# M. R. Garey's research while affiliated with Columbia University and other places

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## Publications (89)

We consider the one-dimensional bin packing problem under the discrete uniform distributions U{j, k}, 1 1, in which the bin capacity is k and item sizes are chosen uniformly from the set 2, . . . , j}. Note that for 0 < u = j/k 1 this is a discrete version of the previously studied continuous uniform distribution U(0, u], where the bin capacity is...

We consider the one-dimensional bin packing problem with unit-capacity bins and item sizes chosen according to the discrete uniform distribution U{j, k}, 1 < j less than or equal to k, where each item size in {1/k, 2/k,...,j/k} has probability 1/j of being chosen. Note that for fixed j, k as m --> infinity the discrete distributions U{mj, mk} appro...

We consider the classical scheduling problem in which a given collection of tasks with lengths t1, t2,...,t(n) are to be run on two processors, subject to specified precedence constraints among the tasks, so as to minimize the completion time of the last-finishing task, the so-called makespan of the schedule. A schedule is said to be nonpreemptive...

We consider one-processor scheduling problems having the following form: Tasks T1, T2,..., TN are given, with each Ti having a specified length li and a preferred starting time ai or, equivalently, a preferred completion time bi. The tasks are to be scheduled nonpreemptively i.e., a task cannot be split on a single processor to begin as close to th...

The fractional covering number τ * of a hypergraph H=(V,E) is defined to be the minimum possible value of ∑ x∈V t(x) where t ranges over all functions t: V→ℝ which satisfy ∑ x∈e t(x)≥1 for all edges e∈E. In the case of ordinary graphs G, it is known that 2τ * (G) is always an integer. By contrast, it is shown (among other things) that for any ratio...

We study a variety of NP-hard bin packing problems under a divisibility constraint that generalizes the often encountered situation in which all item sizes are powers of 2. For ordinary one-dimensional bin packing, we show that First Fit Decreasing produces optimal packings under this restriction, and that if in addition the largest item size divid...

We study the problem of scheduling n given jobs on m uniform processors to minimize expected makespan (maximum finishing time). Job execution times are not known in advance, but are known to be exponentially distributed, with identical rate parameters depending solely on the executing processor. For m=2 and 3, we show that there exist optimal sched...

We study the problem of scheduling n given jobs on m uniform processors to minimize expected makespan (maximum finishing time). Job execution times are not known in advance, but are known to be exponentially distributed, with identical rate parameters depending solely on the executing processor. For m = 2 and 3, we show that there exist optimal sch...

We study the problem of orienting all the undirected edges of a mixed multigraph so as to preserve reachability. Extending work by Robbins and by Boesch and Tindell, we develop a linear-time algorithm to test whether there is an orientation that preserves strong connectivity and to construct such an orientation whenever possible. This algorithm mak...

The FIRST FIT DECREASING algorithm for bin packing has long been famous for its guarantee that no packing it generates will use more than times the optimal number of bins. We present a simple modified version that has essentially the same running time, should perform at least as well on average, and yet provides a guarantee of .

We consider a problem of scheduling file transfers in a network so as to minimize overall finishing time. Although the general problem is NP-complete, we identify polynomial time solvable special cases and derive good performance bounds for several natural approximation algorithms, assuming the existence of a central controller. We also show how th...

It is shown that given a collection F of functions from a finite set D to itself, one can, in polynomial time, find a composition f of functions in F for which the size of f (D) is minimized. This is to be contrasted with the fact that it is PSPACE-complete to determine whether a specific function f is a composition of functions in F. The running t...

The main question addressed in this article is the following: If t edges are removed from a (t + 1) edge-connected graph G having diameter D, how large can the diameter of the resulting graph be? (The diameter of a graph is the maximum, over all pairs of vertices, of the length of the shortest path joining those vertices.) We provide bounds on this...

