Polynomial-time Approximation Algorithm Research Articles

First-order reasoning and efficient semi-algebraic proofs

Semi-algebraic proof systems such as sum-of-squares (SoS) have attracted a lot of attention due to their relation to approximation algorithms: constant degree semi-algebraic proofs lead to conjecturally optimal polynomial-time approximation algorithms for important NP-hard optimization problems. Motivated by the need to allow a more streamlined and uniform framework for working with SoS proofs than the restrictive propositional level, we initiate a systematic first-order logical investigation into the kinds of reasoning possible in algebraic and semi-algebraic proof systems. Specifically, we develop first-order theories that capture in a precise manner constant degree algebraic and semi-algebraic proof systems: every statement of a certain form that is provable in our theories translates into a family of constant degree polynomial calculus or SoS refutations, respectively; and using a reflection principle, the converse also holds.This places algebraic and semi-algebraic proof systems in the established framework of bounded arithmetic, while providing theories corresponding to systems that vary quite substantially from the usual propositional-logic ones.We give examples of how our semi-algebraic theory proves statements such as the pigeonhole principle, we provide a separation between algebraic and semi-algebraic theories, and we describe initial attempts to go beyond these theories by introducing extensions that use the inequality symbol, identifying along the way which extensions lead outside the scope of constant degree SoS. Moreover, we prove new results for propositional proofs, and specifically extend Berkholz's dynamic-by-static simulation of polynomial calculus (PC) by SoS to PC with the radical rule.

A fully polynomial time approximation scheme for a stop schedule of a transportation line without branches

For a transportation line without branches, passenger service scheduling is mainly concerned with the determination of stop schedules for planned service trips. A mixed-integer programming method is proposed to maximize customer satisfaction by minimizing the total number of in-vehicle passengers during trips. A heuristic algorithm is proposed for a given number of service trips, in which the error bound of the heuristic is restricted by the number of stations. When the given number of service trips is sufficient to provide express services for all origin–destination pairs, the heuristic provides an optimal solution. However, the error bound of the heuristic approaches infinity as the number of stations increases. A fully polynomial time-approximation algorithm is proposed, where the number of service trips is a decision variable. Finally, the proposed algorithm was applied to the stop-schedule problem of a river bus operating along the Han River in Seoul, Korea.

Approximation Algorithms for Preference Aggregation Using CP-Nets

This paper studies the design and analysis of approximation algorithms for aggregating preferences over combinatorial domains, represented using Conditional Preference Networks (CP-nets). Its focus is on aggregating preferences over so-called swaps, for which optimal solutions in general are already known to be of exponential size. We first analyze a trivial 2-approximation algorithm that simply outputs the best of the given input preferences, and establish a structural condition under which the approximation ratio of this algorithm is improved to 4/3. We then propose a polynomial-time approximation algorithm whose outputs are provably no worse than those of the trivial algorithm, but often substantially better. A family of problem instances is presented for which our improved algorithm produces optimal solutions, while, for any ε, the trivial algorithm cannot attain a (2- ε)-approximation. These results may lead to the first polynomial-time approximation algorithm that solves the CP-net aggregation problem for swaps with an approximation ratio substantially better than 2.

Approximation Algorithms for Fair k-median Problem without Fairness Violation

The fair k-median problem is one of the important clustering problems. The current best approximation ratio is 4.675 for this problem with 1-fairness violation, which was proposed by Bercea et al. [APPROX-RANDOM'2019]. To our best knowledge, there is no available approximation algorithm for this problem without any fairness violation in doubling metrics. In this paper, we consider the fair k-median problem in doubling metrics and general metrics. We provide the first QPTAS for the fair k-median problem based on the hierarchical decomposition and dynamic programming process in doubling metrics. For applying a dynamic programming process to solve this problem, the distances from portals to facilities cannot be directly enumerated since each client may not be assigned to its closest open facility. To overcome the difficulties caused by the fairness constraints, we construct an auxiliary graph and use minimum weighted perfect matching to get the cost between the portals of each block and the ones in its children. In order to satisfy the fairness constraints, we bound the fairness constraints of each open facility in the leaves of the split-tree based on the relation between the subproblem and the subproblems of its children. To obtain the assignment of the given instance and remove the fairness violation, we construct a new b-value min-cost max-flow model based on the set of open facilities. For the fair k-median problem in general metrics, we present a polynomial-time approximation algorithm with ratio O(log⁡k). Our approximation algorithm for the fair k-median problem in doubling metrics is the first result for the corresponding problem without any fairness violation in doubling metrics.

