The Complexity of Promise SAT on Non-Boolean Domains

While 3-SAT is NP-hard, 2-SAT is solvable in polynomial time. Austrin et al. [SICOMP’17] proved a result known as “(2+ɛ)-SAT is NP-hard.” They showed that the problem of distinguishing k -CNF formulas that are g -satisfiable (i.e., some assignment satisfies at least g literals in every clause) from those that are not even 1-satisfiable is NP-hard if g/k < 1/2 and is in P otherwise. We study a generalisation of SAT on arbitrary finite domains, with clauses that are disjunctions of unary constraints, and establish analogous behaviour. Thus, we give a dichotomy for a natural fragment of promise constraint satisfaction problems ( PCSPs ) on arbitrary finite domains. The hardness side is proved using the algebraic approach via a new general NP-hardness criterion on polymorphisms, which is based on a gap version of the Layered Label Cover problem. We show that previously used criteria are insufficient—the problem hence gives an interesting benchmark of algebraic techniques for proving hardness of approximation in problems such as PCSPs.

A Combinatorial Optimization Model for Emergency Resource Allocation after Disasters

Disasters occur over a short or long period of time and cause large-scale harm to humans, infrastructure, as well as the ecosystem every year. Immediate response after a disaster helps minimize its impact on life and property. Therefore, it is crucial to have an emergency response system ready to handle any emergency that may come up after a disaster. In this paper, a model is proposed to optimize the distribution of emergency services at disaster-struck points. Due to the NP-hardness of the problem, two metaheuristic algorithms, Particle Swarm Optimization and Cuckoo Search Optimization have been used to dynamically allocate the available resources based on the given situation. The proposed model uses the distance between the emergency location and the emergency service provider, and the severity of the emergency as the main metrics for scoring any considered solution. The conducted experiments demonstrate that the model provides effective, efficient, and dynamic allocation service at emergency locations in simulated disaster situations.

Designing screen layout in multimedia applications through integer programming and metaheuristic

Binding audiovisual content into multimedia applications requires the specification of each media item, including its size and position, to define a screen layout. The multimedia application author must plan the application’s screen layout (ASL), considering a variety of screen sizes where the application shall be executed. An ASL that maximizes the area occupied by media items on the screen is essential, given that screen space is a valuable asset for media broadcasters. In this paper, we introduce the Application Screen Layout Optimization Problem, and present its NP-hardness. Besides, two integer programming formulations and an Iterated Local Search (ILS) metaheuristic are proposed to solve it. The efficiency of the proposed methods is evaluated, showing that the metaheuristic achieves better results and is at least 12 times faster, on average, than the mathematical formulations. Also, the proposed approaches were compared to a layout design algorithm, showing their effectiveness.

Multiplicative Parameterization Above a Guarantee

Parameterization above a guarantee is a successful paradigm in Parameterized Complexity. To the best of our knowledge, all fixed-parameter tractable problems in this paradigm share an additive form defined as follows. Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k + g(I) . Here, g(I) is usually a lower bound on the minimum size of a solution. Since its introduction in 1999 for M AX SAT and M AX C UT (with g(I) being half the number of clauses and half the number of edges, respectively, in the input), analysis of parameterization above a guarantee has become a very active and fruitful topic of research. We highlight a multiplicative form of parameterization above (or, rather, times) a guarantee: Given an instance ( I,k ) of some (parameterized) problem π with a guarantee g(I) , decide whether I admits a solution of size at least (or at most) k · g(I) . In particular, we study the Long Cycle problem with a multiplicative parameterization above the girth g(I) of the input graph, which is the most natural guarantee for this problem, and provide a fixed-parameter algorithm. Apart from being of independent interest, this exemplifies how parameterization above a multiplicative guarantee can arise naturally. We also show that, for any fixed constant ε > 0, multiplicative parameterization above g(I) 1+ε of Long Cycle yields para-NP-hardness, thus our parameterization is tight in this sense. We complement our main result with the design (or refutation of the existence) of fixed-parameter algorithms as well as kernelization algorithms for additional problems parameterized multiplicatively above girth.