This paper updates a survey [53] written about 3 years ago. All of the results mentioned there are covered here as well. However, as a major justification for this second edition we shall be presenting many new results, some of which represent important advances. As a measure of the impressive amount of research in just 3 years, the present referen...

In this paper we consider a problem related to questions of optimal circuit layout: Given a graph or network, how can we embed it in a planar surface so as to minimize the number of edge-crossings? We show that this problem is NP-complete, and hence there is not likely to be any efficient way to design an optimal embedding.

Motivated by potential applications to computer storage allocation, we generalize the classical one-dimensional bin packing model to include dynamic arrivals and departures of items over time. Within this setting, we prove close upper and lower bounds on the worst-case performance of the commonly used First Fit packing algorithm, and, using adversa...

A basic problem of deterministic scheduling theory is that of scheduling n unit-length tasks on m identical processors subject to precedence constraints so as to meet a given overall deadline. T. C. Hu’s classic “level algorithm” can be used to solve this problem in linear time if the precedence constraints have the form of an in-forest or an out-f...

We consider a problem of scheduling file transfers in a network so as to minimize overall finishing time, which we formalize as a problem of scheduling the edges of a weighted multigraph. Although the general problem is NP-complete, we identify polynomial time solvable special eases and derive good performance bounds for several natural approximati...

Suppose we are given a set L of rectangular items and wish to pack them into identical rectangular bins, so that no two items overlap and so that the number of bins used is minimized. This generalization of the standard one-dimensional bin packing problem models problems arising in a variety of applications, from truck loading to the design of VLSI...

T. Parsons proposed and partially analyzed the following pursuit-evasion problem on graphs: A team of searchers traverse the edges of a graph G in pursuit of a fugitive, who moves along the edges of the graph with complete knowledge of the locations of the pursuers. What is the smallest number s(G) of searchers that will suffice for guaranteeing ca...

The basic problem considered is that of scheduling n unit-time tasks, with arbitrary release times and deadlines, so as to minimize the maximum task completion time. Previous work has shown that this problem can be solved rather easily when all release times are integers. We are concerned with the general case in which noninteger release times are...

Bin packing problems, in which one is asked to pack items of various sizes into bins so as to optimize some given objective function, arise in a wide variety of contexts and have been studied extensively during the past ten years, primarily with the goal of finding fast “approximation algorithms” that construct near-optimal packings. Beginning with...

We analyze several “level-oriented” algorithms for packing rectangles into a unit-width, infinite-height bin so as to minimize the total height of the packing. For the three algorithms we discuss, we show that the ratio of the height obtained by the algorithm to the optimal height is asymptotically bounded, respectively, by 2, 1.7, and 1.5. The lat...

The word problem for products of symmetric groups, the circular arc graph coloring problem, and the circle graph coloring problem, as well as several related problems, are proved to be $NP$-complete. For any fixed number K of colors, the problem of determining whether a given circular arc graph is K-colorable is shown to be solvable in polynomial t...

The m-chromatic number Xm(G) of a graph G = (V, E) is the least integer k such that there exists a mapping f : V → {S ⊆ c {1,2,..., k} : |S| = m} having the property that f (u) f∩f(u) = Θ whenever {u,v } E E. This is a generalization of the standard notion of chromatic number and arises in connection with mobile telephone frequency assignments. Ans...

We consider the game of Checkers generalized to an N × N board. Although certain properties of positions are efficiently computable (e.g., can Black jump all of White's pieces in a single move?), the general question, given a position, of whether a specified player can force a win against best play by his opponent, is shown to be PSPACE-hard. Under...

Let A be a finite set of points in a Euclidean space Er and M a minimum spanning tree (MST) for A, regarded as a subset of Er. If A′ ⊂ Er is a finite subset of M, then M is a spanning tree for A ∪ A′, but in general M will no longer be an MST. However, if A′ is chosen to be the set of all the midpoints of the line segments of M, then M will be an M...

