Abstract An instance of the non-preemptive tree packing problem consists of an undirected graph $$G=(V,E)$$ G = ( V , E ) together with a weight w(e) for every edge $$e\in E$$ e ∈ E . The goal is to activate every edge e for some time interval of length w(e), such that the activated edges keep G connected for the longest possible overall time. We derive a variety of results on this problem. The problem is strongly NP-hard even on graphs of treewidth 2, and it does not allow a polynomial time approximation scheme (unless P=NP). Furthermore, we discuss the performance of a simple greedy algorithm, and we construct and analyze a number of parameterized and exact algorithms.

Abstract We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance $$\delta$$ δ from each other. We investigate the complexity of this problem in terms of the rational parameter $$\delta$$ δ . The problem is polynomially solvable, if the numerator of $$\delta$$ δ is 1 or 2, while all other cases turn out to be NP-hard.

The feasible solutions in the robust recoverable selection problem are subsets of size p that are to be selected from a ground set of size n. The objective is to construct a feasible solution in two sequential stages with two separate (but interleaved) cost structures. The fastest algorithm for this problem in the literature up to now has quadratic running time. We improve on this by developing an algorithm with linear running time.

Abstract We discuss a game theoretic variant of the subset sum problem, in which two players compete for a common resource represented by a knapsack. Each player owns a private set of items, players pack items alternately, and each player either wants to maximize the total weight of his own items packed into the knapsack or to minimize the total weight of the items of the other player. We show that finding the best packing strategy against a hostile or a selfish adversary is PSPACE-complete, and that against these adversaries the optimal reachable item weight for a player cannot be approximated within any constant factor (unless P=NP). The game becomes easier when the adversary is short-sighted and plays greedily: finding the best packing strategy against a greedy adversary is NP-complete in the weak sense. This variant forms one of the rare examples of pseudo-polynomially solvable problems that have a PTAS, but do not allow an FPTAS (unless P=NP).

We analyze two classic variants of the T RAVELING S ALESMAN P ROBLEM ( TSP ) using the toolkit of fine-grained complexity. Our first set of results is motivated by the B ITONIC TSP problem: given a set of n points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in O ( n 2 ) time. While the near-quadratic dependency of similar dynamic programs for L ONGEST C OMMON S UBSEQUENCE and D ISCRETE F réchet D istance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in O ( n log 2 n ) time and its bottleneck version in O ( n log 3 n ) time. In the more general pyramidal TSP problem, the points to be visited are labeled 1,… , n and the sequence of labels in the solution is required to have at most one local maximum. Our algorithms for the bitonic (bottleneck) TSP problem also work for the pyramidal TSP problem in the plane. Our second set of results concerns the popular k - OPT heuristic for TSP in the graph setting. More precisely, we study the k - OPT decision problem, which asks whether a given tour can be improved by a k - OPT move that replaces k edges in the tour by k new edges. A simple algorithm solves k - OPT in O ( n k ) time for fixed k . For 2- OPT , this is easily seen to be optimal. For k =3, we prove that an algorithm with a runtime of the form Õ( n 3−ɛ ) exists if and only if A LL -P AIRS S HORTEST P ATHS in weighted digraphs has such an algorithm. For general k - OPT , it is known that a runtime of f ( k ) · n o ( k / log k ) would contradict the Exponential Time Hypothesis. The results for k =2,3 may suggest that the actual time complexity of k - OPT is Θ ( n k ). We show that this is not the case, by presenting an algorithm that finds the best k -move in O ( n ⌊ 2 k /3 ⌋+1 ) time for fixed k ≥ 3. This implies that 4- OPT can be solved in O ( n 3 ) time, matching the best-known algorithm for 3- OPT . Finally, we show how to beat the quadratic barrier for k =2 in two important settings, namely, for points in the plane and when we want to solve 2- OPT repeatedly.

Abstract It is well known that any set of n intervals in $$\mathbb {R} ^1$$ R1 admits a non-monochromatic coloring with two colors and a conflict-free coloring with three colors. We investigate generalizations of this result to colorings of objects in more complex 1-dimensional spaces, namely so-called tree spaces and planar network spaces.

We study a continuous facility location problem on a graph where all edges have unit length and where the facilities may also be positioned in the interior of the edges. The goal is to position as many facilities as possible subject to the condition that any two facilities have at least distance delta from each other. We investigate the complexity of this problem in terms of the rational parameter delta. The problem is polynomially solvable, if the numerator of delta is 1 or 2, while all other cases turn out to be NP-hard.

