Approximation Algorithms and Linear Programming Relaxations for Scheduling Problems Related to Min-Sum Set Cover

Many of the currently best-known approximation algorithms for NP-hard optimization problems are based on Linear Programming (LP) and Semi-definite Programming (SDP) relaxations. Given its power, this class of algorithms seems to contain the most favourable candidates for outperforming the current state-of-the-art approximation guarantees for NP-hard problems, for which there still exists a gap between the inapproximability results and the approximation guarantees that we know how to achieve in polynomial time. In this thesis, we address both the power and the limitations of these relaxations, as well as the connection between the shortcomings of these relaxations and the inapproximability of the underlying problem. In the first part, we study the limitations of LP relaxations of well-known graph problems such as the Vertex Cover problem and the Independent Set problem. We prove that any small LP relaxation for the aforementioned problems, cannot have an integrality gap strictly better than $2$ and $\omega(1)$, respectively. Furthermore, our lower bound for the Independent Set problem also holds for any SDP relaxation. Prior to our work, it was only known that such LP relaxations cannot have an integrality gap better than $1.5$ for the Vertex Cover Problem, and better than $2$ for the Independent Set problem. In the second part, we study the so-called knapsack cover inequalities that are used in the current best relaxations for numerous combinatorial optimization problems of covering type. In spite of their widespread use, these inequalities yield LP relaxations of exponential size, over which it is not known how to optimize exactly in polynomial time. We address this issue and obtain LP relaxations of quasi-polynomial size that are at least as strong as that given by the knapsack cover inequalities. In the last part, we show a close connection between structural hardness for k-partite graphs and tight inapproximability results for scheduling problems with precedence constraints. This connection is inspired by a family of integrality gap instances of a certain LP relaxation. Assuming the hardness of an optimization problem on k-partite graphs, we obtain a hardness of $2-\varepsilon$ for the problem of minimizing the makespan for scheduling with preemption on identical parallel machines, and a super constant inapproximability for the problem of scheduling on related parallel machines. Prior to this result, it was only known that the first problem does not admit a PTAS, and the second problem is NP-hard to approximate within a factor strictly better than 2, assuming the Unique Games Conjecture.

We consider a single machine scheduling problem with multiple maintenance activities, where the maintenance duration function is of the linear form f(t) = a+bt with a ≥ 0 and b > 1. We propose an approximation algorithm named FFD-LS2I with a worst-case bound of 2 for problem. We also show that there is no polynomial time approximation algorithm with a worst-case bound less than 2 for the problem with b ≥ 0 unless P = NP, which implies that the FFD-LS2I algorithm is the best possible algorithm for the case b > 1 and that the FFD-LS algorithm, which is proposed in the literature, is the best possible algorithm for the case b ≤ 1 both from the worst-case bound point of view.

We study the generalized min sum set cover (GMSSC) problem, wherein given a collection of hyperedges $E$ with arbitrary covering requirements $k_e$, the goal is to find an ordering of the vertices to minimize the total cover time of the hyperedges; a hyperedge $e$ is considered covered by the first time when $k_e$ many of its vertices appear in the ordering. We give a $4.642$ approximation algorithm for GMSSC, coming close to the best possible bound of $4$, already for the classical special case (with all $k_e=1$) of min sum set cover (MSSC) studied by Feige, Lovász and Tetali, and improving upon the previous best known bound of $12.4$ due to Im, Sviridenko and van der Zwaan. Our algorithm is based on transforming the LP solution by a suitable kernel and applying randomized rounding. This also gives an LP-based $4$ approximation for MSSC. As part of the analysis of our algorithm, we also derive an inequality on the lower tail of a sum of independent Bernoulli random variables, which might be of independent interest and broader utility. Another well-known special case is the min sum vertex cover (MSVC) problem, in which the input hypergraph is a graph and $k_e = 1$, for every edge. We give a $16/9$ approximation for MSVC, and show a matching integrality gap for the natural LP relaxation. This improves upon the previous best $1.999946$ approximation of Barenholz, Feige and Peleg. (The claimed $1.79$ approximation result of Iwata, Tetali and Tripathi for the MSVC turned out have an unfortunate, seemingly unfixable, mistake in it.) Finally, we revisit MSSC and consider the $\ell_p$ norm of cover-time of the hyperedges. Using a dual fitting argument, we show that the natural greedy algorithm achieves tight, up to NP-hardness, approximation guarantees of $(p+1)^{1+1/p}$, for all $p\ge 1$. For $p=1$, this gives yet another proof of the $4$ approximation for MSSC.

