Polynomial Time Approximation Scheme Research Articles

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.

Approximability of the complementarily symmetric Holant problems on cubic graphs

We study the approximability of boolean Holant problems with complementarily symmetric constraint functions on cubic graphs. We identify a regime in which the problem admits an FPRAS (fully polynomial-time randomized approximation scheme) and specify conditions under which the problem is hard to approximate. This is the first attempt to classify the approximability of Holant problems with asymmetric constraint functions on cubic graphs.

Models, Algorithms and Approximation Results for a Bi-Level Synchronized Knapsack Problem

We describe here a bi-level Knapsack problem which involves an Eater/Feeder interaction between 2 Knapsack systems and expresses the collaboration between a local resource provider and a consumer. We first describe the SFEK: Synchronized Feeder/Eater Knapsack problem in an accurate way, and set an ILP model, enhanced with additional valid cuts. Next we focus on the complexity issue. We design a Dynamic Programming algorithm which allows us to state that SFEK is pseudo-polynomial and derive for this problem a polynomial time approximation scheme. We conclude with a brief discussion about the Collaborative issue.

The Complexity of Aggregates over Extractions by Regular Expressions

Regular expressions with capture variables, also known as regex-formulas, extract relations of spans (intervals identified by their start and end indices) from text. In turn, the class of regular document spanners is the closure of the regex formulas under the Relational Algebra. We investigate the computational complexity of querying text by aggregate functions, such as sum, average, and quantile, on top of regular document spanners. To this end, we formally define aggregate functions over regular document spanners and analyze the computational complexity of exact and approximate computation. More precisely, we show that in a restricted case, all studied aggregate functions can be computed in polynomial time. In general, however, even though exact computation is intractable, some aggregates can still be approximated with fully polynomial-time randomized approximation schemes (FPRAS).

On the complexity of the Cable-Trench Problem

The Cable-Trench Problem (CTP) is a common generalization of the Single-Source Shortest Paths Problem (SSSP) and the Minimum Spanning Tree Problem (MST): given an edge-weighted graph with a special root vertex and parameters τ,γ≥0, the goal is to find a spanning tree that minimizes the total edge costs plus the total cost of the paths from each vertex to the root, scaled by τ and γ, respectively. While it is well known that both SSSP and MST can be solved in polynomial time, CTP is NP-hard. We show that computing an approximate solution with factor less than 1.000475 is NP-hard, thus ruling out a polynomial-time approximation scheme, unless P=NP.We also consider the more general Steiner Cable-Trench Problem (SCTP), for which only a given subset of terminal vertices must be spanned by a solution. The tree might include non-terminal vertices, known as Steiner vertices, although only paths from terminals to the root are considered in the total cost. For this problem, we present a (2.88+ϵ)-approximation based on a counting argument, for any ϵ>0; also, we give a simple parameterized algorithm with the number of terminals as parameter.

Peak Demand Minimization via Sliced Strip Packing

We study the Non-preemptive Peak Demand Minimization (NPDM) problem, where we are given a set of jobs, specified by their processing times and energy requirements. The goal is to schedule all jobs within a fixed time period such that the peak load (the maximum total energy requirement at any time) is minimized. This problem has recently received significant attention due to its relevance in smart-grids. Theoretically, the problem is related to the classical strip packing problem (SP). In SP, a given set of axis-aligned rectangles must be packed into a fixed-width strip, such that the height of the strip is minimized. NPDM can be modeled as strip packing with slicing and stacking constraint: each rectangle may be cut vertically into multiple slices and the slices may be packed into the strip as individual pieces. The stacking constraint forbids solutions where two slices of the same rectangle are intersected by the same vertical line. Non-preemption enforces the slices to be placed in contiguous horizontal locations (but may be placed at different vertical locations). We obtain a (5/3+varepsilon )-approximation algorithm for the problem. We also provide an asymptotic efficient polynomial-time approximation scheme (AEPTAS) which generates a schedule for almost all jobs with energy consumption (1+varepsilon ) {textrm{OPT}}. The remaining jobs fit into a thin container of height 1. This AEPTAS is used as a subroutine to acquire the (5/3+varepsilon )-approximation algorithm. The previous best result for NPDM was a 2.7-approximation based on FFDH (Ranjan et al., in: 2015 IEEE symposium on computers and communication (ISCC), pp 758–763, IEEE, 2015). One of our key ideas is providing several new lower bounds on the optimal solution of a geometric packing, which could be useful in other related problems. These lower bounds help us to obtain approximative solutions based on Steinberg’s algorithm in many cases. In addition, we show how to split schedules generated by the AEPTAS into few segments and to rearrange the corresponding jobs to insert the thin container mentioned above, such that it does not exceed the bound of (5/3+varepsilon ) {textrm{OPT}}.