We algorithmize the structural characterization for claw-free graphs by Chudnovsky and Seymour. Building on this result, we show that D ominating S et on claw-free graphs is (i) fixed-parameter tractable and (ii) even possesses a polynomial kernel. To complement these results, we establish that D ominating S et is unlikely to be fixed-parameter tractable on the slightly larger class of graphs that exclude K 1,4 as an induced subgraph ( K 1,4 -free graphs). We show that our algorithmization can also be used to show that the related C onnected D ominating S et problem is fixed-parameter tractable on claw-free graphs. To complement that result, we show that C onnected D ominating S et is unlikely to have a polynomial kernel on claw-free graphs and is unlikely to be fixed-parameter tractable on K 1,4 -free graphs. Combined, our results provide a dichotomy for D ominating S et and C onnected D ominating S et on K 1,ℓ -free graphs and show that the problem is fixed-parameter tractable if and only if ℓ ≤ 3.

Voting is almost never done in void, as usually there are some relations between the alternatives on which the voters vote on. These relations shall be taken into consideration when selecting a winning committee of some given multiwinner election. As taking into account all possible relations between the alternatives is generally computationally intractable, in this paper we consider classes of alternatives; intuitively, the number of classes is significantly smaller than the number of alternatives, and thus there is some hope in reaching computational tractability. We model both intraclass relations and interclass relations by functions, which we refer to as synergy functions, and study the computational complexity of identifying the best committee, taking into account those synergy functions. Our model accommodates both positive and negative relations between alternatives; further, our efficient algorithms can also deal with a rich class of diversity wishes, which we show how to model using synergy functions.

The graph similarity problem, also known as approximate graph isomorphism or graph matching problem, has been extensively studied in the machine learning community, but has not received much attention in the algorithms community: Given two graphs G,H of the same order n with adjacency matrices A_G,A_H, a well-studied measure of similarity is the Frobenius distance dist(G,H):=min_{pi}|A_G^{pi}-A_H|_F, where pi ranges over all permutations of the vertex set of G, where A_G^pi denotes the matrix obtained from A_G by permuting rows and columns according to pi, and where |M |_F is the Frobenius norm of a matrix M. The (weighted) graph similarity problem, denoted by GSim (WSim), is the problem of computing this distance for two graphs of same order. This problem is closely related to the notoriously hard quadratic assignment problem (QAP), which is known to be NP-hard even for severely restricted cases. It is known that GSim (WSim) is NP-hard; we strengthen this hardness result by showing that the problem remains NP-hard even for the class of trees. Identifying the boundary of tractability for WSim is best done in the framework of linear algebra. We show that WSim is NP-hard as long as one of the matrices has unbounded rank or negative eigenvalues: hence, the realm of tractability is restricted to positive semi-definite matrices of bounded rank. Our main result is a polynomial time algorithm for the special case where the associated (weighted) adjacency graph for one of the matrices has a bounded number of twin equivalence classes. The key parameter underlying our algorithm is the clustering number of a graph; this parameter arises in context of the spectral graph drawing machinery.

We discuss the computational complexity, the approximability, the algorithmics and the combinatorics of the open shop scheduling problem. We summarize the most important results from the literature and explain their main ideas, we sketch the most beautiful proofs, and we also list a number of open problems.

We introduce the dynamic conflict-free coloring problem for a set S of intervals in R 1 with respect to points, where the goal is to maintain a conflict-free coloring for S under insertions and deletions. We investigate trade-offs between the number of colors used and the number of intervals that are recolored upon insertion or deletion of an interval. Our results include: - a lower bound on the number of recolorings as a function of the number of colors, which implies that with O(1) recolorings per update the worst-case number of colors is Ω(logn/loglogn) , and that any strategy using O(1/ε) colors needs Ω(εn ε ) recolorings; - a coloring strategy that uses O(logn) colors at the cost of O(logn) recolorings, and another strategy that uses O(1/ε) colors at the cost of O(n ε /ε) recolorings; - stronger upper and lower bounds for special cases. We also consider the kinetic setting where the intervals move continuously (but there are no insertions or deletions); here we show how to maintain a coloring with only four colors at the cost of three recolorings per event and show this is tight.

