Polynomial Time Approximation Scheme Research Articles

Affine optimal k-proper connected edge colorings

AbstractWe introduce affine optimalk-proper connected edge colorings as a variation on Fujita’s notion of optimalk-proper connected colorings (Fujita in Optim Lett 14(6):1371–1380, 2020. https://doi.org/10.1007/s11590-019-01442-9) with applications to the frequency assignment problem. Here, for a simple undirected graph G with edge set $$E_G$$ E G , such a coloring corresponds to a decomposition of $$E_G$$ E G into color classes $$C_1, C_2, \ldots , C_n$$ C 1 , C 2 , … , C n , with associated weights $$w_1, w_2, \ldots , w_n$$ w 1 , w 2 , … , w n , minimizing a specified affine function $${\mathcal {A}}\, {:=}\,\sum _{i=1}^{n} \left( w_i \cdot |C_i|\right)$$ A : = ∑ i = 1 n w i · | C i | , while also ensuring the existence of k vertex disjoint proper paths (i.e., simple paths with no two adjacent edges in the same color class) between all pairs of vertices. In this context, we define $$\zeta _{{\mathcal {A}}}^k(G)$$ ζ A k ( G ) as the minimum possible value of $${\mathcal {A}}$$ A under a k-proper connectivity requirement. For any fixed number of color classes, we show that computing $$\zeta _{{\mathcal {A}}}^k(G)$$ ζ A k ( G ) is treewidth fixed parameter tractable. However, we also show that determining $$\zeta _{{\mathcal {A}}^{\prime }}^k(G)$$ ζ A ′ k ( G ) with the affine function $${\mathcal {A}}^{\prime } \, {:=}\,0 \cdot |C_1| + |C_2|$$ A ′ : = 0 · | C 1 | + | C 2 | is NP-hard for 2-connected planar graphs in the case where $$k = 1$$ k = 1 , cubic 3-connected planar graphs for $$k = 2$$ k = 2 , and k-connected graphs $$\forall k \ge 3$$ ∀ k ≥ 3 . We also show that no fully polynomial-time randomized approximation scheme can exist for approximating $$\zeta _{{\mathcal {A}}^{\prime }}^k(G)$$ ζ A ′ k ( G ) under any of the aforementioned constraints unless $$NP=RP$$ N P = R P .

Algorithms for the ferromagnetic Potts model on expanders

Abstract We give algorithms for approximating the partition function of the ferromagnetic $q$ -color Potts model on graphs of maximum degree $d$ . Our primary contribution is a fully polynomial-time approximation scheme for $d$ -regular graphs with an expansion condition at low temperatures (that is, bounded away from the order-disorder threshold). The expansion condition is much weaker than in previous works; for example, the expansion exhibited by the hypercube suffices. The main improvements come from a significantly sharper analysis of standard polymer models; we use extremal graph theory and applications of Karger’s algorithm to count cuts that may be of independent interest. It is #BIS-hard to approximate the partition function at low temperatures on bounded-degree graphs, so our algorithm can be seen as evidence that hard instances of #BIS are rare. We also obtain efficient algorithms in the Gibbs uniqueness region for bounded-degree graphs. While our high-temperature proof follows more standard polymer model analysis, our result holds in the largest-known range of parameters $d$ and $q$ .

Information Design for Congestion Games with Unknown Demand

We study a novel approach to information design in the standard traffic model of network congestion games. It captures the natural condition that the demand is unknown to the users of the network. A principal (e.g., a mobility service) commits to a signaling strategy, observes the realized demand and sends a (public) signal to agents (i.e., users of the network). Based on the induced belief about the demand, the users then form an equilibrium. We consider the algorithmic goal of the principal: Compute a signaling scheme that minimizes the expected total cost of the induced equilibrium. We concentrate on single-commodity networks and affine cost functions, for which we obtain the following results. First, we devise a fully polynomial-time approximation scheme (FPTAS) for the case that the demand can only take two values. It relies on several structural properties of the cost of the induced equilibrium as a function of the updated belief about the distribution of demands. We show that this function is piecewise linear for any number of demands, and monotonic for two demands. Second, we give a complete characterization of the graph structures for which it is optimal to fully reveal the information about the realized demand. This signaling scheme turns out to be optimal for all cost functions and probability distributions over demands if and only if the graph is series-parallel. Third, we propose an algorithm that computes the optimal signaling scheme for any number of demands whose time complexity is polynomial in the number of supports that occur in a Wardrop equilibrium for some demand. Finally, we conduct a computational study that tests this algorithm on real-world instances.