Multitasking scheduling with shared processing

AbstractRecently, the problem of multitasking scheduling has raised a lot of interest in the service industries. Hall et al. (Discrete Applied Mathematics, 2016) proposed a shared processing multitasking scheduling model which allows a team to continue to work on the primary tasks while processing the routinely scheduled activities as they occur. With a team being modeled as a single machine, the processing sharing of the machine is achieved by allocating a fraction of the processing capacity to routine jobs and the remaining fraction, which we denote as sharing ratio, to the primary jobs. In this paper, we generalize this model to parallel machines and allow the fraction of the processing capacity assigned to routine jobs to vary from one to another. The objectives are minimizing makespan and minimizing the total completion time of primary jobs. We show that for both objectives, there is no polynomial time approximation algorithm unless P=NP if the sharing ratios are arbitrary for all machines. Then we consider the problems where the sharing ratios on some machines have a constant lower bound. For each objective, we analyze the performance of the classical scheduling algorithms and their variations and then develop a polynomial time approximation scheme when the number of machines is a constant.

On the Multilayer Planning of Filterless Optical Networks With OTN Encryption

With enhanced cost-effectiveness, filterless optical networks (FONs) have been considered as a promising candidate for future optical infrastructure. However, as the transmission in FON relies on the “select-and-broadcast” scenario, it is more vulnerable to eavesdropping. Therefore, encrypting the communications in FONs will be indispensable, and this can be realized by introducing the optical transport network (OTN) encryption technologies that leverage high-speed encryption cards (ECs) to protect the integrity of OTN payload frames. In this paper, we study the problem of security-aware multilayer planning of FONs with OTN encryption. We first formulate a mixed integer linear programming (MILP) model (i.e., w-MILP) to solve the problem exactly. Then, to reduce the time complexity of problem-solving, we transform w-MILP into two correlated MILP models for establishing fiber trees for an FON (t-MILP) and planning flows in the fiber trees (s-MILP), respectively. The optimization in t-MILP is further transformed into a weighted set partitioning problem, which can be solved time-efficiently. As for s-MILP, we propose a polynomial-time approximation algorithm based on linear programming (LP) relaxation and randomized rounding. Extensive simulations verify the performance of our proposals.

Personalised incentives with constrained regulator's budget

We consider a regulator driving individual choices towards increasing social welfare by providing personal incentives. We formalise and solve this problem by maximising social welfare under a budget constraint. The personalised incentives depend on the alternatives available to each individual and on her preferences. A polynomial time approximation algorithm computes a policy within few seconds. We analytically prove that it is boundedly close to the optimum. We efficiently calculate the curve of social welfare achievable for each value of budget within a given range. This curve can be useful for the regulator to decide the appropriate amount of budget to invest. We extend our formulation to enforcement, taxation and non-personalised-incentive policies. We analytically show that our personalised-incentive policy is also optimal within this class of policies and construct close-to-optimal enforcement and proportional tax-subsidy policies. We then compare analytically and numerically our policy with other state-of-the-art policies. Finally, we simulate a large-scale application to mode choice to reduce CO2 emissions.

Minimizing cost for influencing target groups in social network: A model and algorithmic approach

Stimulated by practical applications arising from economics, viral marketing, and elections, this paper studies the problem of Groups Influence with Minimum cost (GIM), which aims to find a seed set with the smallest cost that can influence all target groups in a social network, where each user is assigned a cost and a score and a group of users is influenced if the total score of influenced users in the group is at least a certain threshold. As the group influence function, defined as the number of influenced groups or users, is neither submodular nor supermodular, theoretical bounds on the quality of solutions returned by the well-known greedy approach may not be guaranteed.In this work, two efficient algorithms with theoretical guarantees for tackling the GIM problem, named Groups Influence Approximation (GIA) and Exact Groups Influence (EGI), are proposed. GIA is a bi-criteria polynomial-time approximation algorithm and EGI is an (almost) exact algorithm; both can return good approximate solutions with high probability. The novelty of our approach lies in two aspects. Firstly, a novel group reachable reverse sample concept is proposed to estimate the group influence function within an error bound. Secondly, a framework algorithmic is designed to find serial candidate solutions with checking theoretical guarantees at the same time. Besides theoretical results, extensive experiments conducted on real social networks show our algorithms’ performance. In particular, both EGI and GIA provide the solution quality several times better, while GIA is up to 800 times faster than the state-of-the-art algorithms.

An Approximation Algorithm for the Nearest Decomposable Polynomial in the Hamming Distance

A univariate polynomial f is decomposable if it is the composition f = g ( h ) of polynomials g and h whose degrees are at least two. We consider the nearest decomposable polynomial to a given polynomial f in the Hamming distance. We propose a polynomial-time approximation algorithm for the nearest decomposable polynomial and analyze the quality of the output.