NP-Hardness and Approximation Algorithms for Iterative Pricing on Social Networks with Externalities

We study how a monopolist seller should price an indivisible product iteratively to the consumers who are connected by a known link-weighted directed social network. For two consumers [Formula: see text] and [Formula: see text], there is an arc directed from [Formula: see text] to [Formula: see text] if and only if [Formula: see text] is a fashion leader of [Formula: see text]. Assuming complete information about the network, the seller offers consumers a sequence of prices over time and the goal is to obtain the maximum revenue. We assume that the consumers buy the product as soon as the seller posts a price not greater than their valuations of the product. The product’s value for a consumer is determined by three factors: a fixed consumer specified intrinsic value and a variable positive (resp. negative) externality that is exerted from the consumer’s out(resp. in)-neighbours. The setting of positive externality is that the influence of fashion leaders on a consumer is the total weight of links from herself to her fashion leaders who have owned the product, and more fashion leaders of a consumer owning the product will increase the influence (external value) on the consumer. And the setting of negative externalities is that the product’s value of showing off for a consumer is the total weight of links from her followers who do not own the product to herself, and more followers of a consumer owning the product will decrease this external value for the consumer. We confirm that finding an optimal iterative pricing is NP-hard even for acyclic networks with maximum total degree [Formula: see text] and with all intrinsic values zero. We design a greedy algorithm which achieves [Formula: see text]-approximation for networks with all intrinsic values zero and show that the approximation ratio [Formula: see text] is tight. Complementary to the hardness result, we design a [Formula: see text]-approximation algorithm for Barabási–Albert networks.

Exact Methods and Heuristics for Order Acceptance Scheduling Problem under Time-of-Use Costs and Carbon Emissions

This research focuses on an Order Acceptance Scheduling (OAS) problem on a single machine under time-of-use (TOU) tariffs and taxed carbon emissions periods with the objective to maximize total profit minus tardiness penalties and environmental costs. Due to the NP-hardness of the considered problem especially in presence of sequence-dependent setup-times, two fix-and-relax (FR) heuristics based on different time-indexed (TI) formulations are proposed. A metaheuristic based on the Dynamic Island Model (DIM) framework is also employed to tackle this optimization problem. These approached methods show promising results both in terms of solution quality and solving time compared to state-of-the-art exact solving approaches.

An Analysis of Solutions to the 2D Bin Packing Problem and Additional Complexities

The Bin Packing problem in 2 space is an NP-Hard combinatorial problem in optimization of packing and arrangement of objects in a given space. It has a wide variety of applications ranging from logistics in retail industries to resource allocation in cloud computing. In this paper, we discuss the mathematical formulation of this problem. Furthermore, we analyse its time complexity, its NP-Hardness and some of its stochastic solutions with their efficiencies. We then propose additional complexities that would make the problem more fit for industrial use and discuss in depth the domains in which it might prove to be useful. We conclude while suggesting areas of improvement in operations research on this subject.

An Analysis of Solutions to the 2D Bin Packing Problem and Additional Complexities

The Bin Packing problem in 2 space is an NP-Hard combinatorial problem in optimization of packing and arrangement of objects in a given space. It has a wide variety of applications ranging from logistics in retail industries to resource allocation in cloud computing. In this paper, we discuss the mathematical formulation of this problem. Furthermore, we analyse its time complexity, its NP-Hardness and some of its stochastic solutions with their efficiencies. We then propose additional complexities that would make the problem more fit for industrial use and discuss in depth the domains in which it might prove to be useful. We conclude while suggesting areas of improvement in operations research on this subject.

Interference-free Walks in Time: Temporally Disjoint Paths

We investigate the computational complexity of finding temporally disjoint paths or walks in temporal graphs. There, the edge set changes over discrete time steps and a temporal path (resp. walk) uses edges that appear at monotonically increasing time steps. Two paths (or walks) are temporally disjoint if they never use the same vertex at the same time; otherwise, they interfere. This reflects applications in robotics, traffic routing, or finding safe pathways in dynamically changing networks. On the one extreme, we show that on general graphs the problem is computationally hard. The "walk version" is W[1]-hard when parameterized by the number of routes. However, it is polynomial-time solvable for any constant number of walks. The "path version" remains NP-hard even if we want to find only two temporally disjoint paths. On the other extreme, restricting the input temporal graph to have a path as underlying graph, quite counterintuitively, we find NP-hardness in general but also identify natural tractable cases.