This paper addresses the Minimum Linear Ordering Problem (MLOP): Given a nonnegative set function f on a finite set V, find a linear ordering on V such that the sum of the function values for all the suffixes is minimized. This problem generalizes well-known problems such as the Minimum Linear Arrangement, Min Sum Set Cover, Minimum Latency Set Cover, and Multiple Intents Ranking. Extending a result of Feige, Lovasz, and Tetali (2004) on Min Sum Set Cover, we show that the greedy algorithm provides a factor 4 approximate optimal solution when the cost function f is supermodular. We also present a factor 2 rounding algorithm for MLOP with a monotone submodular cost function, using the convexity of the Lovasz extension. These are among very few constant factor approximation algorithms for NP-hard minimization problems formulated in terms of submodular/supermodular functions. In contrast, when f is a symmetric submodular function, the problem has an information theoretic lower bound of 2 on the approximability.

We discuss the problem of sequencing precedence-constrained jobs on a single machine to minimize the average weighted completion time. This problem has attracted much attention in the mathematical programming community since Sidney’s pioneering work in 1975. We look at the problem from a polyhedral perspective and uncover a relation between Sidney’s decomposition theorem and different linear programming relaxations. More specifically, we present a generalization of Sidney’s result, which particularly allows us to reason that virtually all known 2-approximation algorithms comply with his decomposition. Moreover, we establish a connection between the single-machine scheduling problem and the vertex cover problem. Indeed, in the special case of series-parallel precedence constraints, we prove that the sequencing problem can be seen as a special case of vertex cover. We also argue that this result is true for general precedence constraints if one can show that a certain integer program represents a valid formulation of the sequencing problem. Finally, we provide a characterization of the active inequalities of a linear programming relaxation in completion time variables.KeywordsSchedule ProblemCompletion TimeSingle MachinePrecedence ConstraintVertex CoverThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Dendritic cells in the skin and mucosa: what's new.

We introduce a novel versatile phosphoramidite building block for the modification of oligonucleotides (ONs) with acyl hydrazides on the 5′- or 3′-terminus, or both. The reaction of these hydrazide functionalized ONs with 4-methoxyphenylaldehyde is demonstrated for solution derivatization. Hydrazides are considered nowadays as promising reactants, which show enhanced reactivity at neutral and slightly acidic conditions and higher stability of yielding products as compared to the aliphatic amines, which are broadly used for ONs derivatization. Our method to introduce hydrazides into ONs employs a phosphoramidite modifier designed to split, during ammonia or lithium hydroxide treatment, into two hydrazides via β-elimination of a central bis-2-carbonylethoxysulfone unit. It allows the creation of ONs derivatized with a hydrazide moiety at the 5′-, 3′- and both 5′- and 3′-termini, as well as two different hydrazide containing ONs at the same time, viz. in one sequence on the same solid support. In latter case one can, for example, synthesize two hydrazide containing ONs, where one is 5′-modified and second one is 3′-modified. This work was supported by German Research Council (DFG) and the Federal Ministry of Education and Research (BMBF, Germany). S.I. Antsypovich gratefully acknowledges a stipend and his fellowship from German Academic Exchange Service (DAAD) and Alexander von Humboldt Foundation (AvH-Stiftung, Germany), respectively.

Do Hemoglobin Concentration and Anemia Prevalence Differ Between Capillary and Venous Blood and Between Analysis Methods?

Introduction/BackgroundHigh-grade serous ovarian cancer (HGSOC) is the most lethal gynecologic cancer. This tumor spreads following peritoneal fluid (PF) dynamics, being the most representative biofluid of the tumor micro-environment (TME). In...