Scheduling coflows for minimizing the total weighted completion time in heterogeneous parallel networks

Coflow is a network abstraction used to represent communication patterns in data centers. The coflow scheduling problem encountered in large data centers is a challenging NP-hard problem. Many previous studies on coflow scheduling mainly focus on the single-core model. However, with the growth of data centers, this single-core model is no longer sufficient. This paper addresses the coflow scheduling problem within heterogeneous parallel networks, which feature an architecture consisting of multiple network cores running in parallel. In this paper, two polynomial-time approximation algorithms are developed for the flow-level scheduling problem and the coflow-level scheduling problem in heterogeneous parallel networks, respectively. For the flow-level scheduling problem, the proposed algorithm achieves an approximation ratio of O(log⁡m/log⁡log⁡m) when all coflows are released at arbitrary times, where m represents the number of network cores. On the other hand, in the coflow-level scheduling problem, the proposed algorithm achieves an approximation ratio of O(m(log⁡m/log⁡log⁡m)2) when all coflows are released at arbitrary times. Moreover, we propose a heuristic algorithm for the flow-level scheduling problem. Simulation results using synthetic traffic traces validate the performance of our algorithms and show improvements over the previous algorithm.

Building a small and informative phylogenetic supertree

We combine two fundamental optimization problems related to the construction of phylogenetic trees called maximum rooted triplets consistency and minimally resolved supertree into a new problem, which we call q-maximum rooted triplets consistency (q-MAXRTC). It takes as input a set R of rooted, binary phylogenetic trees with three leaves each and asks for a phylogenetic tree with exactly q internal nodes that contains the largest possible number of trees from R. We prove that q-MAXRTC is NP-hard to approximate within a constant, develop polynomial-time approximation algorithms for different values of q, and show experimentally that representing a phylogenetic tree by one having much fewer nodes typically does not destroy too much branching information. To demonstrate the algorithmic advantage of using trees with few internal nodes, we also propose a new algorithm for computing the rooted triplet distance that is faster than the existing algorithms when restricted to such trees.

D-Optimal Data Fusion: Exact and Approximation Algorithms

We study the D-optimal Data Fusion (DDF) problem, which aims to select new data points, given an existing Fisher information matrix, so as to maximize the logarithm of the determinant of the overall Fisher information matrix. We show that the DDF problem is NP-hard and has no constant-factor polynomial-time approximation algorithm unless P = NP. Therefore, to solve the DDF problem effectively, we propose two convex integer-programming formulations and investigate their corresponding complementary and Lagrangian-dual problems. Leveraging the concavity of the objective functions in the two proposed convex integer-programming formulations, we design an exact algorithm, aimed at solving the DDF problem to optimality. We further derive a family of submodular valid inequalities and optimality cuts, which can significantly enhance the algorithm performance. We also develop scalable randomized-sampling and local-search algorithms with provable performance guarantees. Finally, we test our algorithms using real-world data on the new phasor-measurement-units placement problem for modern power grids, considering the existing conventional sensors. Our numerical study demonstrates the efficiency of our exact algorithm and the scalability and high-quality outputs of our approximation algorithms. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: Y. Li and W. Xie were supported in part by Division of Civil, Mechanical and Manufacturing Innovation [Grant 2046414] and Division of Computing and Communication Foundations [Grant 2246417]. J. Lee was supported in part by Air Force Office of Scientific Research [Grants FA9550-19-1-0175 and FA9550-22-1-0172]. M. Fampa was supported in part by Conselho Nacional de Desenvolvimento Científico e Tecnológico [Grants 305444/2019-0 and 434683/2018-3]. F. Qiu and R. Yao were supported in part by the U.S. Department of Energy Advanced Grid Modeling Program under [Grant DE-OE0000875]. Supplemental Material: The e-companion is available at https://doi.org/10.1287/ijoc.2022.0235 .

Scheduling double-track gantry cranes to minimize the overall loading/ unloading time

In this paper, we consider the gantry crane scheduling problem at a single storage block where a total of [Formula: see text] gantry cranes are mounted on double tracks so that cranes on different tracks can pass each other while those on the same track cannot. Containers at the storage block are divided into bays and each bay of containers has to be loaded/unloaded together due to the same shipping destination or the same customer. To minimize the overall loading/ unloading time of containers, we first formulate the problem to a mixed integer linear programming (MILP) model, and compute the optimal solutions of small instances by the Gurobi solver. Then we design several heuristic algorithms and test their efficiency and performance by a series of large instances. In particular, we present a polynomial time approximation algorithm for the case where all but one gantry crane is mounted on the same track. We show that the algorithm has a worst case ratio of [Formula: see text], outperforming the partition-based algorithms in the single-track scenario.