The NP-completeness of a computational problem ~s frequently taken to unply its "mtractabthty" However, there are certain NP-complete problems mvolvmg numbers, such as PARTITION and KNAPSACK, which are considered by many practitioners to be tractable The reason for this IS that, although no algontluns for solvmg them in tune bounded by a polynomial...

We present a linear-time algorithm for sparse symmetric matrices which converts a matrix into pentadiagonal form ("bandwidth 2"), whenever it is possible to do so using simultaneous row and column permutations. On the other hand when an arbitrary integer k and graphG are given, we show that it is NP-complete to determine whether or not there exists...

We consider one of the basic, well-studied problems of scheduling theory, that of nonpreemptively scheduling n independent tasks on m identical, parallel processors with the objective of minimizing the “makespan,” i.e., the total timespan required to process all the given tasks. Because this problem is $NP$-complete and apparently intractable in ge...

One approach to coping with the apparent difficulty of many schedule-optimization problems, such as occur in machine shops and computer processing, is to devise efficient algorithms that find schedules guaranteed to be ″near-optimal.″ An introduction to this approach is given by describing its application to a well-known multiprocessor scheduling m...

Given a set $\mathcal{T} = \{ T_1 ,T_2 , \cdots ,T_n \} $ of tasks, each $T_i$ having execution time 1, an integer start-time $s_i \geqq 0$ and a deadline $d_i > 0$, along with precedence constraints among the tasks, we examine the problem of determining whether there exists a schedule on two identical processors that executes each task in the time...

A basic problem of deterministic scheduling theory is that of scheduling n equal length tasks on m identical processors subject to precedence constraints. Although the general problem of finding a schedule which minimizes makespan is NP-complete, the important special case with ″treelike″ precedence constraints can be solved by a well-known algorit...

It is shown that the problem of computing Steiner minimal trees for general planar point sets is inherently at least as difficult as any of the $NP$-complete problems (a well known class of computationally intractable problems). This effectively destroys any hope for finding an efficient algorithm for this problem.

An optimum rectilinear Steiner tree for a set A of points in the plane is a tree which interconnects A using horizontal and vertical lines of shortest possible total length. Such trees correspond to single net wiring patterns on printed backplanes which minimize total wire length. We show that the problem of determining this minimum length, given A...

We consider an abstract partitioning problem which has applications to the design of standard libraries of multipurpose units (such as multi-purpose circuit cards) and to storage allocation in the presence of users with conflicting demands. Although the general problem of finding minimum cost partitions appears to be very difficult, we describe a d...

For a tree T on n vertices, let D(T)=(dij) denote the distance matrix of T, i.e., dij(T) is just the length of the unique path then the ith vertex and the jth vertex of T. Denote by ΔT(x) the characteristic polynom, of D(T), so that ΔT(x) = det(D(T) xl). In this paper, we investigate a number of properties of ΔT(x). In particular, we find simple ex...

We consider the problem of determining whether a planar, cubic, triply-connected graph G has a Hamiltonian circuit. We show that this problem is NP-complete. Hence the Hamiltonian circuit problem for this class of graphs, or any larger class containing all such graphs, is probably computationally intractable.

Given a set @@@@ = {T1,T2,···,Tn} of tasks, with each Ti having execution time 1 and a deadline di > 0, and a set of precedence constraints which restrict allowable schedules, the problem of determining whether there exists a schedule using two processors in which each task is completed before its deadline is examined. An efficient algorithm for fi...

We show that the STEINER TREE problem and TRAVELING SALESMAN problem for points in the plane are NP-complete when distances are measured either by the rectilinear (Manhattan) metric or by a natural discretized version of the Euclidean metric. Our proofs also indicate that the problems are NP-hard if the distance measure is the (unmodified) Euclidea...

NP-complete problems form an extensive equivalence class of combinatorial problems for which no nonenumerative algorithms are known. The first result shows that determining a shortest-length schedule in an m-machine flowshop is NP-complete for m greater than equivalent to 3. (For m equals 2, there is an efficient algorithm for finding such schedule...

