Polynomial Running Time Research Articles

A Near-Optimal Solution to a Two-Dimensional Cutting Stock Problem

We present an asymptotic fully polynomial approximation scheme for strip-packing, or packing rectangles into a rectangle of fixed width and minimum height, a classical NP-hard cutting-stock problem. The algorithm, based on a new linear-programming relaxation, finds a packing of n rectangles whose total height is within a factor of (1 + ε) of optimal (up to an additive term), and has running time polynomial both in n and in 1/ε.

Algorithms for Colouring Random k-colourable Graphs

The k-colouring problem is to colour a given k-colourable graph with k colours. This problem is known to be NP-hard even for fixed k [ges ] 3. The best known polynomial time approximation algorithms require nδ (for a positive constant δ depending on k) colours to colour an arbitrary k-colourable n-vertex graph. The situation is entirely different if we look at the average performance of an algorithm rather than its worst-case performance. It is well known that a k-colourable graph drawn from certain classes of distributions can be k-coloured almost surely in polynomial time.In this paper, we present further results in this direction. We consider k-colourable graphs drawn from the random model in which each allowed edge is chosen independently with probability p(n) after initially partitioning the vertex set into k colour classes. We present polynomial time algorithms of two different types. The first type of algorithm always runs in polynomial time and succeeds almost surely. Algorithms of this type have been proposed before, but our algorithms have provably exponentially small failure probabilities. The second type of algorithm always succeeds and has polynomial running time on average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work as long as p(n) [ges ] n−1+ε where ε is a constant greater than 1/4.

Efficient Algorithms for Protein Sequence Design and the Analysis of Certain Evolutionary Fitness Landscapes

Protein sequence design is a natural inverse problem to protein structure prediction: given a target structure in three dimensions, we wish to design an amino acid sequence that is likely fold to it. A model of Sun, Brem, Chan, and Dill casts this problem as an optimization on a space of sequences of hydrophobic (H) and polar (P) monomers; the goal is to find a sequence that achieves a dense hydrophobic core with few solvent-exposed hydrophobic residues. Sun et al. developed a heuristic method to search the space of sequences, without a guarantee of optimality or near-optimality; Hart subsequently raised the computational tractability of constructing an optimal sequence in this model as an open question. Here we resolve this question by providing an efficient algorithm to construct optimal sequences; our algorithm has a polynomial running time, and performs very efficiently in practice. We illustrate the implementation of our method on structures drawn from the Protein Data Bank. We also consider extensions of the model to larger amino acid alphabets, as a way to overcome the limitations of the binary H/P alphabet. We show that for a natural class of arbitrarily large alphabets, it remains possible to design optimal sequences efficiently. Finally, we analyze some of the consequences of this sequence design model for the study of evolutionary fitness landscapes. A given target structure may have many sequences that are optimal in the model of Sun et al.; following a notion raised by the work of J. Maynard Smith, we can ask whether these optimal sequences are "connected" by successive point mutations. We provide a polynomial-time algorithm to decide this connectedness property, relative to a given target structure. We develop the algorithm by first solving an analogous problem expressed in terms of submodular functions, a fundamental object of study in combinatorial optimization.

The vertex set of a [formula omitted]-polytope is strongly [formula omitted]-enumerable

In this paper, we discuss the computational complexity of the following enumeration problem: given a rational convex polyhedron P defined by a system of linear inequalities, output each vertex of P. It is still an open question whether there exists an algorithm for listing all vertices in running time polynomial in the input size and the output size. Informally speaking, a linear running time in the output size leads to the notion of P-enumerability introduced by Valiant (1979). The concept of strong P-enumerability additionally requires an output independent space complexity of the respective algorithm. We give such an algorithm for polytopes all of whose vertices are among the vertices of a polytope combinatorially equivalent to the hypercube. As a very important special case, this class of polytopes contains all 0 1 -polytopes. Our implementation based on the commercial LP solver CPLEX is superior to general vertex enumeration algorithms. We give an example how simplifications of our algorithm lead to efficient enumeration of combinatorial objects.

