Develop Polynomial Time Algorithms Research Articles

Joint pricing and inventory management with deterministic demand and costly price adjustment

We analyze a joint inventory and pricing model of a single product over a finite planning horizon with deterministic demand. In this model, an ordering quantity and a price are decided simultaneously at the beginning of each period, demand of the period depends on the price, and a price adjustment cost is incurred if the price is changed from the previous period. We develop polynomial time algorithms to maximize the total profit and discuss their computational complexity.

Independent Directed Acyclic Graphs for Resilient Multipath Routing

In order to achieve resilient multipath routing, we introduce the concept of independent directed acyclic graphs (IDAGs) in this paper. Link-independent (node-independent) DAGs satisfy the property that any path from a source to the root on one DAG is link-disjoint (node-disjoint) with any path from the source to the root on the other DAG. Given a network, we develop polynomial-time algorithms to compute link-independent and node-independent DAGs. The algorithm developed in this paper: 1) provides multipath routing; 2) utilizes all possible edges; 3) guarantees recovery from single link failure; and 4) achieves all these with at most one bit per packet as overhead when routing is based on destination address and incoming edge. We show the effectiveness of the proposed IDAGs approach by comparing key performance indices to that of the independent trees and multiple pairs of independent trees techniques through extensive simulations.

Assessing the Vulnerability of the Fiber Infrastructure to Disasters

Communication networks are vulnerable to natural disasters, such as earthquakes or floods, as well as to physical attacks, such as an electromagnetic pulse (EMP) attack. Such real-world events happen in specific geographical locations and disrupt specific parts of the network. Therefore, the geographical layout of the network determines the impact of such events on the network's connectivity. In this paper, we focus on assessing the vulnerability of (geographical) networks to such disasters. In particular, we aim to identify the most vulnerable parts of the network. That is, the locations of disasters that would have the maximum disruptive effect on the network in terms of capacity and connectivity. We consider graph models in which nodes and links are geographically located on a plane. First, we consider a simplistic bipartite graph model and present a polynomial-time algorithm for finding a worst-case vertical line segment cut. We then generalize the network model to graphs with nodes at arbitrary locations. We model the disaster event as a line segment or a disk and develop polynomial-time algorithms that find a worst-case line segment cut and a worst-case circular cut. Finally, we obtain numerical results for a specific backbone network, thereby demonstrating the applicability of our algorithms to real-world networks. Our novel approach provides a promising new direction for network design to avert geographical disasters or attacks.

Capacity of wireless networks under SINR interference constraints

A fundamental problem in wireless networks is to estimate their throughput capacity--given a set of wireless nodes and a set of connections, what is the maximum rate at which data can be sent on these connections. Most of the research in this direction has focused either on random distributions of points, or has assumed simple graph-based models for wireless interference. In this paper, we study the capacity estimation problem using a realistic Signal to Interference Plus Noise Ratio (SINR) model for interference, on arbitrary wireless networks without any assumptions on node distributions. The problem becomes much more challenging for this setting, because of the non-locality of the SINR model. Recent work by Moscibroda et al. (IEEE INFOCOM 2006, ACM MobiHoc 2006) has shown that the throughput achieved by using SINR models can differ significantly from that obtained by using graph-based models. In this work, we develop polynomial time algorithms to provably approximate the throughput capacity of wireless network under the SINR model.

Reconstruction of interval graphs

The graph reconstruction conjecture is a long-standing open problem in graph theory. There are many algorithmic studies related to it, such as DECK CHECKING, LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING. We study these algorithmic problems by limiting the graph class to interval graphs. Since we can solve GRAPH ISOMORPHISM for interval graphs in polynomial time, DECK CHECKING for interval graphs is easily done in polynomial time. Since the number of interval graphs that can be obtained from an interval graph by adding a vertex and edges incident to it can be exponentially large, developing polynomial time algorithms for LEGITIMATE DECK, PREIMAGE CONSTRUCTION, and PREIMAGE COUNTING on interval graphs is not trivial. We present that these three problems are solvable in polynomial time on interval graphs.

Stochastic lot-sizing with backlogging: computational complexity analysis

For a multi-stage stochastic programming problem, one approach is to explore a scenario tree based formulation for the problem and solve the formulation efficiently. There has been significant research progress on how to generate scenario trees in the literature. However, there is limited work on analyzing the computational complexity of the scenario-tree based formulation that considers the number of tree nodes as the input size. In this paper, we use stochastic lot-sizing problems as examples to study the computational complexity issues for the scenario-tree based formulations. We develop production path properties and a general dynamic programming framework based on these properties. The dynamic programming framework allows us to show that the optimal value function is piecewise linear and continuous, which enables us to develop polynomial time algorithms for several different problems, including those with backlogging and varying capacities under certain conditions. As special cases, we develop polynomial time algorithms that run in $${\mathcal{O}(n^2)}$$ and $${\mathcal{O}(n^2T\,{\rm log}\,n)}$$ times, respectively for stochastic uncapacitated and constant capacitated lot-sizing problems with backlogging, regardless of the scenario tree structure.

