Polynomial Time Research Articles

Spanning trees for many different numbers of leaves

Let $G$ be a connected graph and $L(G)$ the set of all integers $k$ such that $G$ contains a spanning tree with exactly $k$ leaves. We show that for a connected graph $G$, the set $L(G)$ is contiguous. It follows from work of Chen, Ren, and Shan that every connected and locally connected $n$-vertex graph -- this includes triangulations -- has a spanning tree with at least $n/2 + 1$ leaves, so by a classic theorem of Whitney and our result, in any plane $4$-connected $n$-vertex triangulation one can find for any integer $k$ which is at least $2$ and at most $n/2 + 1$ a spanning tree with exactly $k$ leaves (and each of these trees can be constructed in polynomial time). We also prove that there exist infinitely many $n$ such that there is a plane $4$-connected $n$-vertex triangulation containing a spanning tree with $2n/3$ leaves, but no spanning tree with more than $2n/3$ leaves.

The Compact Genetic Algorithm Struggles on Cliff Functions

AbstractEstimation of distribution algorithms (EDAs) are general-purpose optimizers that maintain a probability distribution over a given search space. This probability distribution is updated through sampling from the distribution and a reinforcement learning process which rewards solution components that have shown to be part of good quality samples. The compact genetic algorithm (cGA) is a non-elitist EDA able to deal with difficult multimodal fitness landscapes that are hard to solve by elitist algorithms. We investigate the cGA on the Cliff function for which it was shown recently that non-elitist evolutionary algorithms and artificial immune systems optimize it in expected polynomial time. We point out that the cGA faces major difficulties when solving the Cliff function and investigate its dynamics both experimentally and theoretically. Our experimental results indicate that the cGA requires exponential time for all values of the update strength 1/K. We show theoretically that, under sensible assumptions, there is a negative drift when sampling around the location of the cliff. Experiments further suggest that there is a phase transition for K where the expected optimization time drops from $$n^{\Theta (n)}$$ n Θ ( n ) to $$2^{\Theta (n)}$$ 2 Θ ( n ) .

Minimum-Energy Scheduling of Flexible Job-Shop Through Optimization and Comprehensive Heuristic

This study considers a flexible job-shop scheduling problem where energy cost savings are the primary objective and where the classical objective of the minimization of the make-span is replaced by the satisfaction of due times for each job. An original two-level mixed-integer formulation of this optimization problem is proposed, where the processed flows of material and their timing are explicitly considered. Its exact solution is discussed, and, considering its computational complexity, a comprehensive heuristic, balancing energy performance and due time constraint satisfaction, is developed to provide acceptable solutions in polynomial time to the minimum-energy flexible job-shop scheduling problem, even when considering its dynamic environment. The proposed approach is illustrated through a small-scale example.

On variants of a load-balancing problem with unit-load jobs

AbstractIn this paper, we reconsider an offline load-balancing problem with unit-time jobs that require one unit from a common resource throughout their execution. In the unit-time case, the jobs have to be assigned to time-slots such that a separable convex function of the load of the resource has to be minimized. Variants of this problem have been studied extensively in the literature under different names. We briefly discuss these problems and give a new implementation for one of them with a better worst-case time complexity than any of the known methods. We also consider the more general preemptive problem in which the execution of the jobs can be interrupted and resumed later. Furthermore, we divide the time horizon into disjoint time intervals, and for each interval, a separable convex cost function is given. The jobs have to be scheduled within their feasible intervals preemptively such that the total cost is minimized, where the cost is determined separately for each interval by the corresponding cost function. We show how to solve this problem in polynomial time by a single minimum-cost-flow computation. For the preemptive problem with one cost function only, we propose a proprietary algorithm for finding a feasible solution which is optimal for any convex cost function. We also present some qualitative computational results.

