Deterministic Algorithm for Coded Cooperative Data Exchange

We consider a cooperative data exchange problem with the goal of generating a secret key. Specifically, we investigate the number of public transmissions required for a set of clients to agree on a secret key with probability one, subject to the constraint that it remains private from an eavesdropper. Although the problems are closely related, we prove that secret key generation with fewest linear transmissions is NP-hard, while it is known that the analogous problem in traditional cooperative data exchange can be solved in polynomial time. In doing this, we completely characterize the best-possible performance of linear coding schemes, and also prove that linear codes can be strictly suboptimal.

We consider the problem of finding the minimum sum-rate strategy in cooperative data exchange systems that do not allow packet-splitting (NPS-CDE). In an NPS-CDE system, there are a number of geographically close cooperative clients who send packets to help the others recover a packet set. A minimum sum-rate strategy is the strategy that achieves universal recovery (the situation when all the clients recover the whole packet set) with the the minimal sum-rate (the total number of transmissions). We propose an iterative merging (IM) algorithm that recursively merges client sets based on a lower estimate of the minimum sum-rate and updates to the value of the minimum sum-rate. We also show that a minimum sum-rate strategy can be learned by allocating rates for the local recovery in each merged client set in the IM algorithm. We run an experiment to show that the complexity of the IM algorithm is lower than that of the existing deterministic algorithm when the number of clients is lower than 94.

We consider the bin packing problem with d different item sizes s i and item multiplicities a i , where all numbers are given in binary encoding. This problem formulation is also known as the one-dimensional cutting stock problem . In this work, we provide an algorithm that, for constant d , solves bin packing in polynomial time. This was an open problem for all d\ge 3 . In fact, for constant d our algorithm solves the following problem in polynomial time: Given two d -dimensional polytopes P and Q , find the smallest number of integer points in P whose sum lies in Q . Our approach also applies to high multiplicity scheduling problems in which the number of copies of each job type is given in binary encoding and each type comes with certain parameters such as release dates, processing times, and deadlines. We show that a variety of high multiplicity scheduling problems can be solved in polynomial time if the number of job types is constant.

Understanding spatial correlation is vital in many fields including epidemiology and social science. Lee, Meeks and Pettersson (Stat. Comput. 2021) recently demonstrated that improved inference for areal unit count data can be achieved by carrying out modifications to a graph representing spatial correlations; specifically, they delete edges of the planar graph derived from border-sharing between geographic regions in order to maximise a specific objective function. In this paper we address the computational complexity of the associated graph optimisation problem. We demonstrate that this problem cannot be solved in polynomial time unless P = NP; we further show intractability for two simpler variants of the problem. We follow these results with two parameterised algorithms that exactly solve the problem in polynomial time in restricted settings. The first of these utilises dynamic programming on a tree decomposition, and runs in polynomial time if both the treewidth and maximum degree are bounded. The second algorithm is restricted to problem instances with maximum degree three, as may arise from triangulations of planar surfaces, but is an FPT algorithm when the maximum number of edges that can be removed is taken as the parameter.

In this work, we study the computational perspective of network coding, focusing on two issues. First, we address the computational complexity of finding a network code for acyclic multicast networks. Second, we address the issue of reducing the amount of computation performed by the network nodes. In particular, we consider the problem of finding a network code with the minimum possible number of encoding nodes, i.e., nodes that generate new packets by combining the packets received over incoming links. We present a deterministic algorithm that finds a feasible network code for a multicast network over an underlying graph G(V, E) in time O(|E|kh+|V|k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> +h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">4</sup> k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> (k+h)), where k is the number of destinations and h is the number of packets. This improves the best known running time of O(|E|kh+|V|k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> (k+h)) of Jaggi et al. (2005) in the typical case of large communication graphs. In addition, our algorithm guarantees that the number of encoding nodes in the obtained network code is bounded by O(h <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">3</sup> k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> ). Next, we address the problem of finding a network code with the minimum number of encoding nodes in both integer and fractional coding networks. We prove that in the majority of settings this problem is NP-hard. However, we show that if h=O(1) and k=O(1) and the underlying communication graph is acyclic, then there exists an algorithm that solves this problem in polynomial time.

We present a quantum polynomial time attack against the GMMSSZ branching program obfuscator of Garg et al. (TCC’16), when instantiated with the GGH13 multilinear map of Garg et al. (EUROCRYPT’13). This candidate obfuscator was proved secure in the weak multilinear map model introduced by Miles et al. (CRYPTO’16). Our attack uses the short principal ideal solver of Cramer et al. (EUROCRYPT’16), to recover a secret element of the GGH13 multilinear map in quantum polynomial time. We then use this secret element to mount a (classical) polynomial time mixed-input attack against the GMMSSZ obfuscator. The main result of this article can hence be seen as a classical reduction from the security of the GMMSSZ obfuscator to the short principal ideal problem (the quantum setting is then only used to solve this problem in polynomial time). As an additional contribution, we explain how the same ideas can be adapted to mount a quantum polynomial time attack against the DGGMM obfuscator of Döttling et al. (ePrint 2016), which was also proved secure in the weak multilinear map model.