Approximation Scheme for Weighted Metric Clustering via Sherali-Adams

Motivated by applications to classification problems on metric data, we study Weighted Metric Clustering problem: given a metric d over n points and a k x k symmetric matrix A with non-negative entries, the goal is to find a k-partition of these points into clusters C1,...,Ck, while minimizing the sum of A[i,j] * d(u,v) over all pairs of clusters Ci and Cj and all pairs of points u from Ci and v from Cj. Specific choices of A lead to Weighted Metric Clustering capturing well-studied graph partitioning problems in metric spaces, such as Min-Uncut, Min-k-Sum, Min-k-Cut, and more. Our main result is that Weighted Metric Clustering admits a polynomial-time approximation scheme (PTAS). Our algorithm handles all the above problems using the Sherali-Adams linear programming relaxation. This subsumes several prior works, unifies many of the techniques for various metric clustering objectives, and yields a PTAS for several new problems, including metric clustering on manifolds and a new family of hierarchical clustering objectives. Our experiments on the hierarchical clustering objective show that it better captures the ground-truth structural information compared to the popular Dasgupta's objective.

Competition among Pairwise Lottery Contests

We investigate a two-stage competitive model involving multiple contests. In this model, each contest designer chooses two participants from a pool of candidate contestants and determines the biases. Contestants strategically distribute their efforts across various contests within their budget. We first show the existence of a pure strategy Nash equilibrium (PNE) for the contestants, and propose a fully polynomial-time approximation scheme to compute an approximate PNE. In the scenario where designers simultaneously decide the participants and biases, the subgame perfect equilibrium (SPE) may not exist. Nonetheless, when designers' decisions are made in two substages, the existence of SPE is established. In the scenario where designers can hold multiple contests, we show that the SPE always exists under mild conditions and can be computed efficiently.

The Near Exact Bin Covering Problem

AbstractWe present a new generalization of the bin covering problem that is known to be a strongly NP-hard problem. In our generalization there is a positive constant $$\varDelta $$ Δ , and we are given a set of items each of which has a positive size. We would like to find a partition of the items into bins. We say that a bin is near exact covered if the total size of items packed into the bin is between 1 and $$1+\varDelta $$ 1 + Δ . Our goal is to maximize the number of near exact covered bins. If $$\varDelta =0$$ Δ = 0 or $$\varDelta >0$$ Δ > 0 is given as part of the input, our problem is shown here to have no approximation algorithm with a bounded asymptotic approximation ratio (assuming that $$P\ne NP$$ P ≠ N P ). However, for the case where $$\varDelta >0$$ Δ > 0 is seen as a constant, we present an asymptotic fully polynomial time approximation scheme (AFPTAS) that is our main contribution.

Rescheduling with New Orders Under Bounded Disruption

Rescheduling problems arise when unpredicted events occur, such as the arrival of new orders. These new jobs should be integrated in a proper way in the existing schedule of the so-called old jobs, with the aim of minimizing an objective function for the joint set of jobs. To avoid a major disruption of the original schedule, each old job is not allowed to deviate from its original completion time by more than a certain threshold. Filling a gap in the existing literature, we consider the minimization of the total weighted completion time. The resulting rescheduling problem is shown to be weakly NP-hard and several properties of the structure of an optimal schedule are derived. These can be used for the construction of an exact dynamic programming algorithm with pseudo-polynomial running time. A fully polynomial time approximation scheme is obtained from the dynamic program by three different scaling and reduction steps. Finally, for the minimization of the number of late jobs a strong NP-hardness result is derived. History: Accepted by Erwin Pesch, Area Editor for Heuristic Search & Approximation Algorithms. Funding: This work was partially supported by the Ministero dell’Istruzione, dell’Università e della Ricerca [Award TESUN-83486178370409 finanziamento dipartimenti di eccellenza CAP. 1694 TIT. 232 ART. 6]. U. Pferschy acknowledges support by the Field of Excellence COLIBRI at the University of Graz.

Coordinating scheduling and rejection decisions in a two-machine flow shop scheduling problem

We study a two-machine flow shop scheduling problem where any operation can be rejected at a certain cost. A solution for such a problem requires two sets of decisions. The first involves the partition of the set of operations into two subsets: the set of operations that are accepted for scheduling in the shop, and the set of rejected operations. The second decision involves scheduling the set of accepted operations in the shop. The objective is to find a solution that minimizes the sum of the makespan and the total rejection cost. We prove that the problem is NP-hard even if all processing operations have identical processing times and identical rejection costs on either one of the two machines. We show, however, that the problem is fixed parameterized tractable with respect to a parameter that combine the number of different processing times on both machines with the number of different rejection costs on one out of the two machines. We also provide a pseudo-polynomial time algorithm for the problem, which we then convert into a fully polynomial time approximation scheme. This is achieved by dividing the problem into a set of subproblems and deriving a fully polynomial time approximation scheme for each one of them, separately. Finally, we present an integer linear programming formulation of the problem and two simple 2-approximation algorithms.