On the complexity of the dominating induced matching problem in hereditary classes of graphs

The dominating induced matching problem, also known as efficient edge domination, is the problem of determining whether a graph has an induced matching that dominates every edge of the graph. This problem is known to be NP-complete. We study the computational complexity of the problem in special graph classes. In the present paper, we identify a critical class for this problem (i.e., a class lying on a “boundary” separating difficult instances of the problem from polynomially solvable ones) and derive a number of polynomial-time results. In particular, we develop polynomial-time algorithms to solve the problem for claw-free graphs and convex graphs.

Deterministic risk control for cost-effective network connections

This paper considers the minimum connection problem in networks with uncertain data. In such a network it is assumed that one can establish a link e by paying a cost c e in a given interval [ c e − , c e + ] while taking a risk ( c e + − c e ) / ( c e + − c e − ) of link failure. We develop polynomial time algorithms for minimum cost network connection with paths or spanning trees under risk-sum constraints.

MULTI-TERMINAL NETWORK CONNECTEDNESS ON SERIES-PARALLEL NETWORKS

Network operation may require that a specified number k of nodes be able to communicate via paths consisting of operating edges and nodes. In an environment of node and edge failure, this leads to associated reliability measures. When the k nodes are known in advance, this has been widely studied as k-terminal reliability; when the k nodes are chosen uniformly at random, this has been studied as k-resilience. A third notion, when it suffices to have anyk nodes communicate, is related to the expected size of the largest component in the network. We generalize these three measures to the probability that given h nodes chosen in advance and i nodes chosen at random, they appear in a component of size at least k = h + i + j. As expected, for general networks, for most choices of (h, i, j) the computation is #P-complete and hence unlikely to admit a polynomial time algorithm. We develop polynomial time algorithms in the special case that the network is series-parallel, which subsume and generalize earlier methods for k-terminal reliability and k-resilience.

Logistics scheduling to minimize inventory and transport costs

We study a logistics scheduling problem where a manufacturer receives raw materials from a supplier, manufactures products in a factory, and delivers the finished products to a customer. The supplier, factory and customer are located at three different sites. The objective is to minimize the sum of work-in-process inventory cost and transport cost, which includes both supply and delivery costs. For the special case of the problem where all the jobs have identical processing times, we show that the inventory cost function can be unified into a common expression for various batching schemes. Based on this characteristic and other optimal properties, we develop an O( n) algorithm to solve this case. For the general problem, we examine several special cases, identify their optimal properties, and develop polynomial-time algorithms to solve them optimally.

Two-echelon requirements planning with pricing decisions

We consider a two-level uncapacitated lot-sizing problem where production, inventory carrying, transportation, and pricing decisions are integrated to maximize total profits. We show how this problem, under many different revenue functions and production, inventory holding, and transportation cost structures can be solved in polynomial time. As a byproduct, we develop polynomial-time algorithms for generalizations of single-level lot-sizing problems with pricing as well.

Computing correlated equilibria in multi-player games

We develop polynomial-time algorithms for finding correlated equilibria—a well-studied notion of rationality that generalizes the Nash equilibrium—in a broad class of succinctly representable multiplayer games, encompassing graphical games, anonymous games, polymatrix games, congestion games, scheduling games, local effect games, as well as several generalizations. Our algorithm is based on a variant of the existence proof due to Hart and Schmeidler, and employs linear programming duality, the ellipsoid algorithm, Markov chain steady state computations, as well as application-specific methods for computing multivariate expectations over product distributions. For anonymous games and graphical games of bounded tree-width, we provide a different polynomial-time algorithm for optimizing an arbitrary linear function over the set of correlated equilibria of the game. In contrast to our sweeping positive results for computing an arbitrary correlated equilibrium, we prove that optimizing over correlated equilibria is NP-hard in all of the other classes of games that we consider.

Center location problems on tree graphs with subtree-shaped customers

We consider the p-center problem on tree graphs where the customers are modeled as continua subtrees. We address unweighted and weighted models as well as distances with and without addends. We prove that a relatively simple modification of Handler’s classical linear time algorithms for unweighted 1- and 2-center problems with respect to point customers, linearly solves the unweighted 1- and 2-center problems with addends of the above subtree customer model. We also develop polynomial time algorithms for the p-center problems based on solving covering problems and searching over special domains.