It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of Simple Max Cut (Max Cut with edge weights restricted to value...

Graph coloring problems, in which one would like to color the vertices of a given graph with a small number of colors so that no two adjacent vertices receive the same color, arise in many applications, including various scheduling and partitioning problems. In this paper the complexity and performance of algorithms which construct such colorings a...

We examine the computational complexity of scheduling problems associated with a certain abstract model of a multiprocessing system. The essential elements of the model are a finite number of identical processors, a finite set of tasks to be executed, a partial order constraining the sequence in which tasks may be executed, a finite set of limited...

A proposed method for testing printed circuit boards for the existence of possible (undesired) short circuits transforms the test minimization problem into one of finding minimum vertex colorings of certain special graphs, called line-of-sight graphs. Under certain assumptions on the possible types of short circuits, we analyze the structure of suc...

One well-studied model ofa multiprocessing system involves a fixed number n ofidentical abstract processors, a finite set oftasks to be executed, each requiring a specified amount ofcomputation time, and a partial ordering on the tasks which requires certain tasks to be completed before certain others can be initiated. The nonpreemptive operation o...

An algorithm for constructing an optimal prefix code of n equiprobable words over r unequal cost coding letters is given. The discussion is in terms of rooted labeled trees. The algorithm consists of two parts. The first one is an extension algorithm which constructs a prefix code of n words. This code is either optimal or is a ″good″ approximation...

It is frequently of interest to represent a given graph G as a subgraph of a graph H which has some special structure. A particularly useful class of graphs in which to embed G is the class of n-dimensional cubes. This has found applications, for example, in coding theory, data transmission, and linguistics. In this note, we study the structure of...

In machine fault-location, medical diagnosis, species identification, and computer decisionmaking, one is often required to identify some unknown object or condition, belonging to a known set of M possibilities, by applying a sequence of binary-valued tests, which are selected from a given set of available tests. One would usually prefer such a tes...

The following abstract problem models several practical problems in computer science and operations research: given a list L of real numbers between 0 and l, place the elements of L into a minimum number $L^ * $ of “bins” so that no bin contains numbers whose sum exceeds l. Motivated by the likelihood that an excessive amount of computation will be...

An algorithm is given for constructing a binary tree of minimum weighted path length for n nonnegative weights under the constraint that no path length exceed a given bound L. The number of operations required is proportional to $Ln^2 $. Such problems, which impose an additional constraint on the usual Huffman tree, arise in many applications, incl...

Given a finite set of n items, containing at least one defective item, and, for each item, the probability that it is defective, it is desired to determine a group-testing procedure which isolates a single defective with a minimum expected number of tests. We prove that such an optimal procedure can be found by constructing an optimal “alphabetic b...

It is widely believed that showing a problem to be NP-complete is tantamount to proving its computational intractability. In this paper we show that a number of NP-complete problems remain NP-complete even when their domains are substantially restricted. First we show the completeness of SIMPLE MAX CUT (MAX CUT with edge weights restricted to value...

A number of authors (of. [12],[6], [7],[3],[11],[4],[5],[9]) have recently been concerned with scheduling problems associated with a certain model of an abstract multiprocessing system (to be described in the next section) and, in particular, with bounds on the worst-case behavior of this system as a function of the way in which the inputs are allo...

Binary identification problems model a variety of actual problems, all containing the requirement that one construct a testing procedure for identifying a single unknown object belonging to a known finite set of possibilities. They arise in connection with machine fault-location, medical diagnosis, species identification, and computer programming....

Given some unknown object belonging to a known finite set of n possibilities, it is required to determine its identity by successive comparisons with each of the possibilities. Associating with each of these possibilities a testing cost and a probability that it is identical to the unknown object, we would like to obtain such a testing procedure wh...

We consider economical representations for the path information in a directed graph. A directed graph $G^t $ is said to be a transitive reduction of the directed graph G provided that (i) $G^t $ has a directed path from vertex u to vertex v if and only if G has a directed path from vertex u to vertex v, and (ii) there is no graph with fewer arcs th...