We study the parameterized complexity of dominating sets in geometric intersection graphs. In one dimension, we investigate intersection graphs induced by translates of a fixed pattern Q that consists of a finite number of intervals and a finite number of isolated points. We prove that Dominating Set on such intersection graphs is polynomially solvable whenever Q contains at least one interval, and whenever Q contains no intervals and for any two point pairs in Q the distance ratio is rational. The remaining case where Q contains no intervals but does contain an irrational distance ratio is shown to be NP-complete and contained in FPT (when parameterized by the solution size). In two and higher dimensions, we prove that Dominating Set is contained in W[1] for intersection graphs of semi-algebraic sets with constant description complexity. This generalizes known results from the literature. Finally, we establish W[1]-hardness for a large class of intersection graphs.

We analyze two classic variants of the Traveling Salesman Problem using the toolkit of fine-grained complexity. Our first set of results is motivated by the Bitonic TSP problem: given a set of $n$ points in the plane, compute a shortest tour consisting of two monotone chains. It is a classic dynamic-programming exercise to solve this problem in $O(n^2)$ time. While the near-quadratic dependency of similar dynamic programs for Longest Common Subsequence and Discrete Frechet Distance has recently been proven to be essentially optimal under the Strong Exponential Time Hypothesis, we show that bitonic tours can be found in subquadratic time. More precisely, we present an algorithm that solves bitonic TSP in $O(n \log^2 n)$ time and its bottleneck version in $O(n \log^3 n)$ time. Our second set of results concerns the popular $k$-OPT heuristic for TSP in the graph setting. More precisely, we study the $k$-OPT decision problem, which asks whether a given tour can be improved by a $k$-OPT move that replaces $k$ edges in the tour by $k$ new edges. A simple algorithm solves $k$-OPT in $O(n^k)$ time for fixed $k$. For 2-OPT, this is easily seen to be optimal. For $k=3$ we prove that an algorithm with a runtime of the form $\tilde{O}(n^{3-ε})$ exists if and only if All-Pairs Shortest Paths in weighted digraphs has such an algorithm. The results for $k=2,3$ may suggest that the actual time complexity of $k$-OPT is $Θ(n^k)$. We show that this is not the case, by presenting an algorithm that finds the best $k$-move in $O(n^{\lfloor 2k/3 \rfloor + 1})$ time for fixed $k \geq 3$. This implies that 4-OPT can be solved in $O(n^3)$ time, matching the best-known algorithm for 3-OPT. Finally, we show how to beat the quadratic barrier for $k=2$ in two important settings, namely for points in the plane and when we want to solve 2-OPT repeatedly.

We consider a bilevel integer programming model that extends the classic 0–1 knapsack problem in a very natural way. The model describes a Stackelberg game where the leader’s decision interdicts a subset of the knapsack items for the follower. As this interdiction of items substantially increases the difficulty of the problem, it prevents the application of the classical methods for bilevel programming and of the specialized approaches that are tailored to other bilevel knapsack variants. Motivated by the simple description of the model, by its complexity, by its economic applications, and by the lack of algorithms to solve it, we design a novel viable way for computing optimal solutions. Finally, we present extensive computational results that show the effectiveness of the new algorithm on instances from the literature and on randomly generated instances.

We study a scheduling problem where two agents (each equipped with a private set of jobs) compete to perform their respective jobs on a common single machine. Each agent wants to keep the weighted sum of completion times of his jobs below a given (agent-dependent) bound. This problem is known to be NP-hard, even for quite restrictive settings of the problem parameters. We consider parameterized versions of the problem where one of the agents has a small number of jobs (and where this small number constitutes the parameter). The problem becomes much more tangible in this case, and we present three positive algorithmic results for it. Our study is complemented by showing that the general problem is NP-complete even when one agent only has a single job.

We introduce a graph-theoretic vertex dissolution model that applies to a number of redistribution scenarios, such as gerrymandering in political districting or work balancing in an online situation. The central aspect of our model is the deletion of certain vertices and the redistribution of their load to neighboring vertices in a completely balanced way. We investigate how the underlying graph structure, the knowledge of which vertices should be deleted, and the relation between old and new vertex loads influence the computational complexity of the underlying graph problems. Our results establish a clear borderline between tractable and intractable cases.