On the quantum security of high-dimensional RSA protocol

Abstract The idea of extending the classical RSA protocol using algebraic number fields was introduced by Takagi and Naito (Construction of RSA cryptosystem over the algebraic field using ideal theory and investigation of its security. Electron Commun Japan Part III Fund Electr Sci. 2000;83:19–29). Recently, Zheng et al. proposed the use of the ring of algebraic integers of an algebraic number field and the lattice theory to present a high-dimensional form of RSA. The authors claim that their proposal is post-quantum and is significant both from the theoretical and practical point of view. In this article, we prove that the security of Zheng et al.’s scheme is still based on the factorization problem, and we present a practical quantum attack on this proposed scheme, our attack is a quantum polynomial time algorithm that employs Shor’s algorithm as a subroutine.

Constructing Concise Characteristic Samples for Acceptors of Omega Regular Languages

A characteristic sample for a language $L$ and a learning algorithm $\textbf{L}$ is a finite sample of words $T_L$ labeled by their membership in $L$ such that for any sample $T \supseteq T_L$ consistent with $L$, on input $T$ the learning algorithm $\textbf{L}$ returns a hypothesis equivalent to $L$. Which omega automata have characteristic sets of polynomial size, and can these sets be constructed in polynomial time? We address these questions here. In brief, non-deterministic omega automata of any of the common types, in particular B\"uchi, do not have characteristic samples of polynomial size. For deterministic omega automata that are isomorphic to their right congruence automata, the fully informative languages, polynomial time algorithms for constructing characteristic samples and learning from them are given. The algorithms for constructing characteristic sets in polynomial time for the different omega automata (of types B\"uchi, coB\"uchi, parity, Rabin, Street, or Muller), require deterministic polynomial time algorithms for (1) equivalence of the respective omega automata, and (2) testing membership of the language of the automaton in the informative classes, which we provide.

Optimal approximate minimization of one-letter weighted finite automata

Abstract In this paper, we study the approximate minimization problem of weighted finite automata (WFAs): to compute the best possible approximation of a WFA given a bound on the number of states. By reformulating the problem in terms of Hankel matrices, we leverage classical results on the approximation of Hankel operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory. We solve the optimal spectral-norm approximate minimization problem for irredundant WFAs with real weights, defined over a one-letter alphabet. We present a theoretical analysis based on AAK theory and bounds on the quality of the approximation in the spectral norm and $\ell ^2$ norm. Moreover, we provide a closed-form solution, and an algorithm, to compute the optimal approximation of a given size in polynomial time.

A Long-Memory Model for Multiple Cycles with an Application to the US Stock Market

This paper proposes a long-memory model that includes multiple cycles in addition to the long-run component. Specifically, instead of a single pole or singularity in the spectrum, it allows for multiple poles and, thus, different cycles with different degrees of persistence. It also incorporates non-linear deterministic structures in the form of Chebyshev polynomials in time. Simulations are carried out to analyze the finite sample properties of the proposed test, which is shown to perform well in the case of a relatively large sample with at least 1000 observations. The model is then applied to weekly data on the S&P 500 from 1 January 1970 to 26 October 2023 as an illustration. The estimation results based on the first differenced logged values (i.e., the returns) point to the existence of three cyclical structures in the series, with lengths of approximately one month, one year, and four years, respectively, and to orders of integration in the range (0, 0.20), which implies stationary long memory in all cases.

Fixed-Ratio Approximation Algorithm for the Minimum Cost Cover of a Digraph by Bounded Number of Cycles

We consider polynomial time approximation for the minimum cost cycle cover problem of an edge-weighted digraph, where feasible covers are restricted to have at most k disjoint cycles. In the literature this problem is referred to as Min-k-SCCP. The problem is closely related to classic Traveling Salesman Problem (TSP) and Vehicle Routing Problem (VRP) and has many important applications in algorithms design and operations research. Unlike its unconstrained variant, the Min-k-SCCP is strongly NP-hard even on undirected graphs and remains intractable in very specific settings. For any metric, the problem can be approximated in polynomial time within ratio 2, while in fixed-dimensional Euclidean spaces it admits Polynomial Time Approximation Schemes (PTAS). In the same time, approximation of the more general asymmetric Min-k-SCCP still remains weakly studied. In this paper, we propose the first fixed-ratio approximation algorithm for this problem, which extends the recent breakthrough Svensson-Tarnawski-Vegh and Traub-Vygen results for the Asymmetric Traveling Salesman Problem.

