Sensitivity analysis of the greedy heuristic for binary knapsack problems

Greedy heuristics are a popular choice of heuristics when we have to solve a large variety of NP -hard combinatorial problems. In particular for binary knapsack problems, these heuristics generate good results. If some uncertainty exists beforehand regarding the value of any one element in the problem data, sensitivity analysis procedures can be used to know the tolerance limits within which the value may vary will not cause changes in the output. In this paper we provide a polynomial time characterization of such limits for greedy heuristics on two classes of binary knapsack problems, namely the 0-1 knapsack problem and the subset sum problem. We also study the relation between algorithms to solve knapsack problems and algorithms to solve their sensitivity analysis problems, the conditions under which the sensitivity analysis of the heuristic generates bounds for the toler-ance limits for the optimal solutions, and the empirical behavior of the greedy output when there is a change in the problem data.

Fault-Tolerant Spanners: Better and Simpler

A natural requirement of many distributed structures is fault-tolerance: after some failures, whatever remains from the structure should still be effective for whatever remains from the network. In this paper we examine spanners of general graphs that are tolerant to vertex failures, and significantly improve their dependence on the number of faults $r$, for all stretch bounds. For stretch $k \geq 3$ we design a simple transformation that converts every $k$-spanner construction with at most $f(n)$ edges into an $r$-fault-tolerant $k$-spanner construction with at most $O(r^3 \log n) \cdot f(2n/r)$ edges. Applying this to standard greedy spanner constructions gives $r$-fault tolerant $k$-spanners with $\tilde O(r^{2} n^{1+\frac{2}{k+1}})$ edges. The previous construction by Chechik, Langberg, Peleg, and Roddity [STOC 2009] depends similarly on $n$ but exponentially on $r$ (approximately like $k^r$). For the case $k=2$ and unit-length edges, an $O(r \log n)$-approximation algorithm is known from recent work of Dinitz and Krauthgamer [arXiv 2010], where several spanner results are obtained using a common approach of rounding a natural flow-based linear programming relaxation. Here we use a different (stronger) LP relaxation and improve the approximation ratio to $O(\log n)$, which is, notably, independent of the number of faults $r$. We further strengthen this bound in terms of the maximum degree by using the \Lovasz Local Lemma. Finally, we show that most of our constructions are inherently local by designing equivalent distributed algorithms in the LOCAL model of distributed computation.Comment: 17 page

A Dimension-Adaptive Multi-Index Monte Carlo Method Applied to a Model of a Heat Exchanger

We present an adaptive version of the Multi-Index Monte Carlo method, introduced by Haji-Ali, Nobile and Tempone (2016), for simulating PDEs with coefficients that are random fields. A classical technique for sampling from these random fields is the Karhunen-Lo\`eve expansion. Our adaptive algorithm is based on the adaptive algorithm used in sparse grid cubature as introduced by Gerstner and Griebel (2003), and automatically chooses the number of terms needed in this expansion, as well as the required spatial discretizations of the PDE model. We apply the method to a simplified model of a heat exchanger with random insulator material, where the stochastic characteristics are modeled as a lognormal random field, and we show consistent computational savings