Percolation-like Scaling Exponents for Minimal Paths and Trees in the Stochastic Mean Field Model

In the mean field (or random link) model there are $n$ points and inter-point distances are independent random variables. For $0 < \ell < \infty$ and in the $n \to \infty$ limit, let $\delta(\ell) = 1/n \times$ (maximum number of steps in a path whose average step-length is $\leq \ell$). The function $\delta(\ell)$ is analogous to the percolation function in percolation theory: there is a critical value $\ell_* = e^{-1}$ at which $\delta(\cdot)$ becomes non-zero, and (presumably) a scaling exponent $\beta$ in the sense $\delta(\ell) \asymp (\ell - \ell_*)^\beta$. Recently developed probabilistic methodology (in some sense a rephrasing of the cavity method of Mezard-Parisi) provides a simple albeit non-rigorous way of writing down such functions in terms of solutions of fixed-point equations for probability distributions. Solving numerically gives convincing evidence that $\beta = 3$. A parallel study with trees instead of paths gives scaling exponent $\beta = 2$. The new exponents coincide with those found in a different context (comparing optimal and near-optimal solutions of mean-field TSP and MST) and reinforce the suggestion that these scaling exponents determine universality classes for optimization problems on random points.Comment: 19 page

Shortest-weight paths in random regular graphs

Consider a random regular graph with degree $d$ and of size $n$. Assign to each edge an i.i.d. exponential random variable with mean one. In this paper we establish a precise asymptotic expression for the maximum number of edges on the shortest-weight paths between a fixed vertex and all the other vertices, as well as between any pair of vertices. Namely, for any fixed $d \geq 3$, we show that the longest of these shortest-weight paths has about $\hat{\alpha}\log n$ edges where $\hat{\alpha}$ is the unique solution of the equation $\alpha \log(\frac{d-2}{d-1}\alpha) - \alpha = \frac{d-3}{d-2}$, for $\alpha > \frac{d-1}{d-2}$.Comment: 20 pages. arXiv admin note: text overlap with arXiv:1112.633

Minimum-cost matching in a random graph with random costs

Let $G_{n,p}$ be the standard Erd\H{o}s-R\'enyi-Gilbert random graph and let $G_{n,n,p}$ be the random bipartite graph on $n+n$ vertices, where each $e\in [n]^2$ appears as an edge independently with probability $p$. For a graph $G=(V,E)$, suppose that each edge $e\in E$ is given an independent uniform exponential rate one cost. Let $C(G)$ denote the random variable equal to the length of the minimum cost perfect matching, assuming that $G$ contains at least one. We show that w.h.p. if $d=np\gg(\log n)^2$ then w.h.p. {\bf E}[C(G_{n,n,p})] =(1+o(1))\frac{\p^2}{6p}. This generalises the well-known result for the case $G=K_{n,n}$. We also show that w.h.p. {\bf E}[C(G_{n,p})] =(1+o(1))\frac{\p^2}{12p} along with concentration results for both types of random graph.Comment: Replaces an earlier paper where $G$ was an arbitrary regular bipartite grap

Probabilistic Analysis of Optimization Problems on Generalized Random Shortest Path Metrics

Simple heuristics often show a remarkable performance in practice for optimization problems. Worst-case analysis often falls short of explaining this performance. Because of this, "beyond worst-case analysis" of algorithms has recently gained a lot of attention, including probabilistic analysis of algorithms. The instances of many optimization problems are essentially a discrete metric space. Probabilistic analysis for such metric optimization problems has nevertheless mostly been conducted on instances drawn from Euclidean space, which provides a structure that is usually heavily exploited in the analysis. However, most instances from practice are not Euclidean. Little work has been done on metric instances drawn from other, more realistic, distributions. Some initial results have been obtained by Bringmann et al. (Algorithmica, 2013), who have used random shortest path metrics on complete graphs to analyze heuristics. The goal of this paper is to generalize these findings to non-complete graphs, especially Erd\H{o}s-R\'enyi random graphs. A random shortest path metric is constructed by drawing independent random edge weights for each edge in the graph and setting the distance between every pair of vertices to the length of a shortest path between them with respect to the drawn weights. For such instances, we prove that the greedy heuristic for the minimum distance maximum matching problem, the nearest neighbor and insertion heuristics for the traveling salesman problem, and a trivial heuristic for the $k$-median problem all achieve a constant expected approximation ratio. Additionally, we show a polynomial upper bound for the expected number of iterations of the 2-opt heuristic for the traveling salesman problem.Comment: An extended abstract appeared in the proceedings of WALCOM 201

Gender-based violence and its association with mental health among Somali women in a Kenyan refugee camp: a latent class analysis

background In conflict-affected settings, women and girls are vulnerable to gender-based violence (GBV). GBV is associated with poor long-term mental health such as anxiety, depression and post-traumatic stress disorder (PTSD). Understanding the interaction between current violence and past conflict-related violence with ongoing mental health is essential for improving mental health service provision in refugee camps. Methods Using data collected from 209 women attending GBV case management centres in the Dadaab refugee camps, Kenya, we grouped women by recent experience of GBV using latent class analysis and modelled the relationship between the groups and symptomatic scores for anxiety, depression and PTSD using linear regression. Results Women with past-year experience of intimate partner violence alone may have a higher risk of depression than women with past-year experience of non-partner violence alone (Coef. 1.68, 95% CI 0.25 to 3.11). Conflict-related violence was an important risk factor for poor mental health among women who accessed GBV services, despite time since occurrence (average time in camp was 11.5 years) and even for those with a past-year experience of GBV (Anxiety: 3.48, 1.85–5.10; Depression: 2.26, 0.51–4.02; PTSD: 6.83, 4.21–9.44). Conclusion Refugee women who experienced past-year intimate partner violence or conflict-related violence may be at increased risk of depression, anxiety or PTSD. Service providers should be aware that compared to the general refugee population, women who have experienced violence may require additional psychological support and recognise the enduring impact of violence that occurred before, during and after periods of conflict and tailor outreach and treatment services accordingly

Minimum-weight combinatorial structures under random cost-constraints

Recall that Janson showed that if the edges of the complete graph Kn are assigned exponentially distributed independent random weights, then the expected length of a shortest path between a fixed pair of vertices is asymptotically equal to (log n)/n. We consider analogous problems where edges have not only a random length but also a random cost, and we are interested in the length of the minimumlength structure whose total cost is less than some cost budget. For several classes of structures, we determine the correct minimum length structure as a function of the cost-budget, up to constant factors. Moreover, we achieve this even in the more general setting where the distribution of weights and costs are arbitrary, so long as the density f(x) as x → 0 behaves like cxγ for some γ ≥ 0; previously, this case was not understood even in the absence of cost constraints. We also handle the case where each edge has several independent costs associated to it, and we must simultaneously satisfy budgets on each cost. In this case, we show that the minimum-length structure obtainable is essentially controlled by the product of the cost thresholds