Assume that each of n voters may or may not approve each of m issues. If an agent (the lobby) may influence up to k voters, then the central question of the NP-hard Lobbying problem is whether the lobby can choose the voters to be influenced so that as a result each issue gets a majority of approvals. This problem can be modeled as a simple matrix modification problem: Can one replace k rows of a binary n x m-matrix by k all-1 rows such that each column in the resulting matrix has a majority of 1s? Significantly extending on previous work that showed parameterized intractability (W[2]-completeness) with respect to the number k of modified rows, we study how natural parameters such as n, m, k, or the "maximum number of 1s missing for any column to have a majority of 1s" (referred to as "gap value g") govern the computational complexity of Lobbying. Among other results, we prove that Lobbying is fixed-parameter tractable for parameter m and provide a greedy logarithmic-factor approximation algorithm which solves Lobbying even optimally if m < 5. We also show empirically that this greedy algorithm performs well on general instances. As a further key result, we prove that Lobbying is LOGSNP-complete for constant values g>0, thus providing a first natural complete problem from voting for this complexity class of limited nondeterminism.

We study a motion planning problem where items have to be transported from the top room of a tower to the bottom of the tower, while simultaneously other items have to be transported into the opposite direction. Item sets are moved in two baskets hanging on a rope and pulley. To guarantee stability of the system, the weight difference between the contents of the two baskets must always stay below a given threshold. We prove that it is Pi-2-p-complete to decide whether some given initial situation of the underlying discrete system can lead to a given goal situation. Furthermore we identify several polynomially solvable special cases of this reachability problem, and we also settle the computational complexity of a number of related questions.

We prove strong ${\mathbb {NP}}$ -completeness for the four variants of caching with multi-size pages. These four variants are obtained by choosing either the fault cost or the bit cost model, and by combining it with either a forced or an optional caching policy. This resolves two questions in the area of paging and caching that were open since the 1990s.

The Borda voting rule is a positional scoring rule where, for m candidates, for every vote the first candidate receives m-1 points, the second m-2 points and so on. A Borda winner is a candidate with highest total score. It has been a prominent open problem to determine the computational complexity of UNWEIGHTED COALITIONAL MANIPULATION UNDER BORDA: Can one add a certain number of additional votes (called manipulators) to an election such that a distinguished candidate becomes a winner? We settle this open problem by showing NP-hardness even for two manipulators and three input votes. Moreover, we discuss extensions and limitations of this hardness result.

We study algorithmic problems in multi-stage open shop processing systems that are centered around reachability and deadlock detection questions. We characterize safe and unsafe system states. We show that it is easy to recognize system states that can be reached from the initial state (where the system is empty), but that in general it is hard to decide whether one given system state is reachable from another given system state. We show that the problem of identifying reachable deadlock states is hard in general open shop systems, but is easy in the special case where no job needs processing on more than two machines (by linear programming and matching theory), and in the special case where all machines have capacity one (by graph-theoretic arguments).

We consider the problem of assigning sensors to track targets so as to minimize the expected error in the resulting estimation for target locations. The so-called Focus of Attention problem deals with the special case where every target is tracked by one pair of range sensors. We provide a complete complexity and approximability analysis of the Focus Of Attention problem: We establish its strong NP-hardness, and we construct a polynomial time approximation scheme for it.

Let $P$ be a set of points in $\Reals^d$, and let $\alpha \ge 1$ be a real number. We define the distance between two points $p,q\in P$ as $|pq|^{\alpha}$, where $|pq|$ denotes the standard Euclidean distance between $p$ and $q$. We denote the traveling salesman problem under this distance function by \tsp($d,\alpha$). We design a 5-approximation algorithm for \tsp(2,2) and generalize this result to obtain an approximation factor of $3^{\alpha-1}+\sqrt{6}^{\,\alpha}\!/3$ for $d=2$ and all $\alpha\ge2$. We also study the variant Rev-\tsp\ of the problem where the traveling salesman is allowed to revisit points. We present a polynomial-time approximation scheme for Rev-\tsp$(2,\alpha)$ with $\alpha\ge2$, and we show that Rev-\tsp$(d, \alpha)$ is \apx-hard if $d\ge3$ and $\alpha>1$. The \apx-hardness proof carries over to \tsp$(d, \alpha)$ for the same parameter ranges.