Algorithms for coloring semi-random graphs

The graph coloring problem is to color a given graph with the minimum number of colors. This problem is known to be NP-hard even if we are only aiming at approximate solutions. On the other hand, the best known approximation algorithms require $n^\delta (\delta>0)$ colors even for bounded chromatic k-colorable for fixed k n-vertex graphs. The situation changes dramatically if we look at the average performance of an algorithm rather than its worst case performance. A k-colorable graph drawn from certain classes of distributions can be k-colored almost surely in polynomial time. It is also possible to k-color such random graphs in polynomial average time. In this paper, we present polynomial time algorithms for k-coloring graphs drawn from the semirandom model. In this model, the graph is supplied by an adversary each of whose decisions regarding inclusion of edges is reversed with some probability p. In terms of randomness, this model lies between the worst case model and the usual random model where each edge is chosen with equal probability. We present polynomial time algorithms of two different types. The first type of algorithms w always run in polynomial time and succeed almost surely. Blum and Spencer [J. Algorithms, 19, 204-234 1995] have also obtained independently such algorithms, but our results are based on different proof techniques which are interesting in their own right. The second type of algorithms always succeed and have polynomial running time on the average. Such algorithms are more useful and more difficult to obtain than the first type of algorithms. Our algorithms work for semirandom graphs drawn from a wide range of distributions and work $p \ge n^{-{\alpha (k)}+\epsilon}}$ Where $\alpha(k) = \frac{(2k)}{((k-1)(k+2))}$ and \epsilon is a positive constant.

Single machine group scheduling to minimize mean flow time subject to due date constraints

This paper considers a single machine group scheduling problem. All jobs are classified into groups and the jobs within a group are processed contiguously on the machine. A sequence-independent setup time is incurred between each two consecutively scheduled groups. This paper presents a solution procedure which utilizes Smith's algorithm and a proposed modified Smith's algorithm to find an optimal job sequence and an optimal group sequence which minimizes the mean flow time of jobs subject to the constraint that no jobs are tardy. The complexity of the algorithm is shown to have a polynomial running time in the number of groups and jobs.

Generalized p-Center problems: Complexity results and approximation algorithms

In an earlier paper, two alternative p-Center problems, where the centers serving costumers must be chosen so that exactly one node from each of p prespecified disjoint pairs of nodes is selected, were shown to be NP-complete. This paper considers a generalized version of these problems, in which the nodes from which the p servers are to be selected are partitioned into k sets and the number of servers selected from each set must be within a prespecified range. We refer to these problems as the ‘Set’ p-Center problems. We establish that the triangle inequality (Δ-inequality) versions of these problems, in which the edge weights are assumed to satisfy the triangle inequality, are also NP-complete. We also provide a polynomial time approximation algorithm for the two Δ-inequality Set p-Center problems that is optimal for one of the problems in the sense that no algorithm with polynomial running time can provide a better constant factor performance guarantee, unless P = NP. For the special case ‘alternative’ p-Center problems, which we refer to as the ‘Pair’ p-Center problems, we extend the previous results in several ways. For example, the results mentioned above for the Set p-Center problems also apply to the Pair p-Center problems. Furthermore, we establish and exploit a correspondence between satisfiability and the dominating set type of problems that naturally arise when considering the decision versions of the Pair p-Center problems.

Probabilistic Analysis of a Generalized Bin Packing Problem and Applications

We give a unified probabilistic analysis for a general class of bin packing problems by directly analyzing corresponding mathematical programs. In this general class of packing problems, objects are described by a given number of attribute values. (Some attributes may be discrete; others may be continuous.) Bins are sets of objects, and the collection of feasible bins is merely required to satisfy some general consistency properties. We characterize the asymptotic optimal value as the value of an easily specified linear program whose size is independent of the number of objects to be packed, or as the limit of a sequence of such linear program values. We also provide bounds for the rate of convergence of the average cost to its asymptotic value. The analysis suggests an (a.s.) asymptotically ϵ-optimal heuristic that runs in linear time. The heuristics can be designed to be asymptotically optimal while still running in polynomial time. We also show that in several important cases, the algorithm has both polynomially fast convergence and polynomial running time. This heuristic consists of solving a linear program and rounding its solution up to the nearest integer vector. We show how our results can be used to analyze a general vehicle routing model with capacity and time window constraints.

