Computational Hardness Research Articles

On the Shared Transportation Problem: Computational Hardness and Exact Approach

In our modern societies, a certain number of people do not own a car, by choice or by obligation. For some trips, there is no or few alternatives to the car. One way to make these trips possible for these people is to be transported by others who have already planned their trips. We propose to model this problem using as path-finding problem in a list edge-colored graph. This problem is a generalization of the [Formula: see text]-path problem, studied by Böhmová et al. We consider two optimization functions: minimizing the number of color changes and minimizing the number of colors. We study for the previous problems, the classic complexity (polynomial-case, NP-completeness, hardness of approximation) and parameter complexity (W[2]-hardness) even in restricted cases. We also propose a lower bound for exact algorithm. On the positive side we provide a polynomial-time approximation algorithm and a FPT algorithm.

Computing maximum matchings in temporal graphs

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph G, a temporal graph is represented by assigning a set of integer time-labels to every edge e of G, indicating the discrete time steps at which e is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) (e,t) such that no vertex is matched more than once within any time window of Δ consecutive time slots, where Δ∈N is given. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases, as well as fixed-parameter algorithms with respect to natural parameters and polynomial-time approximation algorithms.

Estimating Fock-state linear optics evolution using coherent states

This paper presents two methods for simulating the interference of bosonic Fock states through linear interferometers using coherent states. The first method repeats the interferometer, injects coherent states in particular modes, and uses symmetric combinations of the outputs to reconstruct the state amplitudes of the Fock-state interference. The second method constructs a new interferometer that can be probed with coherent states on individual inputs to extract the required state amplitudes. The two approaches here show explicitly where the classical computational difficultly arises. In the first approach, the computational hardness is in the measurement post-processing, and in the second approach, it is within the construction of the required state evolution.

Evaluation of Quantum Key Distribution for Secure Satellite-integrated IoT Networks

There has been a dramatic increase in the number of Internet of Things (IoT) devices and their applications. Furthermore, there is a growing impetus to integrate IoT networks on a global scale, using satellites to expand the range of IoT connectivity into geographically remote areas. Ensuring the security of satellite backhaul for IoT networks is thus of paramount importance. The steady advance of quantum computing in recent years threatens to nullify classical cryptographic approaches based on assumptions of computational hardness, motivating the need for post-quantum cryptography. Quantum computing algorithms have been developed that, once a quantum computer of sufficient scale is realised, will be able to break classical cryptosystems efficiently (at polynomial-time complexity). A promising method of securing information against this threat at the physical layer has emerged in the form of quantum key distribution (QKD). QKD exploits the fundamental physical properties of light to guarantee information-theoretic security. Research into the application and standardisation of QKD to secure satellite backhaul, however, is still in its infancy. This paper presents a brief overview of the theoretical basis for QKD, whilst also providing a survey of contemporary QKD protocols. It evaluates the ability of these protocols to secure satellite backhaul in the context of a typical satellite-IoT network architecture. Furthermore, it highlights the vulnerabilities, as well as the technical challenges associated with this endeavour. Finally, it proposes directions for future research and development into protocols and standardisation for the satellite-integrated IoT domain. Several challenges must be overcome before QKD can evolve into a global-scale solution for securing satellite-IoT. Secret key generation rate remains very low in practical demonstrations of trusted-relay QKD satellite architectures. Further research is needed to overcome or mitigate the fundamental rate-distance trade-off before satellite QKD can be considered practicable in an IoT application. Alternatives that do not rely on trusted nodes are contingent on nascent technologies such as quantum repeaters and quantum memory. Whilst in theory QKD provides perfect information-theoretic security, it remains vulnerable to attacks that exploit imperfections in real-world equipment. Further effort is needed to develop QKD protocols that can safeguard against the aforementioned challenges.

Privacy of Synthetic Data: A Statistical Framework

Privacy-preserving data analysis is emerging as a challenging problem with far-reaching impact. In particular, synthetic data are a promising concept toward solving the aporetic conflict between data privacy and data sharing. Yet, it is known that accurately generating private, synthetic data of certain kinds is NP-hard. We develop a statistical framework for differentially private synthetic data, which enables us to circumvent the computational hardness of the problem. We consider the true data as a random sample drawn from a population <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Omega $ </tex-math></inline-formula> according to some unknown density. We then replace <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Omega $ </tex-math></inline-formula> by a much smaller random subset <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Omega ^{\ast}$ </tex-math></inline-formula> , which we sample according to some known density. We generate synthetic data on the reduced space <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\Omega ^{\ast}$ </tex-math></inline-formula> by fitting the specified linear statistics obtained from the true data. To ensure privacy we use the common Laplacian mechanism. Employing the concept of Rényi condition number, which measures how well the sampling distribution is correlated with the population distribution, we derive explicit bounds on the privacy and accuracy provided by the proposed method.

