A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than e e−1 ≈ 1.582 (unless P = N P), which is larger than the inapproximability bound of 1.5 for the original problem.Mathematics Subject Classification Primary 90B35 · 68W25; Secondary 68Q25 · 90C10

In the ordinal Matroid Secretary Problem (MSP), elements from a weighted matroid are presented in random order to an algorithm that must incrementally select a large weight independent set. However, the algorithm can only compare pairs of revealed elements without using its numerical value. An algorithm is α probability-competitive if every element from the optimum appears with probability 1/α in the output. We present a technique to design algorithms with strong probability-competitive ratios, improving the guarantees for almost every matroid class considered in the literature: e.g., we get ratios of 4 for graphic matroids (improving on 2e by Korula and Pál [ICALP 2009]) and of 5.19 for laminar matroids (improving on 9.6 by Ma et al. [THEOR COMPUT SYST 2016]). We also obtain new results for superclasses of k column sparse matroids, for hypergraphic matroids, certain gammoids and graph packing matroids, and a 1 + O( log ρ/ρ) probability-competitive algorithm for uniform matroids of rank ρ based on Kleinberg's 1 + O( 1/ρ) utility-competitive algorithm [SODA 2005] for that class. Our second contribution are algorithms for the ordinal MSP on arbitrary matroids of rank ρ. We devise an O(log ρ) probability-competitive algorithm and an O(log log ρ) ordinal-competitive algorithm, a weaker notion of competitiveness but stronger than the utility variant. These are based on the O(log log ρ) utility-competitive algorithm by Feldman et al. [SODA 2015].

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A natural extension of the makespan minimization problem on unrelated machines is to allow jobs to be partially processed by different machines while incurring an arbitrary setup time. In this paper we present increasingly stronger LP-relaxations for this problem and their implications on the approximability of the problem. First we show that the straightforward LP, extending the approach for the original problem, has an integrality gap of 3 and yields an approximation algorithm of the same factor. By applying a lift-and-project procedure, we are able to improve both the integrality gap and the implied approximation factor to 1 + φ, where φ is the golden ratio. Since this bound remains tight for the seemingly stronger machine configuration LP, we propose a new job configuration LP that is based on an infinite continuum of fractional assignments of each job to the machines. We prove that this LP has a finite representation and can be solved in polynomial time up to any accuracy. Interestingly, we show that our problem cannot be approximated within a factor better than e e−1 ≈ 1.582 (unless P = N P), which is larger than the inapproximability bound of 1.5 for the original problem.Mathematics Subject Classification Primary 90B35 · 68W25; Secondary 68Q25 · 90C10