A polynomial-time approximation scheme for an arbitrary number of parallel identical multi-stage flow-shops

A polynomial-time approximation scheme for an arbitrary number of parallel identical multi-stage flow-shops

Approximation schemes for scheduling jobs on identical parallel machines to minimize the maximum lateness and makespan

We consider a multiobjective scheduling problem, with the aim of minimizing the maximum lateness and the makespan on identical machines, when the number of machines is fixed. This paper  proposes an exact algorithm (based on a dynamic programming) to generate the complete Pareto Frontier in a pseudo-polynomial time. Moreover, four heuristics have been proposed in order to optimize our algorithm. Then, we present a Polynomial Time Approximation Scheme (PTAS) to generate an approximate Pareto Frontier. In this scheme, we use a simplification technique based on the merging of jobs. Furthermore, we present a  Fully Polynomial Time Approximation Scheme (FPTAS) to generate an approximate Pareto Frontier, based on the conversion of the dynamic programming algorithm. The proposed FPTAS is strongly polynomial. Finally, some numerical experiments are provided in order to compare the proposed approaches.

Flow shop scheduling problems with transportation constraints revisited

This paper investigates two flow shop scheduling problems with two machines A and B, and a single transporter V. In the first problem, the transporter V is located at machine A initially, each job has to be processed first on A, then transported to B for further processing. While in the second problem, the transporter V is located at machine B initially, each job needs to be processed first on A, then on B, finally transported to the destination. In both problems, the transporter V can carry up to c (where c is a constant and c≥1) jobs at a time and the objective is to minimize the makespan. For the former problem, the best approximation algorithm guarantees a worst-case ratio bound of (53+ε), while for the latter one, the best approximation algorithm provides a worst-case ratio bound of 2. For each problem, we design a polynomial-time approximation scheme (PTAS).

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.

Pliability and Approximating Max-CSPs

We identify a sufficient condition, treewidth-pliability , that gives a polynomial-time algorithm for an arbitrarily good approximation of the optimal value in a large class of Max-2-CSPs parameterised by the class of allowed constraint graphs (with arbitrary constraints on an unbounded alphabet). Our result applies more generally to the maximum homomorphism problem between two rational-valued structures. The condition unifies the two main approaches for designing a polynomial-time approximation scheme. One is Baker’s layering technique, which applies to sparse graphs such as planar or excluded-minor graphs. The other is based on Szemerédi’s regularity lemma and applies to dense graphs. We extend the applicability of both techniques to new classes of Max-CSPs. However, we prove that the condition cannot be used to find solutions (as opposed to approximating the optimal value) in general. Treewidth-pliability turns out to be a robust notion that can be defined in several equivalent ways, including characterisations via size, treedepth, or the Hadwiger number. We show connections to the notions of fractional-treewidth-fragility from structural graph theory, hyperfiniteness from the area of property testing, and regularity partitions from the theory of dense graph limits. These may be of independent interest. In particular, we show that a monotone class of graphs is hyperfinite if and only if it is fractionally-treewidth-fragile and has bounded degree.

Approximation algorithms for scheduling parallel machines with an energy constraint in green manufacturing

Motivated by current green manufacturing standards, in this paper we study a parallel-machine scheduling model in which the energy cost incurred on each machine is machine-dependent and proportional to the load of the machine. The objective is to determine a production schedule with the minimum makespan subject to the energy constraint that the total energy cost does not exceed a given bound. We provide a technical lemma that enables us to design a very efficient approximation algorithm with a worst-case bound that can arbitrarily approach 43, improving on the existing performance ratio of 33+14≈1.686 in the literature. By introducing the concept of a monotonic schedule, we are able to develop the first polynomial time approximation scheme for this scheduling problem. The scheduling problem studied in this paper is an important special case of the generalized assignment problem (GAP). Our techniques and results bring new insights into research on the GAP.

Scheduling on identical machines with preemption and setup times

In this paper, we address the problem of scheduling jobs on identical machines for minimising the maximum completion time (makespan). Each job requires a sequence-independent setup time, which represents the time needed to prepare the machines for job execution. Then, we introduce a dynamic programme to solve the case with two machines, and show that this problem admits a fully polynomial time approximation scheme. For the case of m machines, we propose heuristics and an adapted genetic algorithm. Some numerical experiments are done to evaluate the proposed algorithms.

