Exact Polynomial-time Algorithm Research Articles

Bounded-Hop Communication Networks

We study the problem of assigning transmission ranges to radio stations in the plane such that any pair of stations can communicate within a bounded number of hops h and the cost of the network is minimized. We consider two settings of the problem: collinear station locations and arbitrary locations. For the case of collinear stations, we introduce the pioneer polynomial-time exact algorithm for any $$\alpha \ge 1$$ and constant h, and thus conclude that the 1D version of the problem, where h is a constant, is in $$\mathcal {P}$$ . Furthermore, we provide a 3 / 2-approximation algorithm for the case where h is an arbitrary number and $$\alpha =1$$ , improving the previously best known approximation ratio of 2. For the case of stations placed arbitrarily in the plane and $$\alpha =1$$ , we first present a $$(1.5+ \varepsilon )$$ -approximation algorithm for a case where a deviation of one hop ( $$h+1$$ hops in total) is acceptable. Then, we show a $$(3+\varepsilon )$$ -approximation algorithm that complies with the exact hop bound. To achieve that, we introduce the following two auxiliary problems, which are of independent interest. The first is the bounded-hop multi-sink range problem, for which we present a PTAS which can be applied to compute a $$(1+\varepsilon )$$ -approximation for the bounded diameter minimum spanning tree, for any $$\varepsilon >0$$ . The second auxiliary problem is the bounded-hop dual-sink pruning problem, for which we show a polynomial-time algorithm. To conclude, we consider the initial bounded-hop all-to-all range assignment problem and show that a combined application of the aforementioned problems yields the $$(3+\varepsilon )$$ -approximation ratio for this problem, which improves the previously best known approximation ratio of $$4(9^{h-2})/(\root h \of {2}-1)$$ .

Evangelism in social networks: Algorithms and complexity

We consider a population of interconnected individuals that, with respect to a piece of information, at each time instant can be subdivided into three (time‐dependent) categories: agnostics, influenced, and evangelists. A dynamical process of information diffusion evolves among the individuals of the population according to the following rules. Initially, all individuals are agnostic. Then, a set of people is chosen from the outside and convinced to start evangelizing, that is, to start spreading the information. When a number of evangelists, greater than a given threshold, communicate with a node v, the node v becomes influenced, whereas, as soon as the individual v is contacted by a sufficiently much larger number of evangelists, it is itself converted into an evangelist and consequently it starts spreading the information. The question is: How to choose a bounded cardinality initial set of evangelists so as to maximize the final number of influenced individuals? We prove that the problem is hard to solve, even in an approximate sense. On the positive side, we present exact polynomial time algorithms for trees and complete graphs. For general graphs, we derive exact parameterized algorithms. We also study the problem when the objective is to select a minimum number of evangelists capable of influencing the whole network. Our motivations to study these problems come from the areas of Viral Marketing and spread of influence in social networks. © 2017 Wiley Periodicals, Inc. NETWORKS, Vol. 71(4), 346–357 2018

A new maximum fault-tolerance barrier-coverage problem in hybrid sensor network and its polynomial time exact algorithm

This paper introduces a new maximum fault-tolerance barrier-coverage problem in hybrid sensor network, which consists of a number of both static ground sensors and fully-controllable mobile sensors. The problem aims to relocate the mobile sensor nodes so that the fault-tolerance of the barrier-coverage of the hybrid sensor network is maximized. The main contribution of this paper is the polynomial time exact algorithm for this new problem.

Guarding monotone art galleries with sliding cameras in linear time

A sliding camera in an orthogonal polygon P—that is, a polygon all of whose edges are axis-parallel—is a point guard g that travels back and forth along an axis-parallel line segment s inside P. A point p in P is guarded by g if and only if there exists a point q on s such that line segment pq is normal to s and contained in P. In the minimum sliding cameras (MSC) problem, the objective is to guard P with the minimum number of sliding cameras.We give a dynamic programming algorithm that solves the MSC problem exactly on monotone orthogonal polygons in O(n) time, improving the 2-approximation algorithm of Katz and Morgenstern (2011). More generally, our algorithm can be used to solve the MSC problem in O(n) time on simple orthogonal polygons P for which the dual graph induced by the vertical decomposition of P is a path. Our results provide the first polynomial-time exact algorithms for the MSC problem on a non-trivial subclass of orthogonal polygons.