Batching and scheduling in a multi-machine flow shop

We study the problem of batching and scheduling n jobs in a flow shop comprising m, m?2, machines. Each job has to be processed on machines 1,?,m in this order. Batches are formed on each machine. A machine dependent setup time precedes the processing of each batch. Jobs of the same batch are processed on each machine sequentially so that the processing time of a batch is equal to the sum of the processing times of the jobs contained in it. Jobs of the same batch formed on machine l become available for a downstream operation on machine l+1 at the same time when the processing of the last job of the batch on machine l has been finished. The objective is to minimize maximum job completion time. We establish several properties of an optimal schedule and develop polynomial time algorithms for important special cases. They are improvements over the existing methods with regard to their generality and time efficiency.

Operational fixed interval scheduling problem on uniform parallel machines

In this study, we address an operational fixed interval scheduling problem on uniform parallel machines. Our objective is to maximize the total weight of the jobs processed. We show that the problem is NP-hard in the strong sense and develop polynomial time algorithms for some special cases. We propose a branch and bound algorithm that employs dominance conditions and tight bounds. The results of our computational tests have revealed that the algorithm returns optimal solutions for problem instances with up to 70 jobs very quickly.

Games on triangulations

We analyze several perfect-information combinatorial games played on planar triangulations. We introduce three broad categories of such games—constructing, transforming, and marking triangulations—and several specific games within each category. In various situations of each game, we develop polynomial-time algorithms to determine who wins a given game position under optimal play, and to find a winning strategy. Along the way, we show connections to existing combinatorial games such as Kayles and Nimstring (a variation on Dots-and-Boxes).

Pareto-optimal patterns in logical analysis of data

Patterns are the key building blocks in the logical analysis of data (LAD). It has been observed in empirical studies and practical applications that some patterns are more “suitable” than others for use in LAD. In this paper, we model various such suitability criteria as partial preorders defined on the set of patterns. We introduce three such preferences, and describe patterns which are Pareto-optimal with respect to any one of them, or to certain combinations of them. We develop polynomial time algorithms for recognizing Pareto-optimal patterns, as well as for transforming an arbitrary pattern to a better Pareto-optimal one with respect to any one of the considered criteria, or their combinations. We obtain analytical representations characterizing some of the sets of Pareto-optimal patterns, and investigate the computational complexity of generating all Pareto-optimal patterns. The empirical evaluation of the relative merits of various types of Pareto-optimality is carried out by comparing the classification accuracy of Pareto-optimal theories on several real life data sets. This evaluation indicates the advantages of “strong patterns”, i.e. those patterns which are Pareto-optimal with respect to the “evidential preference” introduced in this paper.

Machine scheduling with earliness, tardiness and non-execution penalties

The study of scheduling problems with earliness–tardiness (E/T) penalties is motivated by the just-in-time (JIT) philosophy, which supports the notion that earliness, as well as tardiness, should be discouraged. In this work, we consider several scheduling problems. We begin by generalizing a known polynomial time algorithm that calculates an optimal schedule for a given sequence of tasks, on a single machine, assuming that the tasks have distinct E/T penalty weights, distinct processing times and distinct due dates. We then present new results to problems, where tasks have common processing times. We also introduce a new concept in E/T scheduling problems, where we allow the non-execution of tasks and consequently, are penalized for each non-executed task. The notion of task's non-execution, coincides with the JIT philosophy in that every violation or a breach of an agreement, should be penalized. We develop polynomial time algorithms for special cases in E/T scheduling problems with non-execution penalties.

Algorithmic results for ordered median problems

In a recent series of papers a new type of objective function in location theory, called ordered median function, has been introduced and analyzed. This objective function unifies and generalizes most common objective functions used in location theory. In this paper we identify finite dominating sets for these models and develop polynomial time algorithms together with a detailed complexity analysis.

On functional dependencies in q-Horn theories

This paper studies functional dependencies in q-Horn theories, and discusses their use in knowledge condensation. We introduce compact model-based representations of q-Horn theories, analyze the structure of functional dependencies in q-Horn theories, and show that every minimal functional dependency in a q-Horn theory Σ can be expressed either by a single term or by a single clause. We also prove that the set of all minimal functional dependencies in Σ is quasi-acyclic. We then develop polynomial time algorithms for recognizing whether a given functional dependency holds in a q-Horn theory, which is represented either by a q-Horn CNF or as the q-Horn envelope of a set of models. Finally, we show that every q-Horn theory has a unique condensation, and can be condensed in polynomial time.