We consider a planning problem that generalizes Alcuin's river crossing problem to scenarios with arbitrary conflict graphs. This generalization leads to the so-called Alcuin number of the underlying conflict graph. We derive a variety of combinatorial, structural, algorithmical, and complexity theoretical results around the Alcuin number. Our technical main result is an NP-certificate for the Alcuin number. It turns out that the Alcuin number of a graph is closely related to the size of a minimum vertex cover in the graph, and we unravel several surprising connections between these two graph parameters. We provide hardness results and a fixed parameter tractability result for computing the Alcuin number. Furthermore we demonstrate that the Alcuin number of chordal graphs, bipartite graphs, and planar graphs is substantially easier to analyze than the Alcuin number of general graphs.

Abstract This paper introduces the generalized Egghe‐indices as a new family of scientific impact measures for ranking the output of scientific researchers. The definition of this family is strongly inspired by Egghe's well‐known g‐index. The main contribution of the paper is a family of axiomatic characterizations that characterize every generalized Egghe‐index in terms of four axioms.

Abstract For a fixed pattern graph H, let H‐CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. This paper is part I of our study on the computational complexity of the H‐CONTRACTIBILITY problem. We continue a line of research that was started in 1987 by Brouwer and Veldman, and we determine the computational complexity of the H‐CONTRACTIBILITY problem for certain classes of pattern graphs. In particular, we pinpoint the complexity for all graphs H with five vertices except for two graphs, whose polynomial time algorithms are presented in part II. Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP‐complete, the pattern graph H does not have a dominating vertex. © 2007 Wiley Periodicals, Inc. NETWORKS, 2008

Abstract For a fixed pattern graph H, let H‐CONTRACTIBILITY denote the problem of deciding whether a given input graph is contractible to H. This article is part II of our study on the computational complexity of the H‐CONTRACTIBILITY problem. In the first article we pinpointed the complexity for all pattern graphs with five vertices except for two pattern graphs H. Here, we present polynomial time algorithms for these two remaining pattern graphs. Interestingly, in all connected cases that are known to be polynomially solvable, the pattern graph H has a dominating vertex, whereas in all cases that are known to be NP‐complete, the pattern graph H does not have a dominating vertex. © 2008 Wiley Periodicals, Inc. NETWORKS, 2008

We consider the speed scaling problem of minimizing the average response time of a collection of dynamically released jobs subject to a constraint A on energy used. We propose an algorithmic approach in which an energy optimal schedule is computed for a huge A , and then the energy optimal schedule is maintained as A decreases. We show that this approach yields an efficient algorithm for equi-work jobs. We note that the energy optimal schedule has the surprising feature that the job speeds are not monotone functions of the available energy. We then explain why this algorithmic approach is problematic for arbitrary work jobs. Finally, we explain how to use the algorithm for equi-work jobs to obtain an algorithm for arbitrary work jobs that is O (1)-approximate with respect to average response time, given an additional factor of (1 + ϵ) energy.

In a recent paper [International Journal of Uncertainty, Fuzziness, and Knowledge-Based Systems 8:611–618], Yager & Kreinovich characterize a certain proportional division rule in terms of the three axioms (1) symmetry, (2) participant mergability, and (3) continuity. This technical note tightens their characterization: The proportional division rule is already fully characterized by the first two axioms (1) symmetry and (2) participant mergability.

Abstract A dynamic network consists of a directed graph with capacities, costs, and integral transit times on the arcs. In the minimum‐cost dynamic flow problem (MCDFP), the goal is to compute, for a given dynamic network with source s, sink t, and two integers v and T, a feasible dynamic flow from s to t of value v, obeying the time bound T, and having minimum total cost. MCDFP contains as subproblems the minimum‐cost maximum dynamic flow problem, where v is fixed to the maximum amount of flow that can be sent from s to t within time T and the minimum‐cost quickest flow problem, where is T is fixed to the minimum time needed for sending v units of flow from s to t. We first prove that both subproblems are NP‐hard even on two‐terminal series‐parallel graphs with unit capacities. As main result, we formulate a greedy algorithm for MCDFP and provide a full characterization via forbidden subgraphs of the class 𝒢 of graphs, for which this greedy algorithm always yields an optimum solution (for arbitrary choices of problem parameters). 𝒢 is a subclass of the class of two‐terminal series‐parallel graphs. We show that the greedy algorithm solves MCDFP restricted to graphs in 𝒢 in polynomial time. © 2004 Wiley Periodicals, Inc.