Fast Algorithm for Cyber-Attack Estimation and Attack Path Extraction Using Attack Graphs with AND/OR Nodes

This paper studies the security issues for cyber–physical systems, aimed at countering potential malicious cyber-attacks. The main focus is on solving the problem of extracting the most vulnerable attack path in a known attack graph, where an attack path is a sequence of steps that an attacker can take to compromise the underlying network. Determining an attacker’s possible attack path is critical to cyber defenders as it helps identify threats, harden the network, and thwart attacker’s intentions. We formulate this problem as a path-finding optimization problem with logical constraints represented by AND and OR nodes. We propose a new Dijkstra-type algorithm that combines elements from Dijkstra’s shortest path algorithm and the critical path method. Although the path extraction problem is generally NP-hard, for the studied special case, the proposed algorithm determines the optimal attack path in polynomial time, O(nm), where n is the number of nodes and m is the number of edges in the attack graph. To our knowledge this is the first exact polynomial algorithm that can solve the path extraction problem for different attack graphs, both cycle-containing and cycle-free. Computational experiments with real and synthetic data have shown that the proposed algorithm consistently and quickly finds optimal solutions to the problem.

Polynomial Time Convergence of the Iterative Evaluation of Datalogo Programs

Polynomial Time Convergence of the Iterative Evaluation of Datalogo Programs

Complexity Results on the Decomposition of a Digraph into Directed Linear Forests and Out-Stars

We consider two decomposition problems in directed graphs. We say that a digraph is $k$-bounded for some $k \in \mathbb{Z}_{\geq 1}$ if each of its connected components contains at most $k$ arcs. For the first problem, a directed linear forest is a collection of vertex-disjoint directed paths and we consider the problem of decomposing a given digraph into a $k$-bounded and an $\ell$-bounded directed linear forest for some fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. We give a full dichotomy for this problem by showing that it can be solved in polynomial time if $k+\ell \leq 3$ and is NP-complete otherwise. This answers a question of Campbell, Hörsch, and Moore. For the second problem, we say that an out-galaxy is a vertex-disjoint collection of out-stars. Again, we give a full dichotomy of when a given digraph can be edge-decomposed into a $k$-bounded and an $\ell$-bounded out-galaxy for fixed $k,\ell \in \mathbb{Z}_{\geq 1}\cup \{\infty\}$. More precisely, we show that the problem can be solved in polynomial time if $\min\{k,\ell\}\in \{1,\infty\}$ and is NP-complete otherwise.

Circuit and graver walks and linear and integer programming

We show that a circuit walk from a given feasible point of a given linear program to an optimal point can be computed in polynomial time using only linear algebra operations and the solution of the single given linear program. We also show that a Graver walk from a given feasible point of a given integer program to an optimal point is polynomial time computable using an integer programming oracle, but without such an oracle, it is hard to compute such a walk even if an optimal solution to the given program is given as well. Combining our oracle algorithm with recent results on sparse integer programming, we also show that Graver walks from any point are polynomial time computable over matrices of bounded tree-depth and subdeterminants.

Polynomial time algorithms to find Pareto optimal schedules of bicriteria lot scheduling problems with splitable jobs on a single parallel-batch machine

Polynomial time algorithms to find Pareto optimal schedules of bicriteria lot scheduling problems with splitable jobs on a single parallel-batch machine

Backwards-reachability for cooperating multi-pushdown systems

A cooperating multi-pushdown system consists of a tuple of pushdown systems that can delegate the execution of recursive procedures to sub-tuples; control returns to the calling tuple once all sub-tuples finished their task. This allows the concurrent execution since disjoint sub-tuples can perform their task independently. Because of the concrete form of recursive descent into sub-tuples, the content of the multi-pushdown does not form an arbitrary tuple of words, but can be understood as a Mazurkiewicz trace. For such systems, we prove that the backwards reachability relation efficiently preserves recognizability, generalizing a result and proof technique by Bouajjani et al. for single-pushdown systems. It follows that the reachability relation is decidable for cooperating multi-pushdown systems in polynomial time and the same holds, e.g., for safety and liveness properties given by recognizable sets of configurations.