Multiple facility location on a network with linear reliability order of edges

We study the problem of locating facilities on the nodes of a network to maximize the expected demand serviced. The edges of the input graph are subject to random failure due to a disruptive event. We consider a special type of failure correlation. The edge dependency model assumes that the failure of a more reliable edge implies the failure of all less reliable ones. Under this dependency model called Linear Reliability Order (LRO) we give two polynomial time exact algorithms. When two distinct LRO’s exist, we prove the total unimodularity of a linear programming formulation. In addition, we show that minimizing the sum of facility opening costs and expected cost of unserviced demand under two orderings reduces to a matching problem. We prove NP-hardness of the three orderings case and show that the problem with an arbitrary number of orderings generalizes the deterministic maximum coverage problem. When a demand point can be covered only if a facility exists within a distance limit, we show that the problem is NP-hard even for a single ordering.

On the most imbalanced orientation of a graph

We study the problem of orienting the edges of a graph such that the minimum over all the vertices of the absolute difference between the outdegree and the indegree of a vertex is maximized. We call this minimum the imbalance of the orientation, i.e. the higher it gets, the more imbalanced the orientation is. The studied problem is denoted by $${{\mathrm{\textsc {MaxIm}}}}$$ . We first characterize graphs for which the optimal objective value of $${{\mathrm{\textsc {MaxIm}}}}$$ is zero. Next we show that $${{\mathrm{\textsc {MaxIm}}}}$$ is generally NP-hard and cannot be approximated within a ratio of $$\frac{1}{2}+\varepsilon $$ for any constant $$\varepsilon >0$$ in polynomial time unless $$\texttt {P}=\texttt {NP}$$ even if the minimum degree of the graph $$\delta $$ equals 2. Then we describe a polynomial-time approximation algorithm whose ratio is almost equal to $$\frac{1}{2}$$ . An exact polynomial-time algorithm is also derived for cacti. Finally, two mixed integer linear programming formulations are presented. Several valid inequalities are exhibited with the related separation algorithms. The performance of the strengthened formulations is assessed through several numerical experiments.

Approximation Algorithms for Optimization of Combinatorial Dynamical Systems

We consider an optimization problem for a dynamical system whose evolution depends on a collection of binary decision variables. We develop scalable approximation algorithms with provable suboptimality bounds to provide computationally tractable solution methods even when the dimension of the system and the number of the binary variables are large. The proposed method employs a linear approximation of the objective function such that the approximate problem is defined over the feasible space of the binary decision variables, which is a discrete set. To define such a linear approximation, we propose two different variation methods: one uses continuous relaxation of the discrete space and the other uses convex combinations of the vector field and running payoff. The approximate problem is a 0-1 linear program, which can be solved by existing polynomial-time exact or approximation algorithms, and does not require the solution of the dynamical system. Furthermore, we characterize a sufficient condition ensuring the approximate solution has a provable suboptimality bound. We show that this condition can be interpreted as the concavity of the objective function or that of a reformulated objective function.

Polynomial Time Algorithms for Variants of Graph Matching on Partial k-Trees

Abstract In this paper, we deal with two variants of graph matching, the graph isomorphism with restriction and the prefix set of graph isomorphism. The former problem is known to be NP-complete, whereas the latter problem is known to be GI-complete. We propose polynomial time exact algorithms for these problems on partial k-trees.

The label cut problem with respect to path length and label frequency

Given a graph with labels defined on edges and a source-sink pair (s,t), the Labels-tCut problem asks for a minimum number of labels such that the removal of edges with these labels disconnects s and t. Similarly, the Global Label Cut problem asks for a minimum number of labels to disconnect G itself. For these two problems, we identify two useful parameters, i.e., lmax, the maximum length of any s-t path (only applies to Labels-tCut), and fmax, the maximum number of appearances of any label in the graph (applies to the two problems). We show that lmax=2 and fmax=2 are two complexity thresholds for Labels-tCut. Furthermore, we give (i) an O⁎(ck) time parameterized algorithm for Labels-tCut with lmax bounded from above, where parameter k is the number of labels in a solution, and c is a constant with lmax−1<c<lmax, (ii) a combinatorial lmax-approximation algorithm for Labels-tCut, and (iii) a polynomial time exact algorithm for Global Label Cut with fmax bounded from above.