On the computational hardness of the code equivalence problem in cryptography

&lt;p style='text-indent:20px;'&gt;Code equivalence is a well-known concept in coding theory. Recently, literature saw an increased interest in this notion, due to the introduction of protocols based on the hardness of finding the equivalence between two linear codes. In this paper, we analyze the security of code equivalence, with a special focus on the hardest instances, in the interest of cryptographic usage. Our work stems from a thorough review of existing literature, identifies the various types of solvers for the problem, and provides a precise complexity analysis, where previously absent. Furthermore, we are able to improve on the state of the art, providing more efficient algorithm variations, for which we include numerical simulation data. In the end, the goal of this paper is to provide a complete, single point of access, which can be used as a tool for designing schemes that rely on the code equivalence problem.&lt;/p&gt;

A Computational Complexity Perspective on Segmentation as a Cognitive Subcomputation.

Computational feasibility is a widespread concern that guides the framing and modeling of natural and artificial intelligence. The specification of cognitive system capacities is often shaped by unexamined intuitive assumptions about the search space and complexity of a subcomputation. However, a mistaken intuition might make such initial conceptualizations misleading for what empirical questions appear relevant later on. We undertake here computational-level modeling and complexity analyses of segmentation - a widely hypothesized subcomputation that plays a requisite role in explanations of capacities across domains, such as speech recognition, music cognition, active sensing, event memory, action parsing, and statistical learning - as a case study to show how crucial it is to formally assess these assumptions. We mathematically prove two sets of results regarding computational hardness and search space size that may run counter to intuition, and position their implications with respect to existing views on thesubcapacity.

Integer Factorization: Why Two-Item Joint Replenishment Is Hard

Joint replenishment problems constitute an important class of models in inventory management. They exhibit aspects of possible coordination among multiple products to save costs. Their computational complexity had been open even if there are just two products that need to be synced. In “Integer factorization: Why two-item joint replenishment is hard,” Schulz and Telha present a simple framework based on integer factorization to establish the computational hardness of two variants of the joint replenishment problem with two items. Whereas difficult to solve in practice and not believed to be solvable in polynomial time, integer factorization is not as difficult as NP-complete problems. The authors show that a similar technique can be used to show even the NP-completeness of one variant of the joint replenishment problem (again with just two items).

Disordered systems insights on computational hardness

In this review article we discuss connections between the physics of disordered systems, phase transitions in inference problems, and computational hardness. We introduce two models representing the behavior of glassy systems, the spiked tensor model and the generalized linear model. We discuss the random (non-planted) versions of these problems as prototypical optimization problems, as well as the planted versions (with a hidden solution) as prototypical problems in statistical inference and learning. Based on ideas from physics, many of these problems have transitions where they are believed to jump from easy (solvable in polynomial time) to hard (requiring exponential time). We discuss several emerging ideas in theoretical computer science and statistics that provide rigorous evidence for hardness by proving that large classes of algorithms fail in the conjectured hard regime. This includes the overlap gap property, a particular mathematization of clustering or dynamical symmetry-breaking, which can be used to show that many algorithms that are local or robust to changes in their input fail. We also discuss the sum-of-squares hierarchy, which places bounds on proofs or algorithms that use low-degree polynomials such as standard spectral methods and semidefinite relaxations, including the Sherrington–Kirkpatrick model. Throughout the manuscript we present connections to the physics of disordered systems and associated replica symmetry breaking properties.

Embedding Divisor and Semi-Prime Testability in f-Vectors of Polytopes

We obtain computational hardness results for f-vectors of polytopes by exhibiting reductions of the problems DIVISOR and SEMI-PRIME TESTABILITY to problems on f-vectors of polytopes. Further, we show that the corresponding problems for f-vectors of simplicial polytopes are polytime solvable. The situation where we prove this computational difference (conditioned on standard conjectures on the density of primes and on \(P \ne NP \)) is when the dimension d tends to infinity and the number of facets is linear in d.

Blockchain-Based Voting Considered Harmful?