Machine scheduling with restricted rejection: An Application to task offloading in cloud–edge collaborative computing

With the burgeoning of the Internet of everything, the amount of data generated by edge devices increases dramatically. In order to relieve the huge pressure of the could computing center, a popular computing scheme, called edge computing, is to select and process part of the computation tasks on edge servers of the network. In this paper we model the task offloading problem motivated by the popular Cloud–Edge Collaborative Computing Frame as a parallel-machine scheduling problem with restricted job rejection. We present an easy-to-implement heuristic with worst-case bound analysis and polynomial time approximation schemes for the general problem and some of the important special cases.

Display optimization under the multinomial logit choice model: Balancing revenue and customer satisfaction

In this paper, we consider an assortment optimization problem in which a platform must choose pairwise disjoint sets of assortments to offer across a series of T stages. Arriving customers begin their search process in the first stage, and progress sequentially through the stages until their patience expires, at which point they make a multinomial logit–based purchasing decision from among all products they have viewed throughout their search process. The goal is to choose the sequential displays of product offerings to maximize expected revenue. Additionally, we impose stage‐specific constraints that ensure that as each customer progresses farther and farther through the T stages, there is a minimum level of “desirability” met by the collections of displayed products. We consider two related measures of desirability: purchase likelihood and expected utility derived from the offered assortments. In this way, the offered sequence of assortments must be both high earning and well liked, which breaks from the traditional assortment setting, where customer‐centric considerations are generally not explicitly accounted for. We show that our assortment problem of interest is strongly NP‐Hard, thus ruling out the existence of a fully polynomial‐time approximation scheme (FPTAS). From an algorithmic standpoint, as a warm‐up, we develop a simple constant factor approximation scheme in which we carefully stitch together myopically selected assortments for each stage. Our main algorithmic result consists of a polynomial‐time approximation scheme (PTAS), which combines a handful of structural results related to the make‐up of the optimal assortment sequence within an approximate dynamic programming framework. We also provide an additional approximation scheme, which, under mild assumptions, can handle a cardinality constraint that enforces that an exact number of new products are introduced at each stage. Using an extensive set of numerical experiments, we demonstrate that both algorithms exhibit excellent practical performance, producing sequences of assortments that are, on average, always within 2% of optimal.

An FPTAS for budgeted laminar matroid independent set

We study the budgeted laminar matroid independent set problem. The input is a ground set, where each element has a cost and a non-negative profit, along with a laminar matroid over the elements and a budget. The goal is to select a maximum profit independent set of the matroid whose total cost is bounded by the budget. We present a fully polynomial-time approximation scheme (FPTAS) for the problem, improving the previous efficient PTAS for this matroid family.

Fair allocation algorithms for indivisible items under structured conflict constraints

We consider the fair allocation of indivisible items to several agents with additional conflict constraints. These are represented by a conflict graph where each item corresponds to a vertex of the graph and edges in the graph represent incompatible pairs of items which should not be allocated to the same agent. This setting combines the issues of Partition and Independent Set and can be seen as a partial coloring of the conflict graph. In the resulting optimization problem, each agent has its own valuation function for the profits of the items. We aim at maximizing the lowest total profit obtained by any of the agents. In a previous paper, this problem was shown to be strongly NP-hard for several well-known graph classes, e.g., bipartite graphs and their line graphs. On the other hand, it was shown that pseudo-polynomial time algorithms exist for the classes of chordal graphs, cocomparability graphs, biconvex bipartite graphs, and graphs of bounded treewidth. In this contribution, we extend this line of research by developing pseudo-polynomial time algorithms that solve the problem for the class of convex bipartite conflict graphs, graphs of bounded clique-width, and graphs of bounded tree-independence number. The algorithms are based on dynamic programming and also permit fully polynomial-time approximation schemes (FPTAS).

Inapproximability of Positive Semidefinite Permanents and Quantum State Tomography

AbstractMatrix permanents are hard to compute or even estimate in general. It had been previously suggested that the permanents of Positive Semidefinite (PSD) matrices may have efficient approximations. By relating PSD permanents to a task in quantum state tomography, we show that PSD permanents are NP-hard to approximate within a constant factor, and so admit no polynomial-time approximation scheme (unless P = NP). We also establish that several natural tasks in quantum state tomography, even approximately, are NP-hard in the dimension of the Hilbert space. These state tomography tasks therefore remain hard even with only logarithmically few qubits.