Production and interplant batch delivery scheduling: Dominance and cooperation

In this paper, we consider scheduling problems in a supply chain with two agents, a manufacturer and a third-party logistics (3PL) provider. The manufacturer has to process a set of jobs at the upstream stage and at the downstream stage. The 3PL provider is in charge of transportation of semi-finished products from the upstream stage to the downstream stage. The manufacturer's objective is to minimize makespan Cmax and the 3PL provider's objective is to minimize transportation cost TC. We investigate three scenarios, corresponding to different types of contract: (i) decentralized scenario with strict responsiveness; (ii) decentralized scenario with flexible responsiveness; (iii) centralized (cooperative) scenario. We provide exact polynomial-time algorithms or prove the NP-completeness of the scheduling problems in these three scenarios. Moreover, we evaluate and compare various scenarios through a large set of computational experiments. The results show that cooperation may bring significant benefits to both actors. The benefit for the 3PL provider is particularly high if compared to situations in which the manufacturing sequence is fixed. Also the manufacturer can benefit from relaxing the job-by-job responsiveness constraint in favor of an integrated schedule which appropriately accounts for the role of the 3PL provider.

Collapse of the Hierarchy of Constant-Depth Exact Quantum Circuits

We study the quantum complexity class $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 of quantum operations implementable exactly by constant-depth polynomial-size quantum circuits with unbounded fan-out gates. Our main result is that the quantum OR operation is in $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0, which is an affirmative answer to the question posed by HOyer and źpalek. In sharp contrast to the strict hierarchy of the classical complexity classes: $${\mathsf{NC}^{0} \subsetneq \mathsf{AC}^{0} \subsetneq \mathsf{TC}^{0}}$$NC0źAC0źTC0, our result with HOyer and źpalek's one implies the collapse of the hierarchy of the corresponding quantum ones: $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}=\mathsf{QAC}^\mathsf{0}_\mathsf{f}=\mathsf{QTC}^\mathsf{0}_\mathsf{f}}$$QNCf0=QACf0=QTCf0. Then, we show that there exists a constant-depth subquadratic-size quantum circuit for the quantum threshold operation. This allows us to obtain a better bound on the size difference between the $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 and $${\mathsf{QTC}^\mathsf{0}_\mathsf{f}}$$QTCf0 circuits for implementing the same operation. Lastly, we show that, if the quantum Fourier transform modulo a prime is in $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0, there exists a polynomial-time exact classical algorithm for a discrete logarithm problem using a $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 oracle. This implies that, under a plausible assumption, there exists a classically hard problem that is solvable exactly by a $${\mathsf{QNC}^\mathsf{0}_\mathsf{f}}$$QNCf0 circuit with gates for the quantum Fourier transform.

A new family of facet defining inequalities for the maximum edge-weighted clique problem

This paper considers a family of cutting planes, recently developed for mixed 0–1 polynomial programs and shows that they define facets for the maximum edge-weighted clique problem. There exists a polynomial time exact separation algorithm for these inequalities. The result of this paper may contribute to the development of more efficient algorithms for the maximum edge-weighted clique problem that use cutting planes.

Metric extension operators, vertex sparsifiers and Lipschitz extendability

We study vertex cut and flow sparsifiers that were recently introduced by Moitra (2009), and Leighton and Moitra (2010). We improve and generalize their results. We give a new polynomial-time algorithm for constructing O(log k/ log log k) cut and flow sparsifiers, matching the best known existential upper bound on the quality of a sparsifier, and improving the previous algorithmic upper bound of O(log2 k/ log log k). We show that flow sparsifiers can be obtained from linear operators approximating minimum metric extensions. We introduce the notion of (linear) metric extension operators, prove that they exist, and give an exact polynomialtime algorithm for finding optimal operators. We then establish a direct connection between flow and cut sparsifiers and Lipschitz extendability of maps in Banach spaces, a notion studied in functional analysis since 1930s. Using this connection, we obtain a lower bound of $$\Omega \left( {\sqrt {\log k/\log \log k} } \right)$$ for flow sparsifiers and a lower bound of $$\Omega \left( {\sqrt {\log k} /\log \log k} \right)$$ for cut sparsifiers. We show that if a certain open question posed by Ball in 1992 has a positive answer, then there exist $$\tilde O\left( {\sqrt {\log k} } \right)$$ cut sparsifiers. On the other hand, any lower bound on cut sparsifiers better than $$\tilde \Omega \left( {\sqrt {\log k} } \right)$$ would imply a negative answer to this question.

Covering problems in edge- and node-weighted graphs

This paper discusses the graph covering problem in which a set of edges in an edge- and node-weighted graph is chosen to satisfy some covering constraints while minimizing the sum of the weights. In this problem, because of the large integrality gap of a naive linear programming (LP) relaxation, LP rounding algorithms based on the relaxation yield poor performance. Here we propose a stronger LP relaxation for the graph covering problem. The proposed relaxation is applied to designing primal–dual algorithms for two fundamental graph covering problems: the prize-collecting edge dominating set problem and the multicut problem in trees. Our algorithms are an exact polynomial-time algorithm for the former problem, and a 2-approximation algorithm for the latter problem. These results match the currently known best results for purely edge-weighted graphs.