Simulated Annealing is a Polynomial-Time Approximation Scheme for the Minimum Spanning Tree Problem

We prove that Simulated Annealing with an appropriate cooling schedule computes arbitrarily tight constant-factor approximations to the minimum spanning tree problem in polynomial time. This result was conjectured by Wegener (Automata, Languages and Programming, ICALP, Berlin, 2005). More precisely, denoting by n, m, w_{max }, and w_{min } the number of vertices and edges as well as the maximum and minimum edge weight of the MST instance, we prove that simulated annealing with initial temperature T_0 ge w_{max } and multiplicative cooling schedule with factor 1-1/ell , where ell = omega (mnln (m)), with probability at least 1-1/m computes in time O(ell (ln ln (ell ) + ln (T_0/w_{min }) )) a spanning tree with weight at most 1+kappa times the optimum weight, where 1+kappa = frac{(1+o(1))ln (ell m)}{ln (ell ) -ln (mnln (m))}. Consequently, for any epsilon >0, we can choose ell in such a way that a (1+epsilon )-approximation is found in time O((mnln (n))^{1+1/epsilon +o(1)}(ln ln n + ln (T_0/w_{min }))) with probability at least 1-1/m. In the special case of so-called (1+epsilon )-separated weights, this algorithm computes an optimal solution (again in time O( (mnln (n))^{1+1/epsilon +o(1)}(ln ln n + ln (T_0/w_{min })))), which is a significant speed-up over Wegener’s runtime guarantee of O(m^{8 + 8/epsilon }). Our tighter upper bound also admits the result that in some situations a hybridization of simulated annealing and the {(1 + 1)} EA can lead to stronger runtime guarantees than either algorithm alone.

Approximability and efficient algorithms for constrained fixed-horizon POMDPs with durative actions

Partially Observable Markov Decision Process (POMDP) is a fundamental model for probabilistic planning in stochastic domains. More recently, constrained POMDP and chance-constrained POMDP extend the model allowing constraints to be specified on some aspects of the policy in addition to the objective function. Despite their expressive power, these models assume all actions take a fixed duration, which poses a limitation in modeling real-world planning problems. In this work, we propose a unified model for durative POMDP and its constrained extensions. First, we convert these extensions into an Integer Linear Programming (ILP) formulation, which can be solved using existing solvers in the ILP literature. Second, a heuristic search approach is provided that can efficiently prune the search space, guided by solving successive partial ILP programs. Third, we give a theoretical analysis of the problem: unlike short-horizon POMDPs, with policies of a constant depth, which can be solved in polynomial time, the constrained extensions are NP-Hard even with a planning horizon of two and non-negative rewards. To alleviate that, we propose a Fully Polynomial Time Approximation Scheme (FPTAS) that computes (near) optimal deterministic policies in polynomial time. The FPTAS is among the best achievable in theory in terms of approximation ratio. Finally, evaluation results show that our approach is empirically superior to the state-of-the-art fixed-horizon chance-constrained POMDP solver.

Single Machine Scheduling with Rejection and a Non-Availability Interval to Minimize the Maximum Delivery Completion Time Plus the Total Rejection Cost

In this paper, we consider the single machine scheduling with rejection and a nonavailability interval. Two types of non-availability intervals are studied, namely the machine non-availability (MNA) intervals and the operator non-availability (ONA) intervals. The objective is to minimize the sum of the maximum delivery completion time of accepted jobs and the total rejection cost of rejected jobs. For these two problems, a pseudo-polynomial-time dynamic programming algorithm and a fully polynomial-time approximation scheme (FPTAS) are proposed, respectively.

A Faster Practical Approximation Scheme for the Permanent

The permanent of a matrix has numerous applications but is notoriously hard to compute. While nonnegative matrices admit polynomial approximation schemes based on rapidly mixing Markov chains, the known practical estimators of the permanent rely on importance or rejection sampling. We advance the rejection sampling approach, which provides probabilistic accuracy guarantees, unlike importance sampling. Specifically, we give a novel class of nesting upper bounds and a simple preprocessing method that, in comparison to previous works, enable faster sampling with better acceptance rate; we demonstrate order-of-magnitude improvements with both theoretical and empirical analyses. In addition, we display instances on which our approximation scheme is competitive against state-of-the-art importance sampling based estimators.

An FPTAS for scheduling with resource constraints

We study a basic scheduling problem with resource constraints: A number of jobs need to be scheduled on two parallel identical machines with the objective of minimizing the makespan, subject to the constraint that jobs may require a unit of one of the given renewable resources during their execution. For this NP-hard problem, we develop a fully polynomial-time approximation scheme (FPTAS). Our FPTAS makes a novel use of existing algorithms for the subset-sum problem and the open shop scheduling problem.

Assortment Optimization Under the Multi-Purchase Multinomial Logit Choice Model

Updating a Classic: Assortment Optimization under the Multi-Purchase MNL Model In the paper “Updating a Classic: Assortment Optimization under the Multi-Purchase MNL Model,” our primary contribution resides in proposing the first multi-purchase choice model that can be fully operationalized. Our main algorithmic results consist of two distinct polynomial time approximation schemes (PTAS); the first, and simpler of the two, caters to a setting where each customer may buy only a constant number of products, whereas the second, more nuanced algorithm applies to our multi-purchase model in its general form. Additionally, we study the revenue potential of making assortment decisions that account for multi-purchase behavior in comparison with those that overlook this phenomenon. In particular, we relate both the structure and revenue performance of the optimal assortment under a traditional single-purchase model to that of the optimal assortment in the multi-purchase setting.

