Complete Characterization Of Complexity Research Articles

The Continuous Time-Resource Trade-off Scheduling Problem with Time Windows

We introduce a variant of the cumulative scheduling problem (CuSP) characterized by continuous modes, time windows, and a criterion that involves safety margin maximization. The study of this variant is motivated by the Geospatial based Environment for Optimisation Systems Addressing Fire Emergencies Horizon 2020 Project, which is devoted to the design of evacuation plans in the face of natural disasters and more specifically, wildfire. People and goods have to be transferred from endangered places to safe places, and evacuation planning consists of scheduling evacuee moves along precomputed paths under arc capacities and deadlines. The resulting model is relevant in other contexts, such as project or industrial process scheduling. We consider here several formulations of the continuous time-resource trade-off scheduling problem (CTRTP-TW) with a safety maximization objective. We establish a complete complexity characterization distinguishing polynomial and NP-hard special cases depending on key parameters. We show that the problem with fixed sequencing (i.e., with predetermined overlap or precedence relations between activities) is convex. We then show that the preemptive variant is polynomial, and we propose lower and upper bounds based on this relaxation. A flow-based mixed-integer linear programming formulation is presented, from which a branch-and-cut exact method and an insertion heuristic are derived. An exact dedicated branch-and-bound algorithm is also designed. Extensive computational experiments are carried out to compare the different approaches on evacuation planning instances and on general CTRTP-TW instances. The experiments also show the interest of the continuous model compared with a previously proposed discrete approximation. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms—Discrete. Funding: This work was funded by the Horizon 2020 Marie Skłodowska-Curie Research and Innovation Staff Exchange European Project 691161 GEO-SAFE (Geospatial based Environment for Optimisation Systems Addressing Fire Emergencie). This work has also been supported by ANITI, the Artificial and Natural Intelligence Toulouse Institute. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0142 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0142 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

Dynamic Spectrum Management: A Complete Complexity Characterization

Consider a multi-user multi-carrier communication system where multiple users share multiple discrete subcarriers. To achieve high spectrum efficiency, the users in the system must choose their transmit power dynamically in response to fast channel fluctuations. Assuming perfect channel state information, two formulations for the spectrum management (power control) problem are considered in this paper: the first is to minimize the total transmission power subject to all users’ transmission data rate constraints, and the second is to maximize the min-rate utility subject to individual power constraints at each user. It is known in the literature that both formulations of the problem are polynomial time solvable when the number of subcarriers is one and strongly NP-hard when the number of subcarriers are greater than or equal to three. However, the complexity characterization of the problem when the number of subcarriers is two has been missing for a long time. This paper answers this long-standing open question: both formulations of the problem are strongly NP-hard when the number of subcarriers is two.

Extending partial representations of subclasses of chordal graphs

Chordal graphs are intersection graphs of subtrees of a tree T. We investigate the complexity of the partial representation extension problem for chordal graphs. A partial representation specifies a tree T′ and some pre-drawn subtrees of T′. It asks whether it is possible to construct a representation inside a modified tree T which extends the partial representation (i.e., keeps the pre-drawn subtrees unchanged).We consider four modifications of T′ leading to vastly different problems: (i) T=T′, (ii) T is a subdivision of T′, (iii) T is a supergraph of T′, and (iv) T′ is a topological minor of T. In some cases, it is interesting to consider the complexity even when just T′ is given and no subtree is pre-drawn. Also, we consider three well-known subclasses of chordal graphs: Proper interval graphs, interval graphs and path graphs. We give an almost complete complexity characterization. We further study the parametrized complexity of the problems when parametrized by the number of pre-drawn subtrees, the number of components of the input graph G and the size of the tree T′.We describe an interesting relation with integer partition problems. The problem 3-Partition is used for all NP-completeness reductions. When the space in T′ is limited, partial representation extension of proper interval graphs is “equivalent” to the BinPacking problem.

Comparative Study of Morphometric Properties Characterizing the Complexity of Silicate Pore Networks Probed by Adsorption of Nitrogen and Methanol

In this work, we compare the surface and morphometric properties of the pore networks in four silicas (code names Fr1428, Fr474, Fr1386, and MM1164) with different random porosities using the adsorption isotherms of two different probe adsorbents, nitrogen and methanol. The parent material Fr1428 was a pure silica 25 microm sample. The Fr474 sample was the same one with bonded electroneutral diol groups on its outer surface. Fr1386 was the parent material with bonded electroneutral diol groups on its outer surface and sulfonic groups on its inner surface, and the MM1164 sample was the original sample with external electroneutral diol groups and internal n-octadecyl groups. The properties examined were the specific surface area S(p) and the specific pore volume V(p), the pore connectivity c, the pore anisotropy b, the tortuosity tau, and the lacunarity lambda of the pore network as well as the percentage microporosity. These properties provide a complete characterization of complexity of the porous network. The surface areas of the solids were estimated via the traditional BET plots (S(BET)) and the I-point method (S(I)). The two sets of values S(BET) and S(I) were practically identical and they decrease as the size of the functional group increases. The values of percentage microporosity were also determined by the same I-point method using the variation of the C parameter of the BET equation. The total pore volume V(p) was found to be higher in the case of methanol adsorption, compared to nitrogen, which might be related to increase condensation. The networks of the pores were simulated using a dual site bond model (DSBM) and Monte Carlo (MC) techniques for achieving their proper arrangement into the solids. From the resulting simulating networks, the pore connectivity distributions (PCD) and their mean values c(mean) were estimated and favorably compared to the values of connectivity c(Seaton) determined according to the method of Seaton. Both values decrease with the size of the functional groups and are weakly affected by the adsorbent employed. From the simulation pore network, the mean values of tortuosity tau(mean) were also estimated and found to be lower when N2 was used as adsorbate compared to MeOH. The values of lacunarity lambda, estimated according to the method by Allain and Cloitre using the moving box technique in the DSBM/MC simulation matrix of the pore network, indicate that the distribution of the poreless mass into the matrix increases with the size of the functional group. Finally, the internal relationships observed between the pore anisotropy b and the percentage microporosity as well as between the tortuosity tau and the pore connectivity c are discussed.

Complexity of common subsequence and supersequence problems and related problems

The article examines polynomial-time and intractable longest common subsequence and subword problems and shortest common supersequence and superword problems, both old and new. The results provide a more complete complexity characterization of these problems. Some applications are discussed, as well as the dual problems of common nonsubwords, nonsuperwords, nonsubsequences, and nonsupersequences.