Optimizing for Transfers in a Multi-Vehicle Collection and Delivery Problem

Abstract We address the Collection and Delivery Problem (CDP) with multiple vehicles, such that each collects a set of items at different locations and delivers them to a dropoff point. The goal is to minimize either delivery time or the total distance traveled. We introduce an extension to the CDP: what if a vehicle can transfer items to another vehicle before making the final delivery? By dividing the labor among multiple vehicles, the delivery time and cost may be reduced. However, introducing transfers increases the number of feasible schedules exponentially. In this paper, we investigate this Collection and Delivery Problem with Transfers (CDP-T), discuss its theoretical underpinnings, and introduce a two-approximate polynomial time algorithm to minimize total distance travelled. Furthermore, we show that allowing transfers to take place at any location for the CDP-T results in at most a factor of two improvement. We demonstrate our approximation algorithms on large simulated problem instances. Finally, we deploy our algorithms on robots that transfer and deliver items autonomously in an office building.

Gapped String Indexing in subquadratic space and sublinear query time

In Gapped String Indexing, the goal is to compactly represent a string S of length n such that for any query consisting of two strings P1 and P2, called patterns, and an integer interval [α, β], called gap range, we can quickly find occurrences of P1 and P2 in S with distance in [α, β]. Gapped String Indexing is a central problem in computational biology and text mining and has thus received significant research interest, including parameterized and heuristic approaches. Despite this interest, the best-known time-space trade-offs for Gapped String Indexing are the straightforward O(n) space and O(n + occ) query time or Ω(n2) space and Õ(|P1| + |P2| + occ) query time. We break through this barrier obtaining the first interesting trade-offs with polynomially subquadratic space and polynomially sublinear query time. In particular, we show that, for every 0 ≤ δ ≤ 1, there is a data structure for Gapped String Indexing with either Õ(n2−δ/3) or Õ(n3−2δ) space and Õ(|P1|+ |P2|+ nδ · (occ + 1)) query time, where occ is the number of reported occurrences. As a new fundamental tool towards obtaining our main result, we introduce the Shifted Set Intersection problem: preprocess a collection of sets S1, ..., Sk of integers such that for any query consisting of three integers i, j, s, we can quickly output YES if and only if there exist a ∈ Si and b ∈ Sj with a + s = b. We start by showing that the Shifted Set Intersection problem is equivalent to the indexing variant of 3SUM (3SUM Indexing) [Golovnev et al., STOC 2020]. We then give a data structure for Shifted Set Intersection with gaps, which entails a solution to the Gapped String Indexing problem. Furthermore, we enhance our data structure for deciding Shifted Set Intersection, so that we can support the reporting variant of the problem, i.e., outputting all certificates in the affirmative case. Via the obtained equivalence to 3SUM Indexing, we thus give new improved data structures for the reporting variant of 3SUM Indexing, and we show how this improves upon the state-of-the-art solution for Jumbled Indexing [Chan and Lewenstein, STOC 2015] for any alphabet of constant size σ > 5

Connecting de Bruijn graphs

We study the problem of making a de Bruijn graph (dBG), constructed from a collection of strings, weakly connected while minimizing the total cost of edge additions. The input graph is a dBG that can be made weakly connected by adding edges (along with extra nodes if needed) from the underlying complete dBG. The problem arises from genome reconstruction, where the dBG is constructed from a set of sequences generated from a genome sample by a sequencing experiment. Due to sequencing errors, the dBG is never Eulerian in practice and is often not even weakly connected. We show the following results for a dBG G(V, E) of order k consisting of d weakly connected components: 1. Making G weakly connected by adding a set of edges of minimal total cost is NP-hard. 2. No PTAS exists for making G weakly connected by adding a set of edges of minimal total cost (unless the unique games conjecture fails). We complement this result by showing that there does exist a polynomial-time (2 − 2/d)-approximation algorithm for the problem. 3. We consider a restricted version of the above problem, where we are asked to make G weakly connected by only adding directed paths between pairs of components. We show that making G weakly connected by adding d− 1 such paths of minimal total cost can be done in O(k|V |α(|V |) + |E|) time, where α(·) is the inverse Ackermann function. This improves on the O(k|V | log(|V |) + |E|)-time algorithm proposed by Bernardini et al. [CPM 2022] for the same restricted problem. 4. An ILP formulation of polynomial size for making G Eulerian with minimal total cost

Plasma Concentrations of Long Chain N-3 Fatty Acids in Early and Mid-Pregnancy and Risk of Early Preterm Birth

Background: Fish oil supplementation has been shown to delay spontaneous delivery, but the levels and clinical significance remain uncertain. We examined the association between plasma fatty acids quantified in pregnancy and subsequent risk of early preterm birth. Methods: In a case-control design nested in the Danish National Birth Cohort, we identified 376 early preterm cases (<34 gestational weeks, excluding preeclampsia cases) and 348 random controls. Plasma eicosapentaenoic acid plus docosahexaenoic acid (EPA+DHA% of total fatty acids), were measured twice in pregnancy, at gestation weeks 9 and 25 (medians). Odds ratios and 95% confidence intervals (CI's) for associations between EPA+DHA and early preterm risk were estimated by logistic regression, adjusted for the woman's age, height, pre-pregnancy BMI, parity, smoking, and socioeconomic factors. Hypotheses and analytical plan were defined and archived a priori. Findings: Analysis using restricted cubic splines of the mean of 1st and 2nd sample measurements showed a strong and significant non-linear association (p < 0.0001) in which the risk of early preterm birth steeply increased when EPA+DHA concentrations were lower than 2% and flattened out at higher levels. Women in the lowest quintile (EPA+DHA < 1.6%) had 10.27 times (95% confidence interval 6.80–15.79, p < 0.0001) increased risk, and women in the second lowest quintile had 2.86 (95% CI 1.79–4.59, p < 0.0001) times increased risk, when compared to women in the three aggregated highest quintiles (EPA+DHA ≥ 1.8%). Interpretation: Low plasma concentration of EPA and DHA during pregnancy is a strong risk factor for subsequent early preterm birth in Danish women. Keywords: Early preterm birth, Long chained n-3 fatty acids, Biomarkers, Prospective study, Danish National Birth Cohor