We consider the traveling salesman problem when the cities are points in ℝ d for some fixed d and distances are computed according to geometric distances, determined by some norm. We show that for any polyhedral norm, the problem of finding a tour of maximum length can be solved in polynomial time. If arithmetic operations are assumed to take unit time, our algorithms run in time O ( n f -2 log n ), where f is the number of facets of the polyhedron determining the polyhedral norm. Thus, for example, we have O ( n 2 log n ) algorithms for the cases of points in the plane under the Rectilinear and Sup norms. This is in contrast to the fact that finding a minimum length tour in each case is NP-hard. Our approach can be extended to the more general case of quasi-norms with a not necessarily symmetric unit ball, where we get a complexity of O ( n 2 f -2 log n ).For the special case of two-dimensional metrics with f = 4 (which includes the Rectilinear and Sup norms), we present a simple algorithm with O ( n ) running time. The algorithm does not use any indirect addressing, so its running time remains valid even in comparison based models in which sorting requires Ω( n log n ) time. The basic mechanism of the algorithm provides some intuition on why polyhedral norms allow fast algorithms.Complementing the results on simplicity for polyhedral norms, we prove that, for the case of Euclidean distances in ℝ d for d ≥ 3, the Maximum TSP is NP-hard. This sheds new light on the well-studied difficulties of Euclidean distances.

Random redundant data storage strategies have proven to be a good choice for efficient data storage in multimedia servers. These strategies lead to a retrieval problem in which it is decided for each requested data block which disk to use for its retrieval. In this paper, we give a complexity classification of retrieval problems for random redundant storage.

We provide several non-approximability results for deterministic scheduling problems whose objective is to minimize the total job completion time. Unless 𝒫 =𝒩𝒫, none of the problems under consideration can be approximated in polynomial time within arbitrarily good precision. Most of our results are derived by APX-hardness proofs. We show that, whereas scheduling on unrelated machines with unit weights is polynomially solvable, the problem becomes APX-hard if release dates or weights are added. We further show APX-hardness for scheduling in flow shops, job shops, and open shops. We also investigate the problems of scheduling on parallel machines with precedence constraints and unit processing times, and two variants of the latter problem with unit communication delays; for these problems we provide lower bounds on the worst-case behavior of any polynomial-time approximation algorithm through the gap-reduction technique.

We investigate the approximability of no-wait shop scheduling problems under the makespan criterion. In a flow shop, all jobs pass through the machines in the same ordering. In the more general job shop, the routes of the jobs are job-dependent. We present a polynomial time approximation scheme (PTAS) for the no-wait flow shop problem on any fixed number of machines. Unless P=NP, this result cannot be extended to the job shop problem on a fixed number of machines: We show that the no-wait job shop problem is APX-hard on (i) two machines with at most five operations per job, and on (ii) three machines with at most three operations per job.

We derive results of the following flavor: If a combinatorial optimization problem can be formulated via a dynamic program of a certain structure and if the involved cost and transition functions satisfy certain arithmetical and structural conditions, then the optimization problem automatically possesses a fully polynomial time approximation scheme (FPTAS). Our characterizations provide a natural and uniform approach to fully polynomial time approximation schemes. We illustrate their strength and generality by deducing from them the existence of FPTASs for a multitude of scheduling problems. Many known approximability results follow as corollaries from our main result.

We consider the problem of scheduling a set of n jobs on m identical parallel machines so as to minimize the weighted sum of job completion times. This problem is NP-hard in the strong sense. The best approximation result known so far was a 1 2 (1+ p 2)--approximation algorithm that has been derived by Kawaguchi and Kyan back in 1986. The contribution of this paper is a polynomial time approximation scheme for this problem, which settles a problem that was open for a long time. Moreover, our result constitutes the first known approximation scheme for a strongly NP-hard scheduling problem with minsum objective. 1 Introduction The problem. We consider the following machine scheduling model. We are given a set J of n independent jobs that have to be scheduled on m identical parallel machines or processors. Each job j 2 J is specified by its positive processing requirement p j and by its positive weight w j . In a feasible schedule for J , every job j 2 J is processed for p j time uni...

We consider the problem of minimizing the weighted variance of job completion times on a single machine with job-dependent, agreeable weights. In 1995, Cai derived a fully polynomial time approximation scheme for the special case where the weights of the jobs are polynomially bounded in the number n of jobs. In this article, we completely settle the approximability status of this scheduling problem. We construct a fully polynomial time approximation scheme for the general case, without putting any restrictions on the weights of the jobs.