Blockchains (BC) and Distributed Ledgers (DL) offer favorable properties, especially immutability and decentralization, which are suitable for voting systems’ Bulletin Boards (BB). In recent years, an influx of BC-based voting systems have been observed. Distributing trust among multiple trustees is a crucial reason to adopt BCs and DLs in voting systems. The practical deployment must be decentralized, too, and not just done through virtualizing interconnected systems. As discussed widely, adopting a BC or DL can incur threats to a system that assumed a trusted and centralized Public Bulletin Board (PBB). Therefore, the exploitation of BCs or DLs requires careful consideration of cryptographic mechanisms and the overall system design, as well as the adversary model. Besides these operational necessities, the long-term privacy of ballots is essential. Thus, the key question investigated in this article is: Can BC, and DL-based voting systems be considered harmful? Hence, first (i) requirements of BC-based voting systems are provided, followed by (ii) terminology definitions, and (iii) complemented by the design and implementation of a fully decentralized voting system: Æternum, which achieves Unconditional Privacy (UP) and neither relies on computational hardness assumptions nor on a trusted Trusted Third Party (TTP). Achieving UP is crucial because future adversaries may be able to break hardness assumptions. Æternum does not present a Single-Point-of-Failure (SPoF) either, since (i) the PBB in use is based on a permissioned DL, and (ii) the final tally and proofs can be verified by anyone, without requiring trust in any authority.

Online learning for min-max discrete problems

We study various discrete nonlinear combinatorial optimization problems in an online learning framework. In the first part, we address the computational complexity of designing vanishing regret (and vanishing approximate regret) algorithms. We provide a general reduction showing that many (min-max) polynomial time solvable problems not only do not have a vanishing regret, but also no vanishing approximation α-regret, for some α, unless NP=RP. In particular, for the min-max version of the vertex cover problem, which is solvable in polynomial time in the offline case, our reduction implies that there is no (2−ϵ)-regret online randomized algorithm unless Unique Game is in RP. Besides, we prove that the bound is tight by providing an online efficient algorithm based on the online gradient descent method. In the second part, we turn our attention to online learning algorithms that are based on an offline optimization oracle that, given a set of multiple instances of the problem, is able to compute the optimum static solution that performs best on the set of instances overall. We show that for several min-max (nonlinear) discrete optimization problems, it is strongly NP-hard to solve the offline optimization oracle, even for problems that can be solved in polynomial time in the single-instance static case (e.g. min-max vertex cover, min-max perfect matching, etc.). This also provides a useful insight into the connection between the non-linear nature of some problems and the drastic change of their computational hardness when moved to an online learning setting.

Task-independent metrics of computational hardness predict human cognitive performance

The survival of human organisms depends on our ability to solve complex tasks in the face of limited cognitive resources. However, little is known about the factors that drive the complexity of those tasks. Here, building on insights from computational complexity theory, we quantify the computational hardness of cognitive tasks using a set of task-independent metrics related to the computational resource requirements of individual instances of a task. We then examine the relation between those metrics and human behavior and find that they predict both time spent on a task as well as accuracy in three canonical cognitive tasks. Our findings demonstrate that performance in cognitive tasks can be predicted based on generic metrics of their inherent computational hardness.

Fixation Maximization in the Positional Moran Process

The Moran process is a classic stochastic process that models invasion dynamics on graphs. A single mutant (e.g., a new opinion, strain, social trait etc.) invades a population of residents spread over the nodes of a graph. The mutant fitness advantage δ&gt;=0 determines how aggressively mutants propagate to their neighbors. The quantity of interest is the fixation probability, i.e., the probability that the initial mutant eventually takes over the whole population. However, in realistic settings, the invading mutant has an advantage only in certain locations. E.g., the ability to metabolize a certain sugar is an advantageous trait to bacteria only when the sugar is actually present in their surroundings. In this paper we introduce the positional Moran process, a natural generalization in which the mutant fitness advantage is only realized on specific nodes called active nodes, and study the problem of fixation maximization: given a budget k, choose a set of k active nodes that maximize the fixation probability of the invading mutant. We show that the problem is NP-hard, while the optimization function is not submodular, thus indicating strong computational hardness. We focus on two natural limits. In the limit of δ to infinity (strong selection), although the problem remains NP-hard, the optimization function becomes submodular and thus admits a constant-factor approximation using a simple greedy algorithm. In the limit of δ to 0 (weak selection), we show that we can obtain a tight approximation in O(n^{2×ω}) time, where ω is the matrix-multiplication exponent. An experimental evaluation of the new algorithms along with some proposed heuristics corroborates our results.