New Cryptanalysis of Prime Power RSA with Two Private Exponents

Prime Power RSA is a variant of the RSA scheme due to Takagi with modulus N=prq for r⩾2, where p,q are of the same bit-size. In this paper, we concentrate on one type of Prime Power RSA which assumes e·d≡1modpr−1(p−1)(q−1). Two new attacks on this type of Prime Power RSA are presented when given two pairs of public and private exponents, namely, (e1,d1) and (e2,d2) with the same modulus N. Suppose that d1<Nβ1,d2<Nβ2. In 2015, Zheng and Hu showed that when β1β2<(r−1)3/(r+1)3, N may be factored in probabilistic polynomial time. The first attack of this paper shows that one can obtain the factorization of N in probabilistic polynomial time, provided that β1β2<r/(r+1)3. Later, in the second attack, we improve both the first attack and the attack of Zheng and Hu, and show that the condition β1β2<r(r−1)2/(r+1)3 already suffices to break the Prime Power RSA. By introducing multiple parameters, our lattice constructions take full advantage of known information, and obtain the best known attack. Specifically, we make full use of the information that pr is a divisor of N, while the attack of Zheng and Hu only assumes that pr−1 is a divisor of N. As a consequence, this method implies a better lattice construction, and thus improves the previous attack. The experiments which reach a better upper bound than before also verify it. Our approaches are based on Coppersmith’s method for finding small roots of bivariate modular polynomial equations.

Expected 1.x Makespan-Optimal Multi-Agent Path Finding on Grid Graphs in Low Polynomial Time

Multi-Agent Path Finding (MAPF) is NP-hard to solve optimally, even on graphs, suggesting no polynomial-time algorithms can compute exact optimal solutions for them. This raises a natural question: How optimal can polynomial-time algorithms reach? Whereas algorithms for computing constant-factor optimal solutions have been developed, the constant factor is generally very large, limiting their application potential. In this work, among other breakthroughs, we propose the first low-polynomial-time MAPF algorithms delivering 1-1.5 (resp., 1-1.67) asymptotic makespan optimality guarantees for 2D (resp., 3D) grids for random instances at a very high 1/3 agent density, with high probability. Moreover, when regularly distributed obstacles are introduced, our methods experience no performance degradation. These methods generalize to support 100% agent density. Regardless of the dimensionality and density, our high-quality methods are enabled by a unique hierarchical integration of two key building blocks. At the higher level, we apply the labeled Grid Rearrangement Algorithm (GRA), capable of performing efficient reconfiguration on grids through row/column shuffles. At the lower level, we devise novel methods that efficiently simulate row/column shuffles returned by GRA. Our implementations of GRA-based algorithms are highly effective in extensive numerical evaluations, demonstrating excellent scalability compared to other SOTA methods. For example, in 3D settings, GRA-based algorithms readily scale to grids with over 370,000 vertices and over 120,000 agents and consistently achieve conservative makespan optimality approaching 1.5, as predicted by our theoretical analysis.