Various memory allocation problems can be modeled by the following abstract problem. Given a list A &equil; (&agr;1,&agr;2,...&agr;n,) of real numbers in the range (0, 1], place these in a minimum number of “bins” so that no bin holds numbers summing to more than 1. We let A* be the smallest number of bins into which the numbers of list A may be pl...

## Citations

... However, we conjecture that as t approaches 1, the case becomes worst-case. 5. We prove a strict competitive ratio of 2 for FirstFit where each job has a duration 2 and arrival times 0 and 1 (Theorem 5.12). ...

... The focus of this study is on scheduling applications with the BoTs model in the laas layer in the cloud federation. In the BoTs model, each application is composed of independent tasks and there is no relationship between them [14][15][16]. In this study, we assumed that each application with the BoTs model is independent of tasks and there are heterogeneous tasks in each Bag. ...

... There is a vast literature on stochastic scheduling (see Nino-Mora (2009)). In particular, Coffman Jr et al. (1987) characterize optimal solutions for makespan minimization in scenarios involving 2 or 3 machines and jobs with exponentially distributed processing times. Pinedo (2005) shows the connection between a stochastic flow shop scheduling problem and a deterministic traveling salesman problem; we present a similar result involving 2| ∼ N ( , Σ), , = 0|E[ ] in this work. ...

... The power of approximation algorithms lies in their ability, for some problems, to provide a fast approximation to a solution even when computing the exact solution requires exponential time (assuming (P = NP)). Though approximation algorithms have existed in the literature for some time, Garey, Graham, and Ullman [21] and Johnson [22] both introduced the idea formally in 1973 and 1974, respectively. Since then, the computer science and combinatorics literature has featured many advancements in the field of randomized approximation algorithms. ...

... DARP has been shown to be NP-hard even when all the travel distances are unity (Frederickson et al., 1978). Since SDPDTWP subsumes the NP-hard problem, it is also NP-hard (Garey & Johnson, 1976). Therefore DARP is unlikely to be solved in polynomial time, and a heuristic approach provides the only method to efficiently solve a problem of realistic size. ...

... However, the calculation of extremal values on linear arrangements (i.e. minimum or maximum values of a score), such as the solution to the Minimum Linear Arrangement (MLA) problem in unconstrained arrangements (Garey and Johnson, 1976;Shiloach, 1979;Chung, 1984) or of one of its constrained variants (Iordanskii, 1987;Hochberg and Stallmann, 2003;Gildea and Temperley, 2007;Bommasani, 2020;Alemany-Puig et al., 2022) are not straightforward because the algorithm is complex, it is hard to test or both. Likewise, performing statistical tests and calculating expected values (random baselines) require random sampling methods when exact algorithms/formulae are not known; such sampling is typically done uniformly at random over all possible trees (Ferrer-i-Cancho et al., 2018;Gómez-Rodríguez et al., 2020;Yadav et al., 2019) or random arrangements (Ferrer-i-Cancho et al., 2018;. ...

... Here we establish the hardness of the DDP via the bin packing problem [5]. P . ...

... In fact, an additional requirement for NP-completeness is polynomial-time reduction of another NP-complete problem, which for narrative reasons, is omitted here. See e.g.(Garey and Johnson, 2002). ...

... Berge [3] has studied fractional transversals of hypergraphs, Pulleyblank [11] has studied fractional matching, and Aharoni [1] has studied fractional matching and covers in infinite hypergraphs. Chung, Furedi, Garey and Graham [4] have studied fractional covers of hypergraphs. The fractional chromatic number has been looked at by Larsen, Propp and Ullman as well as by others. ...

Reference: Dominating Function in Fractional Graph

... Using this we derive optimal thresholds and the optimal expected costs for the jobs based on their priorities. We call Lemma 1 the monotonicity result, following Coffman et al. [32] for the slow-server problem. ...