Bag-of-Tasks Scheduling with Rejection in Large Computing Systems

We are given a set of [Formula: see text] identical parallel machines and a set of [Formula: see text] jobs in large computing systems, where each job [Formula: see text] consists of a bag of [Formula: see text] identical tasks with a processing time [Formula: see text], and has a rejection penalty [Formula: see text]. Job [Formula: see text] is either accepted in which case all the [Formula: see text] tasks must be processed by one of the machines, or rejected which incurs a rejection penalty [Formula: see text]. The problem of bag-of-tasks scheduling with rejection is to find a feasible schedule, so as to minimize the makespan plus the total rejection penalty of all rejected jobs. In this paper, we present a polynomial time approximation scheme.

Correlation Clustering and Two-Edge-Connected Augmentation for Planar Graphs

We study two problems. In correlation clustering, the input is a weighted graph, where every edge is labelled either $$\langle +\rangle $$ or $$\langle -\rangle $$ according to whether its endpoints are in the same category or in different categories. The goal is to produce a partition of the vertices into categories that tries to respect the labels of the edges. In two-edge-connected augmentation, the input is a weighted graph and a subset R of edges of the graph. The goal is to produce a minimum weight subset S of edges of the graph, such that for every edge in R, its endpoints are two-edge-connected in $$R\cup S$$ . In this paper, we study these problems under the restriction that the input graph must be planar. We give an approximation-preserving reduction from correlation clustering on planar graphs to two-edge-connected augmentation on planar graphs. We give a polynomial-time approximation scheme (PTAS) for the latter problem, yielding a PTAS for the former problem as well. The approximation scheme employs brick decompositions, which have been used in previous approximation schemes for planar graphs, but the way it uses brick decompositions is fundamentally different from previous uses.

Provably Good Solutions to the Knapsack Problem via Neural Networks of Bounded Size

The development of a satisfying and rigorous mathematical understanding of the performance of neural networks is a major challenge in artificial intelligence. Against this background, we study the expressive power of neural networks through the example of the classical NP-hard knapsack problem. Our main contribution is a class of recurrent neural networks (RNNs) with rectified linear units that are iteratively applied to each item of a knapsack instance and thereby compute optimal or provably good solution values. We show that an RNN of depth four and width depending quadratically on the profit of an optimum knapsack solution is sufficient to find optimum knapsack solutions. We also prove the following tradeoff between the size of an RNN and the quality of the computed knapsack solution: for knapsack instances consisting of n items, an RNN of depth five and width w computes a solution of value at least [Formula: see text] times the optimum solution value. Our results build on a classical dynamic programming formulation of the knapsack problem and a careful rounding of profit values that are also at the core of the well-known fully polynomial-time approximation scheme for the knapsack problem. A carefully conducted computational study qualitatively supports our theoretical size bounds. Finally, we point out that our results can be generalized to many other combinatorial optimization problems that admit dynamic programming solution methods, such as various shortest path problems, the longest common subsequence problem, and the traveling salesperson problem. History: Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. An extended abstract of this article, including Figures 1 – 7 , appeared in the Proceedings of the AAAI Conference on Artificial Intelligence, vol. 35, 7685–7693 ( Hertrich and Skutella 2021 ); see https://ojs.aaai.org/index.php/AAAI/article/view/16939 ; copyright © 2021, Association for the Advancement of Artificial Intelligence. Funding: This work was supported by the Deutsche Forschungsgemeinschaft [Grants DFG-GRK 2434 and EXC-2046/1, Project 390685689] and the H2020 European Research Council [ScaleOpt-757481].

Parallel-batch scheduling with rejection: Structural properties and approximation algorithms

In this paper, we consider a parallel-batch machine scheduling (PBMS) model that unifies several existing scheduling models in the extant literature. In our scheduling model, a given set of jobs with different release dates are scheduled onto a machine that can process multiple jobs simultaneously in a batch. However, the number of jobs in a batch is limited. Each job can be rejected subject to a job-dependent penalty cost, but the total penalty cost of the jobs rejected is required to be no greater than a given bound. The objective is to minimize the weighted sum of the makespan of accepted jobs and the total rejection penalty of rejected jobs. We provide some important structural properties and develop fast approximation algorithms. We also present an efficient polynomial time approximation scheme (EPTAS) with near-linear-time complexity. Our results significantly improve both the time complexities and performance bounds of several existing algorithms and approximation schemes.

Knapsack problems with position-dependent item weights or profits

We consider three new knapsack problems with variable weights or profits of items, where the weight or profit of an item depends on the position of the item in the sequence of items packed in the knapsack. We show how to solve the problems exactly using dynamic programming algorithms with pseudo-polynomial running times and propose fully polynomial-time approximation schemes for their approximate solution.