Hybrid algorithms for approximate belief updating in Bayes nets

Belief updating in Bayes nets, a well-known computationally hard problem, has recently been approximated by several deterministic algorithms and by various randomized approximation algorithms. Deterministic algorithms usually provide probability bounds, but have an exponential runtime. Some randomized schemes have a polynomial runtime, but provide only probability estimates. Randomized algorithms that accumulate high-probability partial instantiations, resulting in probability bounds, are presented. Some of these algorithms are also sampling algorithms. Specifically, a variant of backward sampling, used both as a sampling algorithm and as a randomized enumeration algorithm, is introduced and evaluated. An implicit assumption made in prior work, for both sampling and accumulation algorithms, that query nodes must be instantiated in all the samples, is relaxed. Genetic algorithms can be used as an alternate search component for high-probability instantiations; several methods of applying them to belief updating are presented.

Scheduling with batch setup times and earliness-tardiness penalties

We consider a scheduling model in which several batches of jobs need to be processed by a single machine. During processing, a setup time is incurred whenever there is a switch from processing a job in one batch to a job in another batch. All the jobs in the same batch have a common due date that is either externally given as an input data or internally determined as a decision variable. Two problems are investigated. One problem is to minimize the total earliness and tardiness penalties provided that each due date is externally given. We show that this problem is NP-hard even when there are only two batches of jobs and the two due dates are unrestrictively large. The other problem is to minimize the total earliness and tardiness penalties plus the total due date penalty provided that each due date is a decision variable. We give some optimality properties for this problem with the general case and propose a polynomial dynamic programming algorithm for solving this problem with two batches of jobs. We also consider a special case for both of the problems when the common due dates for different batches are all equal. Under this special case, we give a dynamic programming algorithm for solving the first problem with an unrestrictively large due date and for solving the second problem. This algorithm has a running time polynomial in the number of jobs but exponential in the number of batches.

Bounded Delay Timing Analysis of a Class of CSP Programs

We describe an algebraic technique for performing timing analysis of a class of asynchronous circuits described as CSP programs (including Martin‘s probe operator) with the restrictions that there is no OR-causality and that guard selection is either completely free or mutually exclusive. Such a description is transformed into a safe Petri net with interval time delays specified on the places of the net. The timing analysis we perform determines the extreme separation in time between two communication actions of the CSP program for all possible timed executions of the system. We formally define this problem, propose an algorithm for its solution, and demonstrate polynomial running time on a non-trivial parameterized example. Petri nets with 3000 nodes and 10^16 reachable states have been analyzed using these techniques.

Minimum fault coverage in memory arrays: a fast algorithm and probabilistic analysis

The problem of reconfiguring memory arrays using spare rows and spare columns is known to be NP-complete and has received a great deal of attention in recent years. For reason of cost effectiveness, it is desirable in practice to find minimum reconfiguration solutions. While numerous algorithms have been proposed to find minimum reconfiguration solutions, they all run in worst case exponential time complexities. On the other hand, existing heuristic algorithms with fast polynomial running time cannot guarantee minimum solutions. This paper presents a provably good heuristic algorithm for finding minimum reconfiguration solution. Using random bipartite graphs, we prove that the reconfiguration problem is almost always optimally solvable with our new algorithm in polynomial time for all practical purposes. We also show that our algorithm can be used to estimate the number of spare rows and columns that are required to achieve a given percentage of yield for RRAM's with known defect probabilities.

Precoloring Extension III: Classes of Perfect Graphs

We continue the study of the following general problem on the vertex colourings of graphs. Suppose that some vertices of a graphGare assigned to some colours. Can this ‘precolouring’ be extended to a proper colouring ofGwith at mostkcolours (for some givenk)? Here we investigate the complexity status of precolouring extendibility on some classes of perfect graphs, giving good characterizations (necessary and sufficient conditions) that lead to algorithms with linear or polynomial running time. It is also shown how a larger subclass of perfect graphs can be derived from graphs containing no induced path on four vertices.