Taming the Communication and Computation Complexity of Combinatorial Auctions: The FUEL Bid Language

Combinatorial auctions have found widespread application for allocating multiple items in the presence of complex bidder preferences. The enumerative exclusive OR (XOR) bid language is the de facto standard bid language for spectrum auctions and other applications, despite the difficulties, in larger auctions, of enumerating all the relevant packages or solving the resulting NP-hard winner determination problem. We introduce the flexible use and efficient licensing (FUEL) bid language, which was proposed for radio spectrum auctions to ease both communications and computations compared with XOR-based auctions. We model the resulting allocation problem as an integer program, discuss computational complexity, and conduct an extensive set of computational experiments, showing that the winner determination problem of the FUEL bid language can be solved reliably for large realistic-sized problem instances in less than half an hour on average. In contrast, auctions with an XOR bid language quickly become intractable even for much smaller problem sizes. We compare a sealed-bid FUEL auction to a sealed-bid auction with an XOR bid language and to a simultaneous clock auction. The sealed-bid auction with an XOR bid language incurs significant welfare losses because of the missing bids problem and computational hardness, the simultaneous clock auction leads to a substantially lower efficiency than FUEL because of the exposure problem. This paper was accepted by Axel Ockenfels, behavioral economics and decision analysis. Funding: This work was supported by Deutsche Forschungsgemeinschaft [Grant BI 1057-1/8]. P. Milgrom gratefully acknowledges support from the U.S. National Science Foundation [Grant SES-1947514]. M. Bichler and G. Schwarz was supported by the German Research Foundation [Grants BI 1057 I-9 and 277991500]. Supplemental Material: The data files are available at https://doi.org/10.1287/mnsc.2022.4465 .

Fermion Sampling: A Robust Quantum Computational Advantage Scheme Using Fermionic Linear Optics and Magic Input States

Fermionic Linear Optics (FLO) is a restricted model of quantum computation which in its original form is known to be efficiently classically simulable. We show that, when initialized with suitable input states, FLO circuits can be used to demonstrate quantum computational advantage with strong hardness guarantees. Based on this, we propose a quantum advantage scheme which is a fermionic analogue of Boson Sampling: Fermion Sampling with magic input states. We consider in parallel two classes of circuits: particle-number conserving (passive) FLO and active FLO that preserves only fermionic parity and is closely related to Matchgate circuits introduced by Valiant. Mathematically, these classes of circuits can be understood as fermionic representations of the Lie groups $U(d)$ and $SO(2d)$. This observation allows us to prove our main technical results. We first show anticoncentration for probabilities in random FLO circuits of both kind. Moreover, we prove robust average-case hardness of computation of probabilities. To achieve this, we adapt the worst-to-average-case reduction based on Cayley transform, introduced recently by Movassagh, to representations of low-dimensional Lie groups. Taken together, these findings provide hardness guarantees comparable to the paradigm of Random Circuit Sampling. Importantly, our scheme has also a potential for experimental realization. Both passive and active FLO circuits are relevant for quantum chemistry and many-body physics and have been already implemented in proof-of-principle experiments with superconducting qubit architectures. Preparation of the desired quantum input states can be obtained by a simple quantum circuit acting independently on disjoint blocks of four qubits and using 3 entangling gates per block. We also argue that due to the structured nature of FLO circuits, they can be efficiently certified.

Sparse high-dimensional linear regression. Estimating squared error and a phase transition

We consider a sparse high-dimensional regression model where the goal is to recover a k-sparse unknown binary vector β∗ from n noisy linear observations of the form Y=Xβ∗+W∈Rn where X∈Rn×p has i.i.d. N(0,1) entries and W∈Rn has i.i.d. N(0,σ2) entries. In the high signal-to-noise ratio regime and sublinear sparsity regime, while the order of the sample size needed to recover the unknown vector information-theoretically is known to be n∗:=2klogp/log(k/σ2+1), no polynomial-time algorithm is known to succeed unless n>nalg:=(2k+σ2)logp. In this work, we offer a series of results investigating multiple computational and statistical aspects of the recovery task in the regime n∈[n∗,nalg]. First, we establish a novel information-theoretic property of the MLE of the problem happening around n=n∗ samples, which we coin as an “all-or-nothing behavior”: when n>n∗ it recovers almost perfectly the support of β∗, while if n<n∗ it fails to recover any fraction of it correctly. Second, at an attempt to understand the computational hardness in the regime n∈[n∗,nalg], we prove that at order nalg samples there is an Overlap Gap Property (OGP) phase transition occurring at the landscape of the MLE: for constants c,C>0 when n<cnalg OGP appears in the landscape of MLE while if n>Cnalg OGP disappears. OGP is a geometric “disconnectivity” property, which initially appeared in the theory of spin glasses and is known to suggest algorithmic hardness when it occurs. Finally, using certain technical results obtained to establish the OGP phase transition, we additionally establish various novel positive and negative algorithmic results for the recovery task of interest, including the failure of LASSO with access to n<cnalg samples and the success of a simple local search method with access to n>Cnalg samples.