Computation Offloading, sending computational tasks to more resourceful servers, is becoming a widely-used approach to save limited resources on mobile devices like battery life, storage, processor, etc. Given an application that is partitioned into multiple tasks, the offloading decisions can be made on each of them. However, considering the delay constraint and the extra costs on data transmission and remote computation, it is not trivial to make optimized decisions. Existing works have formulated offloading decision problems as either graph-partitioning or binary integer programming problems. The first approach can solve the problem in polynomial time but is not applicable to delay constraints. The second approach relies on an integer programming solver without a polynomial time guarantee. We provide an algorithm, DTP (Deterministic delay constrained Task Partitioning), to solve the offloading decision problem with delay constraints. DTP gives near-optimal solution and runs in polynomial time in the number of tasks. Going beyond prior work on linear delay constraints that apply only to serial tasks, we generalize the delay constraints to settings where the dependency between tasks can be described by a tree. Furthermore, we provide another algorithm, PTP (Probabilistic delay constrained Task Partitioning), which gives stronger QoS guarantees. Simulation results show that our algorithms are accurate and robust, and scale well with the number of tasks.

Partially-specified systems and specifications are used in formal methods such as stepwise design and query checking. Existing methods consider a setting in which the systems and their correctness are Boolean. In recent years there has been growing interest and need for quantitative formal methods, where systems may be weighted and specifications may be multi valued. Weighted automata, which map input words to a numerical value, play a key role in quantitative reasoning. Technically, every transition in a weighted automaton $\mathcal{A}$ has a cost, and the value $\mathcal{A}$ assigns to a finite word w is the sum of the costs on the transitions participating in the most expensive accepting run of $\mathcal{A}$ on w. We study parameterized weighted containment: given three weighted automata $\mathcal{A}, \mathcal{B}$, and $\mathcal{C}$, with $\mathcal{B}$ being partial, the goal is to find an assignment to the missing costs in $\mathcal{B}$ so that we end up with $\mathcal{B}'$ for which $\mathcal{A} \leq \mathcal{B}' \leq \mathcal{C}$, where ≤ is the weighted counterpart of containment. We also consider a one-sided version of the problem, where only $\mathcal{A}$ or only $\mathcal{C}$ are given in addition to $\mathcal{B}$, and the goal is to find a minimal assignment with which $\mathcal{A} \leq B'$ or, respectively, a maximal one with which $\mathcal{B}' \leq \mathcal{C}$. We argue that both problems are useful in stepwise design of weighted systems as well as approximated minimization of weighted automata. We show that when the automata are deterministic, we can solve the problems in polynomial time. Our solution is based on the observation that the set of legal assignments to k missing costs forms a k-dimensional polytope. The technical challenge is to find an assignment in polynomial time even though the polytope is defined by means of exponentially many inequalities. We do so by using a powerful mathematical tool that enables us to develop a divide-and-conquer algorithm based on a separation oracle for polytopes. For nondeterministic automata, the weighted setting is much more complex, and in fact even non-parameterized containment is undecidable. We are still able to study variants of the problems, where containment is replaced by simulation.

The traveling-salesman problem is one of the classical NP-Complete problems. No current algorithms are available which can solve these problems in polynomial time, that is, the number of steps grows as a polynomial according to the size of the input. The traveling-salesman problem involves a salesman who must make a tour of a number of cities using the shortest path available. For each number of cities n, the number of paths which must be explored is n!, causing this problem to grow exponentially rather than as a polynomial.Three separate algorithms are examined. These are an iterative algorithm, a recursive algorithm, and a branch and bound algorithm. The iterative algorithm generates tours as the permutations of the first n - 1 integers (where n is the number of cities). The recursive algorithm begins with a subtour consisting of the starting city and uses a recursive subroutine to build all tours. The branch and bound algorithm builds a tree which represents tours. Each node of the tree has an associated bound, and when the bound of a node becomes larger than the cost of the best tour found so far, that node is no longer eligible for exploration. This pruning process allows this algorithm to delay exponential growth longer than the other two algorithms do. Two algorithms which develop near optimal solutions are also examined.