W[2]-hardness of precedence constrained K-processor scheduling

It is shown that the Precedence Constrained K-Processor Scheduling problem is hard for the parameterized complexity class W[2]. This means that there does not exist a constant c, such that for all fixed K, the Precedence Constrained K-Processor Scheduling problem can be solved in O( n c ) time, unless an unlikely collapse occurs in the parameterized complexity hierarchy. That is, if the problem can be solved in polynomial time for each fixed K, then it is likely that the degree of the running time polynomial must increase as the number of processors K increases.

Computing the area versus delay trade-off curves in technology mapping

We examine the problem of mapping a Boolean network using gates from a finite size cell library. The objective is to minimize the total gate area subject to constraints on signal arrival time at the primary outputs. Our approach consists of two steps. In the first step, we compute delay functions (which capture gate area-arrival time tradeoffs) at all nodes in the network, and in the second step we generate the mapping solution based on the computed delay functions and the required times at the primary outputs. For a NAND-decomposed tree, subject to load calculation errors, this two-step approach finds the minimum area mapping satisfying a delay constraint if such solution exists. The algorithm has polynomial run time on a node-balanced tree and is easily extended to mapping a directed acyclic graph (DAG). We also show how to account for the wire delays during the delay function computation step. Our results compare favorably with those of MIS2.2 mapper.

Strongly polynomial algorithm for two special minimum concave cost network flow problems

We consider two special cases of the Minimum Concave Cost Network Flow Problem (MCCNFP);the single source uncapacitated MCCNFP where all arc costs, except two, are linear and the uncapacitated MCCNFP with 2 sources and 1 nonlinear arc cost. Due to their specific structure these problems can he solved by efficient strongly polynomial time algorithms. More specifically, the algorithms developed have running times polynomial in the number of nodes and arcs and the number of sinks

On some geometric methods in scheduling theory: a survey

Suppose we are given a closed walk consisting of n steps in m-dimensional space, each step having at most unit length (in a certain norm s). We wish to include the walk into a given region of the space by reordering its steps. It turns out that our ability to solve this problem in polynomial time for certain regions of the space (such as the right triangle or hexagon in the plane, the ball in , R m etc.), enables us to construct approximation algorithms with good performance guarantees and polynomial running time for scheduling flow shops, job shops and other problems of the type, known to be NP-hard. Some other geometric methods are also to be spoken of subject to their application to scheduling problems.

One-machine batching and sequencing of multiple-type items

We consider a single-machine scheduling problem in which a given number of simultaneously available items of different types are to be processed. The items must first be batched and then sequenced before processing begins. Only items of the same type can be batched together. A setup time is incurred whenever a batch of a certain type of item is formed. The flowtime of an item in a batch is defined as the completion time of the batch that contains it. The problem is to find an optimal schedule in terms of the optimal batching and sequencing decisions that minimizes the total item flowtime. We present a dynamic programming algorithm to solve this problem. The algorithm has a running time polynomial in the number of items but exponential in the number of types.

On Well-Partial-Order Theory and Its Application to Combinatorial Problems of VLSI Design

The existence of decision algorithms with low-degree polynomial running times for a number of well-studied graph layout, placement, and routing problems is nonconstructively proved. Some were not previously known to be in $\mathcal{P}$ at all; others were only known to be in $\mathcal{P}$ by way of brute force or dynamic programming formulations with unboundedly high-degree polynomial running times. The methods applied include the recent Robertson–Seymour theorems on the well-partial-ordering of graphs under both the minor and immersion orders. The complexity of search versions of these problems is also briefly addressed.

Fast search algorithms for layout permutation problems

We have previously established the existence of decision algorithms with low-degree polynomial running times for a number of difficult combinatorial problems, including many that can be stated in terms of VLSI layout, placement, embedding and routing. In this paper, we turn our attention to the search complexity of these problems. We introduce a general technique, which we term scaffolding, and illustrate how it is useful in the design of efficient search algorithms.