Tensor clustering with planted structures: Statistical optimality and computational limits

This paper studies the statistical and computational limits of high-order clustering with planted structures. We focus on two clustering models, constant high-order clustering (CHC) and rank-one higher-order clustering (ROHC), and study the methods and theory for testing whether a cluster exists (detection) and identifying the support of cluster (recovery). Specifically, we identify the sharp boundaries of signal-to-noise ratio for which CHC and ROHC detection/recovery are statistically possible. We also develop the tight computational thresholds: when the signal-to-noise ratio is below these thresholds, we prove that polynomial-time algorithms cannot solve these problems under the computational hardness conjectures of hypergraphic planted clique (HPC) detection and hypergraphic planted dense subgraph (HPDS) recovery. We also propose polynomial-time tensor algorithms that achieve reliable detection and recovery when the signal-to-noise ratio is above these thresholds. Both sparsity and tensor structures yield the computational barriers in high-order tensor clustering. The interplay between them results in significant differences between high-order tensor clustering and matrix clustering in literature in aspects of statistical and computational phase transition diagrams, algorithmic approaches, hardness conjecture, and proof techniques. To our best knowledge, we are the first to give a thorough characterization of the statistical and computational trade-off for such a double computational-barrier problem. Finally, we provide evidence for the computational hardness conjectures of HPC detection (via low-degree polynomial and Metropolis methods) and HPDS recovery (via low-degree polynomial method).

Freezing, Bounded-Change and Convergent Cellular Automata

This paper studies three classes of cellular automata from a computational point of view: freezing cellular automata where the state of a cell can only decrease according to some order on states, cellular automata where each cell only makes a bounded number of state changes in any orbit, and finally cellular automata where each orbit converges to some fixed point. Many examples studied in the literature fit into these definitions, in particular the works on cristal growth started by S. Ulam in the 60s. The central question addressed here is how the computational power and computational hardness of basic properties is affected by the constraints of convergence, bounded number of change, or local decreasing of states in each cell. By studying various benchmark problems (short-term prediction, long term reachability, limits) and considering various complexity measures and scales (LOGSPACE vs. PTIME, communication complexity, Turing computability and arithmetical hierarchy) we give a rich and nuanced answer: the overall computational complexity of such cellular automata depends on the class considered (among the three above), the dimension, and the precise problem studied. In particular, we show that all settings can achieve universality in the sense of Blondel-Delvenne-K\r{u}rka, although short term predictability varies from NLOGSPACE to P-complete. Besides, the computability of limit configurations starting from computable initial configurations separates bounded-change from convergent cellular automata in dimension~1, but also dimension~1 versus higher dimensions for freezing cellular automata. Another surprising dimension-sensitive result obtained is that nilpotency becomes decidable in dimension~ 1 for all the three classes, while it stays undecidable even for freezing cellular automata in higher dimension.

Computing Maximum Matchings in Temporal Graphs

Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph $G$, a temporal graph is represented by assigning a set of integer time-labels to every edge $e$ of $G$, indicating the discrete time steps at which $e$ is active. We introduce and study the complexity of a natural temporal extension of the classical graph problem Maximum Matching, taking into account the dynamic nature of temporal graphs. In our problem, Maximum Temporal Matching, we are looking for the largest possible number of time-labeled edges (simply time-edges) $(e,t)$ such that no vertex is matched more than once within any time window of $\Delta$ consecutive time slots, where $\Delta \in \mathbb{N}$ is given. The requirement that a vertex cannot be matched twice in any $\Delta$-window models some necessary recovery period that needs to pass for an entity (vertex) after being paired up for some activity with another entity. We prove strong computational hardness results for Maximum Temporal Matching, even for elementary cases. To cope with this computational hardness, we mainly focus on fixed-parameter algorithms with respect to natural parameters, as well as on polynomial-time approximation algorithms.