Exact and Heuristic Algorithms for Routing AGV on Path with Precedence Constraints

A new problem arises when an automated guided vehicle (AGV) is dispatched to visit a set of customers, which are usually located along a fixed wire transmitting signal to navigate the AGV. An optimal visiting sequence is desired with the objective of minimizing the total travelling distance (or time). When precedence constraints are restricted on customers, the problem is referred to as traveling salesman problem on path with precedence constraints (TSPP-PC). Whether or not it is NP-complete has no answer in the literature. In this paper, we design dynamic programming for the TSPP-PC, which is the first polynomial-time exact algorithm when the number of precedence constraints is a constant. For the problem with number of precedence constraints, part of the input can be arbitrarily large, so we provide an efficient heuristic based on the exact algorithm.

Integrated Scheduling of Production and Distribution with Release Dates and Capacitated Deliveries

This paper investigates an integrated scheduling of production and distribution model in a supply chain consisting of a single machine, a customer, and a sufficient number of homogeneous capacitated vehicles. In this model, the customer places a set of orders, each of which has a given release date. All orders are first processed nonpreemptively on the machine and then batch delivered to the customer. Two variations of the model with different objective functions are studied: one is to minimize the arrival time of the last order plus total distribution cost and the other is to minimize total arrival time of the orders plus total distribution cost. For the former one, we provide a polynomial-time exact algorithm. For the latter one, due to its NP-hard property, we provide a heuristic with a worst-case ratio bound of 2.

How to Pack Trapezoids: Exact and Evolutionary Algorithms

The purposes of this paper are twofold. In the first, we describe an exact polynomial-time algorithm for the pair sequencing problem and show how this method can be used to pack fixed-height trapezoids into a single bin such that interitem wastage is minimized. We then go on to examine how this algorithm can be combined with bespoke evolutionary and local search methods for tackling the multiple-bin version of this problem—one that is closely related to 1-D bin packing. In the course of doing this, a number of ideas surrounding recombination, diversity, and genetic repair are also introduced and analyzed.

Efficient constraint reduction in multistage stochastic programming problems with endogenous uncertainty

Multistage stochastic programming with endogenous uncertainty is a new topic in which the timing of uncertainty realization is decision-dependent. In this case, the number of nonanticipativity constraints (NACs) increases very quickly with the number of scenarios, making the problem computationally intractable. Fortunately, a large number of NACs are typically redundant and their elimination leads to a considerable reduction in the problem size. Identifying redundant NACs has been addressed in the literature only in the special case where the scenario set is equal to the Cartesian product of all possible outcomes for endogenous parameters; however, this is a scarce condition in practice. In this paper, we consider the general case where the scenario set is an arbitrary set; and two approaches, able to identify all redundant NACs, are proposed. The first approach is by mixed integer programming formulation and the second one is an exact polynomial time algorithm. Proving the fact that the proposed algorithm is able to make the uppermost reduction in the number of NACs is another novelty of this paper. Computational results evaluate the efficiency of the proposed approaches.

Robust Metric Inequalities for Network Loading Under Demand Uncertainty

In this paper, we consider the network loading problem under demand uncertainties with static routing, i.e., a single routing scheme based on the fraction of the demands has to be determined. We generalize the class of metric inequalities to the Γ-robust setting and show that they yield a formulation in the capacity space. We describe a polynomial time exact algorithm to separate violated robust metric inequalities as model constraints. Moreover, rounded and tight robust metric inequalities describing the convex hull of integer solutions are presented and separated in a cut-and-branch approach. Computational results using real-life telecommunication data demonstrate the potential of (tight) robust metric inequalities by considering the integrality gaps at the root node and the overall optimality gaps. Average speed-up factors between 2 and 5 for the compact flow and between 3 and 25 for the capacity formulation in the case of mid-sized instances have been achieved by exploiting robust metric inequalities in the solving process.

A polynomial-time algorithm for the preemptive mixed-shop problem with two unit operations per job

In a so-called mixed-shop scheduling problem, the operations of some jobs have to be processed in a fixed order (as in the job-shop problem); the other ones can be processed in an arbitrary order (as in the open-shop problem). In this paper we present a new exact polynomial-time algorithm for the mixed-shop problems with preemptions and at most two unit operations per job.