In the minimum-cost strong network orientation problem (MCSO), we are given an undirected graph G = (V, E) with nonnegative edge lengths 𝓁(e) and a transportation schedule T = {(s1, t1, w1), …, (sk, tk, wk)}, where wi units of weight have to be transported from the source vertex si to the target vertex ti for i = 1, …, k. Let Gσ be a strongly connected orientation of G and let L be the length of the shortest (directed) path from si to ti in Gσ. The goal in the MCSO is to find a strongly connected orientation Gσ such that the overall cost of the orientation given by Σ wiL (sum case) or maxi=1,…,k wiL (bottleneck case) is minimized. The strong network orientation problem is motivated by the practical problem of designing the optimal unidirectional flow path of automated guided vehicles. In this paper, we investigate the MCSO from the algorithmic and complexity points of view and propose a classification scheme. In the first part of the paper, we identify several efficiently solvable cases of the MCSO with sum and bottleneck objective functions which arise if additional restrictions are imposed on the structure of the graph G, the edge lengths 𝓁(e), and/or the transportation schedule T. In the second part, we identify special cases of the MCSO which are NP-hard. © 1999 John Wiley & Sons, Inc. Networks 33: 57–70, 1999

The bottleneck graph partition problem consists of partitioning the vertices of an undirected edge-weighted graph into two equally sized sets such that the maximum edge weight in the cut separating the two sets becomes minimum. In this short note, we present an optimum algorithm for this problem with running time O(n2), where n is the number of vertices in the graph. Our result answers an open problem posed in a recent paper by Hochbaum and Pathria (1996). © 1999 John Wiley & Sons, Inc. Networks 33: 189–191, 1999

We consider the problem of scheduling n jobs that are released over time on a single machine in order to minimize the total flow time. This problem is well known to be NP-complete, and the best polynomial-time approximation algorithms constructed so far had (more or less trivial) worst-case performance guarantees of O(n). In this paper, we present one positive and one negative result on polynomial-time approximations for the minimum total flow time problem: The positive result is the first approximation algorithm with a sublinear worst-case performance guarantee of $O(\sqrt{n})$. This algorithm is based on resolving the preemptions of the corresponding optimum preemptive schedule. The performance guarantee of our approximation algorithm is not far from best possible, as our second, negative result demonstrates: Unless P=NP, no polynomial-time approximation algorithm for minimum total flow time can have a worst-case performance guarantee of $O(n^{1/2-\eps})$ for any $\eps>0$.

In 1975, Kalmanson proved that if the distance matrix in the travelling salesman problem (TSP) fulfills certain combinatorial conditions (that are nowadays called the Kalmanson conditions) then the TSP is solvable in polynomial time [Canad. J. Math., 27 (1995), pp. 1000--1010]. We deal with the problem of deciding, for a given instance of the TSP, whether there is a renumbering of the cities such that the corresponding renumbered distance matrix fulfills the Kalmanson conditions. Two results are derived: first, it is shown that---in case it exists---such a renumbering can be found in polynomial time. Secondly, it is proved that such a renumbering exists if and only if the instance possesses the so-called master tour property. A recently posed question by Papadimitriou is thereby answered in the negative.

We consider the problem of scheduling n jobs that are released over time on a single machine in order to minimize the total ??flow time??. This problem is well-??known to be NP??-complete, and the best polynomial time approximation algorithms constructed so far had (more or less trivial)?? worst-??case performance guarantees of O??(n).????\nIn this paper, we present one positive and one negative result on polynomial time approximations for the minimum total ??flow time problem??. The positive result is the first approxima??tion algorithm with a sublinear worst-??case performance guarantee of O(\\sqrt{n}). This algorithm is based on resolving the preemptions of the corresponding optimum preemptive schedule??. The performance guarantee of our approximation algorithm is not far from best possible as our second, negative result demonstrates.?? Unless P=NP, no polynomial time approxima??tion algorithm for minimum total ??flow time can have a worst-??case performance guarantee of O(n^{1/2 - \\epsilon}) for any \\epsilon > 0.\n ?? ?? ????\nKeywords:?? scheduling, approximation algorithm, worst-??case analysis, total flow time, release time, single machine??.