As the demand of mobile devices (MDs) for data services is explosively increasing, traditional offloading in the cellular networks is facing the contradiction of energy efficiency and quality of service. Device-to-device (D2D) communication is considered as an effective solution. This work investigates a scenario where the MDs have the same demand for common content and they cooperate to deliver it using multicast multihop relaying. We focus on the problem of total power minimization by grouping the MDs in multihop D2D networks, while maintaining the minimum rate requirement of each MD. As the problem is shown to be NP-complete and the optimal solution can not be found efficiently, two greedy algorithms are proposed to solve this problem in polynomial time. Simulation results demonstrate that lots of power can be saved in the content delivery situation using multihop D2D communication, and the proposed algorithms are suitable for different situations with different advantages.

Data exchange deals with inserting data from one database into another database having a different schema. Fagin et al. [2005] have shown that among the universal solutions of a solvable data exchange problem, there exists—up to isomorphism—a unique most compact one, “the core”, and have convincingly argued that this core should be the database to be materialized. They stated as an important open problem whether the core can be computed in polynomial time in the general setting where the mapping between the source and target schemas is given by source-to-target constraints that are arbitrary tuple generating dependencies (tgds) and target constraints consisting of equality generating dependencies (egds) and a weakly acyclic set of tgds. In this article, we solve this problem by developing new methods for efficiently computing the core of a universal solution. This positive result shows that data exchange based on cores is feasible and applicable in a very general setting. In addition to our main result, we use the method of hypertree decompositions to derive new algorithms and upper bounds for query containment checking and computing cores of arbitrary database instances. We also show that computing the core of a data exchange problem is fixed-parameter intractable with respect to a number of relevant parameters, and that computing cores is NP-complete if the rule bodies of target tgds are augmented by a special predicate that distinguishes a null value from a constant data value.

In this paper, we investigate the use of instantly decodable network coding (IDNC) for improving two fundamental performance metrics, namely, the completion time (as a measure of throughput) and mean decoding delay, in multicast cooperative data exchange (CDE) systems, where a group of geographically close clients cooperate with each other to obtain their missing packets. Here, an IDNC scheme is used for the transmissions across these clients. We utilize the stochastic shortest path (SSP) technique to study the minimum mean completion time problem. However, since finding the optimum solution is intractable, we use the obtained formulation to draw some theoretical guidelines to heuristically find solutions that can efficiently reduce the completion time. Second, we formulate the minimum mean decoding delay problem as selecting the appropriate maximal clique over well-structured graphs at the clients, and in order to reduce its complexity, we propose a simple heuristic algorithm for it. The effectiveness of our proposed algorithms is verified through extensive simulations and comparisons with existing techniques.

Most of Wireless Sensor Networks (WSNs) are usually deployed in hostile environments where the communications conditions are not stable and not reliable. Hence, there is a need to design an effective distributed schemes to enable the sensors cooperating in order to recover the sensed data. In this paper, we establish a novel cooperative data exchange (CDE) scheme using instantly decodable network coding (IDNC) across the sensor nodes. We model the problem using the cooperative game theory in partition form. We develop also a distributed merge-and-split algorithm in order to form dynamically coalitions that maximize their utilities in terms of both energy consumption and IDNC delay experienced by all sensors. Indeed, the proposed algorithm enables these sensors to self-organize into stable clustered network structure where all sensors do not have incentives to change the cluster he is part of. Simulation results show that our cooperative scheme allows nodes not only to reduce the energy consumption, but also the IDNC completion time.

We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (2005). More precisely, denoting by $n, m, w_{\max}$, and $w_{\min}$ the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature $T_0 \ge w_{\max}$ and multiplicative cooling schedule with factor $1-1/\ell$, where $\ell = \omega (mn\ln(m))$, with probability at least $1-1/m$ computes in time $O(\ell (\ln\ln (\ell) + \ln(T_0/w_{\min}) ))$ a spanning tree with weight at most $1+\kappa$ times the optimum weight, where $1+\kappa = \frac{(1+o(1))\ln(\ell m)}{\ln(\ell) -\ln (mn\ln (m))}$. Consequently, for any $\epsilon>0$, we can choose $\ell$ in such a way that a $(1+\epsilon)$-approximation is found in time $O((mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n + \ln(T_0/w_{\min})))$ with probability at least $1-1/m$. In the special case of so-called $(1+\epsilon)$-separated weights, this algorithm computes an optimal solution (again in time $O( (mn\ln(n))^{1+1/\epsilon+o(1)}(\ln\ln n + \ln(T_0/w_{\min})))$), which is a significant speed-up over Wegener's runtime guarantee of $O(m^{8 + 8/\epsilon})$.

We propose a general approach to solving some vector subset problems in a Euclidean space that is based on higher-order Voronoi diagrams. In the case of a fixed space dimension, this approach allows us to find optimal solutions to these problems in polynomial time which is better than the runtime of available algorithms.