A separable assignment problem (SAP) is defined by a set of bins and a set of items to pack in each bin; a value, fij, for assigning item j to bin i; and a separate packing constraint for each bin—i.e., for each bin, a family of subsets of items that fit in to that bin. The goal is to pack items into bins to maximize the aggregate value. This class of problems includes the maximum generalized assignment problem (GAP)1 and a distributed caching problem (DCP) described in this paper. Given a β-approximation algorithm for finding the highest value packing of a single bin, we give A polynomial-time LP-rounding based ((1 − 1/e)β)-approximation algorithm. A simple polynomial-time local search (β/(β + 1) − ε)-approximation algorithm, for any ε > 0. Therefore, for all examples of SAP that admit an approximation scheme for the single-bin problem, we obtain an LP-based algorithm with (1 − 1/e − ε)-approximation and a local search algorithm with (½ - ε)-approximation guarantee. Furthermore, for cases in which the subproblem admits a fully polynomial approximation scheme (such as for GAP), the LP-based algorithm analysis can be strengthened to give a guarantee of 1 − 1/e. The best previously known approximation algorithm for GAP is a ½-approximation by Shmoys and Tardos and Chekuri and Khanna. Our LP algorithm is based on rounding a new linear programming relaxation, with a provably better integrality gap. To complement these results, we show that SAP and DCP cannot be approximated within a factor better than 1 − 1/e unless NP ⫅ DTIME(nO(log log n)), even if there exists a polynomial-time exact algorithm for the single-bin problem. We extend the (1 − 1/e)-approximation algorithm to a constant-factor approximation algorithms for a nonseparable assignment problem with applications in maximizing revenue for budget-constrained combinatorial auctions and the AdWords assignment problem. We generalize the local search algorithm to yield a ½ - ε approximation algorithm for the maximum k-median problem with hard capacities.

When the present Humboldt Foundation was founded on 10 December 1953, it was the third time a sponsoring institution had been created in the tradition of the distinguished naturalist, universal genius and South America researcher. He thought it was so important to promote young academic talent that he financed promising young researchers from his own personal fortune. The first Alexander von Humboldt Foundation was established in 1860, a year after the death of its eponym, with the objective of sponsoring young German academics to travel abroad. After losing its capital in the great inflation and ceasing to function in 1923, it was re-established in 1925 with the aim of supporting the studies of foreign academics and doctoral students in Germany.

Optimization and Approximation in Deterministic Sequencing and Scheduling: a Survey

We consider the problem of scheduling unrelated parallel machines subject to release dates so as to minimize the total weighted completion time of jobs. The main contribution of this paper is a provably good convex quadratic programming relaxation of strongly polynomial size for this problem. The best previously known approximation algorithms are based on LP relaxations in time- or interval-indexed variables. Those LP relaxations, however, suffer from a huge number of variables. As a result of the convex quadratic programming approach we can give a very simple and easy to analyze 2-approximation algorithm which can be further improved to performance guarantee 3/2 in the absence of release dates. We also consider preemptive scheduling problems and derive approximation algorithms and results on the power of preemption which improve upon the best previously known results for these settings. Finally, for the special case of two machines we introduce a more sophisticated semidefinite programming relaxation and apply the random hyperplane technique introduced by Goemans and Williamson for the MaxCut problem; this leads to an improved 1.2752-approximation.

Single-machine scheduling under the job rejection constraint

We consider a single-machine scheduling problem. Given some continuous, nondecreasing cost function, we aim to compute a schedule minimizing the weighted total cost, where the cost of each job is determined by the cost function value at its completion time. This problem is closely related to scheduling a single machine with nonuniform processing speed. We show that for piecewise linear cost functions it is strongly NP-hard. The main contribution of this article is a tight analysis of the approximation guarantee of Smith’s rule under any convex or concave cost function. More specifically, for these wide classes of cost functions we reduce the task of determining a worst-case problem instance to a continuous optimization problem, which can be solved by standard algebraic or numerical methods. For polynomial cost functions with positive coefficients, it turns out that the tight approximation ratio can be calculated as the root of a univariate polynomial. We show that this approximation ratio is asymptotically equal to k ( k − 1)/( k + 1) , denoting by k the degree of the cost function. To overcome unrealistic worst-case instances, we also give tight bounds for the case of integral processing times that are parameterized by the maximum and total processing time.