Advances on strictly $$\Delta $$-modular IPs

AbstractThere has been significant work recently on integer programs (IPs) $$\min \{c^\top x :Ax\le b,\,x\in \mathbb {Z}^n\}$$ min { c ⊤ x : A x ≤ b , x ∈ Z n } with a constraint marix A with bounded subdeterminants. This is motivated by a well-known conjecture claiming that, for any constant $$\Delta \in \mathbb {Z}_{>0}$$ Δ ∈ Z > 0 , $$\Delta $$ Δ -modular IPs are efficiently solvable, which are IPs where the constraint matrix $$A\in \mathbb {Z}^{m\times n}$$ A ∈ Z m × n has full column rank and all $$n\times n$$ n × n minors of A are within $$\{-\Delta , \dots , \Delta \}$$ { - Δ , ⋯ , Δ } . Previous progress on this question, in particular for $$\Delta =2$$ Δ = 2 , relies on algorithms that solve an important special case, namely strictly$$\Delta $$ Δ -modular IPs, which further restrict the $$n\times n$$ n × n minors of A to be within $$\{-\Delta , 0, \Delta \}$$ { - Δ , 0 , Δ } . Even for $$\Delta =2$$ Δ = 2 , such problems include well-known combinatorial optimization problems like the minimum odd/even cut problem. The conjecture remains open even for strictly $$\Delta $$ Δ -modular IPs. Prior advances were restricted to prime $$\Delta $$ Δ , which allows for employing strong number-theoretic results. In this work, we make first progress beyond the prime case by presenting techniques not relying on such strong number-theoretic prime results. In particular, our approach implies that there is a randomized algorithm to check feasibility of strictly $$\Delta $$ Δ -modular IPs in strongly polynomial time if $$\Delta \le 4$$ Δ ≤ 4 .

Effect of participant baseline characteristics on adherence to self-monitoring of diet, physical activity and weight in a 12-month weight loss mHealth trial

Abstract Background Self-monitoring (SM) of lifestyle behaviors is the centerpiece of standard behavioral treatment (SBT) for weight loss as SM increases one’s awareness of their behaviors. Extensive evidence demonstrates a significant relationship between adherence to SM and successful weight loss outcomes. Purpose Examine baseline characteristics of participants that are associated with higher adherence to SM of diet, physical activity (PA), and weight in a 12-month mHealth weight loss trial. Methods SMARTER was a single-site, 2-arm randomized controlled trial. We randomized 502 adults with a body mass index (BMI) 27-43 kg/m2 to diet, weight, and PA SM plus tailored feedback messages (SM+FB, n=251) or SM alone (n=251). We provided the SM+FB group feedback messages via the SMARTER app on diet daily, PA every other day and weight weekly. The study provided a wearable PA tracker and smart scale for weight and utilized a smartphone app for dietary SM. Adherence of dietary SM was defined as the % of days with ≥50% of calorie goal recorded and PA SM was % of days with ≥500 steps. Weight SM was the % of days with weight based on the scale record. Linear mixed modeling was applied to assess the monthly percent adherence to SM of diet, PA, and weight over a 12-month period. Our analysis incorporated time effects in months and polynomials of time, treatment group assignment (SM+FB vs. SM), baseline variables of interest (e.g., BMI, age, sex, race, education years, income), and the interactions between treatment and time effects with other variables. For parsimony, non-significant effects were removed. Results The sample (N=502) was 82% white (n=414), 79% female (n=399), and had a mean age of 45.0±14.4 years and BMI of 33.7±4.0 kg/m2. Over 12 months, percent adherence to SM of diet, PA, and weight declined non-linearly. Treatment groups did not demonstrate significant differences. The impact of treatment on the monthly decline in adherence to diet SM was curvilinear and varied significantly over 12 months. Older age was associated with greater adherence to the SM of diet (b=0.72, 95% CI: 0.55-0.88, P < .0001), PA (b=0.50, 95% CI: 0.35-0.66, P < .0001), and weight (b=0.71, 95% CI: 0.57-0.85, P < .0001). Conversely, higher baseline BMI was associated with lower adherence to SM of diet (b=-1.14, 95% CI: -1.73 to -0.55, P < .0001) and PA (b=-0.79, 95% CI: -1.35 to -0.22, P= .006). Other baseline variables were not significantly related to adherence. Conclusion The findings suggest that individuals with a higher BMI at baseline may need additional enhancement strategies to improve their engagement with the digital tools for SM lifestyle which could lead to improved weight outcomes.

Anchored rescheduling problem with non-availability periods

A set of jobs with precedence constraints is given along with a baseline schedule, resulting from the first-stage decision of a robust two-stage scheduling problem. Non-availability periods caused by unpredicted disruptions prevent jobs from being executed on certain time intervals, thus requiring rescheduling. The anchored rescheduling problem with non-availability periods is to find a schedule maximizing the number of so-called anchored jobs whose starting times remain close to the baseline schedule. It is shown to be solvable in polynomial time.