Abstraet. The majority of results in computational learning theory are concerned with concept learning, i.e. with the special case of function learning for classes of functions with range {0, 1}. Much less is known about the theory of learning functions with a larger fange such as Nor IR. In particular relatively few results exist about he general structure of common models for function learning, and there are only very few nontrivial function classes for which positive learning results have been exhibited in any of these models. We introduce in this paper the notion of a binaly branching adversary tree for function learning, which allows us to give a somewhat surprising equivalent characterization f the optimal learning cost for learning a class of real-valued functions (in terms of a max-min definition which does not invoive any "learning " model). Another general structural result of this paper elates the cost for learning a union of function classes to the learning costs for the individual function classes. Furthermore, we exhibit an efficient leaming algorithm for learning convex piecewise linear functions from Rd into IR. Previously, the class of linear functions from 1R d into R was the only class of functions with multi-dimensional domain that was known to be learnable within the rigorous framework of a formal model for on-line leaming. Finally we give a sufficient condition for an arbitrary class 5 ~ of functions from IR into R that allows us to learn the class of all functions that can be written as the pointwise maximum of k functions from 5 r. This allows us to exhibit a number of further nontrivial classes of functions from ~ into R for which there exist eflicient]earning algorithms.

This paper presents the problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that $\Omega (\frac{1}{{d^2 }}) \leqslant R \leqslant \frac{{2\pi }}{d}$ for any graph. Moreover, it is proved that $R = \Theta (\frac{1}{d})$ for many graphs including planar graphs, complete graphs, hypercubes, multidimensional meshes and tori, and other special networks. It is also shown that the problem of deciding if $R = \frac{{2\pi }}{d}$ for a graph is NP-hard for $d = 4$, and by using a counting argument that $R = O(\frac{{\log d}}{{d^2 }})$ for many graphs.

The problem of on-line scheduling a set of independent jobs on m machines is considered. The goal is to minimize the makespan of the schedule. Graham’s List Scheduling heuristic [R. L. Graham, SIAM J. Appl. Math., 17(1969), pp. 416–429] guarantees a worst case performance of $2 - \frac{1} {m}$ for this problem. This worst case bound cannot be improved for $m = 2$ and $m = 3$. For $m \geqslant 4$, approximation algorithms with worst case performance at most $2 - \frac{1}{m} - \varepsilon _m $ are presented, where $\varepsilon _m $ is some positive real depending only on m.

The author presents a sequence of linear-time, bounded-space, on-line, bin-packing algorithms that are based on the “HARMONIC” algorithms ${\text{H}}_k $ introduced by Lee and Lee [J. Assoc. Comput. Mach., 32 (1985), pp. 562–572]. The algorithms in this paper guarantee the worst case performance of ${\text{H}}_k $, whereas they only use $O ( \log \log k )$ instead of k active bins. For $k\geqq 6$, the algorithms in this paper outperform all known heuristics using k active bins. For example, the author gives an algorithm that has worst case ratio less than 17/10 and uses only six active bins.

Given a set of nonintersecting polygonal obstacles in the plane, the link distance between two points s and t is the minimum number of edges required to form a polygonal path connecting s to t that avoids all obstacles. We present an algorithm that computes the link distance (and a corresponding minimum-link path) between two points in time Ο(Eα(n) log2 n) (and space Ο(E)), where n is the total number of edges of the obstacles, E is the size of the visibility graph, and α(n) denotes the extremely slowly growing inverse of Ackermann's function.

Given any natural number d, 0 < ε < 1, let ƒd(ε) denote the smallest integer ƒ such that every range space of Vapnik-Chervonenkis dimension d has an ε-net of size at most ƒ We solve a problem of Haussler and Welzl by showing that if d ≥ 2, then ƒd(ε) > 1/48 d/ε log 1/ ε which is not far from being optimal, if d is fixed and ε → 0. Further, we prove that ƒ1(ε) = max(2,⌈1/ε⌉ - 1), and similar bounds are established for some special classes of range spaces of Vapnik-Chervonenkis dimension three.

The problem of drawing a graph in the plane so that edges appear as straight lines and the minimum angle formed by any pair of incident edges is maximized is studied. The resolution of a layout is defined to be the size of the minimum angle formed by incident edges of the graph, and the resolution of a graph is defined to be the maximum resolution of any layout of the graph. The resolution R of a graph is characterized in terms of the maximum node degree d of the graph by proving that Omega (1/d/sup 2/)>