) and are shown to be secure under a similar complexity assumption. As a prominent application, this paper describes an efficient lossy trapdoor function-based thereon.

On the parameterized complexity of non-monotonic logics

We investigate the application of Courcelle's theorem and the logspace version of Elberfeld et al. in the context of non-monotonic reasoning. Here we formalize the implication problem for propositional sets of formulas, the extension existence problem for default logic, the expansion existence problem for autoepistemic logic, the circumscriptive inference problem, as well as the abduction problem in monadic second order logic and thereby obtain fixed-parameter time and space efficient algorithms for these problems. On the other hand, we exhibit, for each of the above problems, families of instances of a very simple structure that, for a wide range of different parameterizations, do not have efficient fixed-parameter algorithms (even in the sense of the large class XPnu, resp., XLnu) under standard complexity assumptions.

A ciphertext‐policy hidden vector encryption scheme supporting multiuser keyword search

AbstractIn cloud computing, large amount of data can be effectively stored and managed. People could outsource the encrypted data using searchable encryption (SE) for data security and efficient retrieval. However, most existing SE schemes only support the single‐user access, and multiuser searchable encryption is required in many enterprise applications. From the attribute‐based encryption (ABE), we found that the flexibility and usability of encryption schemes can be greatly improved by embedding attribute‐based access policy in the ciphertext. In this paper, by using the idea of ABE, we propose a ciphertext‐policy hidden vector encryption (CPHVE) scheme to support both encryption and search operations for multiple users. In the scheme, a keyword is encrypted with an attribute‐based access policy, which can be searched when the users' attributes satisfy the policy. The security of CPHVE is also defined and proved in this paper. Moreover, the CPHVE scheme is based on standard complexity assumptions on bilinear groups of prime order, thus it is more efficient than the existing schemes. Copyright © 2014 John Wiley &amp; Sons, Ltd.

An identity-based strongly unforgeable signature without random oracles from bilinear pairings

This paper proposes an identity-based (ID-based) signature (IBS) scheme which is strongly unforgeable in the standard model whose security is reduced to the hardness of the computational Diffie-Hellman (CDH) problem in bilinear groups. Currently, the ID-based encryption scheme (IBE) due to Waters is known to be the most practically efficient IBE whose security is guaranteed in the standard model depending on the decisional bilinear Diffie-Hellman (BDH) assumption. While as a solution for an ID-based signature of a total ID-based public key cryptosystem cooperating with Waters IBE, we share Waters’s construction and system parameters to keep a key pair corresponding to each identity unchanged, our IBS needs only one group element for signing of messages and two elements for its randomness as supplementary parameters plus the original system parameters. Accordingly, thanks to requiring about half the system parameters against previous ID-based signatures proved secure without using random oracles under the standard complexity assumptions like CDH or BDH, our IBS is the more suitable for storage and communication requirements. In particular, this covers a stronger security property called strong unforgeability in the standard model in itself without applying any transformation technique.

Notes on a group-oriented setting’s multisigncryption scheme with threshold designcryption

Zhang et al. proposed recently a group-oriented multisigncryption scheme with threshold designcryption and affirmed that their scheme is secure under standard complexity assumptions. However, the note elaborates concrete attacks on its indistinguishability and unforgeability under their security models. We also provide the impossibility result of public ciphertext authenticity for the scheme.