An integer programming model for the deployment of mobile health clinics

Facility location and resource allocation play an essential role in the operational management of mobile health clinics (MHCs) as optimization of both can help medical service providers improve their service coverage under limited resources. In this paper, we consider the issue of maximizing the coverage of immunization services provided by MHCs in vulnerable communities such as school zones and census tracts. Since demand in communities overlaps with neighboring communities, this leads to the so-called mobile facility location with overlapped demand (MFL-OD). To capture the overlapped demand in different communities, we introduce a novel integer programming model with linear and bilinear constraints (BLCIP) and show that the BLCIP model is NP-hard. A polynomial-time approximation algorithm is proposed for the BLCIP, and the approximation ratio of the obtained solution is estimated. We implement the proposed model and approximation algorithm for a case study in the Houston Independent School District and its associated communities. Preliminary experiments demonstrate the efficacy of the proposed model and the efficiency of the proposed approximation method.

Digital Twin Placement for Minimum Application Request Delay With Data Age Targets

Digital twins (DTs) are virtual implementations of physical systems (PSs) and can represent the states of the PSs in realtime. In order to update the DTs with changes in their corresponding PSs, the PSs should regularly send their state information data to the DTs. Each DT must be assigned to an execution server (ES) that processes the forwarded data from its corresponding PS. The output is then made available to applications that are operating at an internet cloud server. In this paper we consider the problem of DT placement such that the maximum data request-response delay experienced by the application over all PSs is minimized, subject to maximum data age target constraints at the DTs and the application server. The problem is first formulated as an integer quadratic program (IQP) and then transformed into a semidefinite program (SDP). The problem is NP-complete. Since exact polynomial solutions are unavailable, several practical polynomial-time approximation algorithms are introduced. The algorithms are designed to give solutions with different trade-offs between the accommodation of the application input timing latency and the achievement of data age targets.

Constraint Optimization over Semirings

Interpretations of logical formulas over semirings (other than the Boolean semiring) have applications in various areas of computer science including logic, AI, databases, and security. Such interpretations provide richer information beyond the truth or falsity of a statement. Examples of such semirings include Viterbi semiring, min-max or access control semiring, tropical semiring, and fuzzy semiring. The present work investigates the complexity of constraint optimization problems over semirings. The generic optimization problem we study is the following: Given a propositional formula phi over n variable and a semiring (K,+, . ,0,1), find the maximum value over all possible interpretations of phi over K. This can be seen as a generalization of the well-known satisfiability problem (a propositional formula is satisfiable if and only if the maximum value over all interpretations/assignments over the Boolean semiring is 1). A related problem is to find an interpretation that achieves the maximum value. In this work, we first focus on these optimization problems over the Viterbi semiring, which we call optConfVal and optConf. We first show that for general propositional formulas in negation normal form, optConfVal and optConf are in FP^NP. We then investigate optConf when the input formula phi is represented in the conjunctive normal form. For CNF formulae, we first derive an upper bound on the value of optConf as a function of the number of maximum satisfiable clauses. In particular, we show that if r is the maximum number of satisfiable clauses in a CNF formula with m clauses, then its optConf value is at most 1/4^(m-r). Building on this we establish that optConf for CNF formulae is hard for the complexity class FP^NP[log]. We also design polynomial-time approximation algorithms and establish an inapproximability for optConfVal. We establish similar complexity results for these optimization problems over other semirings including tropical, fuzzy, and access control semirings.

A simple polynomial-time approximation algorithm for the total variation distance between two product distributions

We give a simple polynomial-time approximation algorithm for the total variation distance between two product distributions.

Networking of Sensors With Fixed Transmission Ranges: Distributed Data Processing Over Graphs

In this letter, we address the data gathering problem in low-power wireless sensor networks (WSNs) with sensors of fixed transmission ranges. We model the NP-hard problem as an integer linear program that can be applied to networks of small sizes. Further, we propose a novel graphical model that apprehends the features of such networks, and design a polynomial-time approximation algorithm using the proposed graphical model for networks of large sizes. The performance is evaluated under data compaction strategies based on compressive sensing. Simulation results demonstrate the superior performance of the proposed method over its peer protocols.

A Simple 2-Approximation for Maximum-Leaf Spanning Tree

For an [Formula: see text]-edge connected simple graph [Formula: see text], finding a spanning tree of [Formula: see text] with the maximum number of leaves is MAXSNP-complete. The problem remains NP-complete even if [Formula: see text] is planar and the maximal degree of [Formula: see text] is at most four. Lu and Ravi gave the first known polynomial-time approximation algorithms with approximation factors [Formula: see text] and [Formula: see text]. Later, they obtained a [Formula: see text]-approximation algorithm that runs in near-linear time. The best known result is Solis-Oba, Bonsma, and Lowski’s [Formula: see text]-time [Formula: see text]-approximation algorithm. We show an alternative simple [Formula: see text]-time [Formula: see text]-approximation algorithm whose analysis is simpler. This paper is dedicated to the cherished memory of our dear friend, Professor Takao Nishizeki.

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.