On the computational tractability of a geographic clustering problem arising in redistricting

Redistricting is the problem of dividing a state into a number $k$ of regions, called districts. Voters in each district elect a representative. The primary criteria are: each district is connected, district populations are equal (or nearly equal), and districts are "compact". There are multiple competing definitions of compactness, usually minimizing some quantity. One measure that has been recently promoted by Duchin and others is number of cut edges. In redistricting, one is given atomic regions out of which each district must be built. The populations of the atomic regions are given. Consider the graph with one vertex per atomic region (with weight equal to the region's population) and an edge between atomic regions that share a boundary. A districting plan is a partition of vertices into $k$ parts, each connnected, of nearly equal weight. The districts are considered compact to the extent that the plan minimizes the number of edges crossing between different parts. Consider two problems: find the most compact districting plan, and sample districting plans under a compactness constraint uniformly at random. Both problems are NP-hard so we restrict the input graph to have branchwidth at most $w$. (A planar graph's branchwidth is bounded by its diameter.) If both $k$ and $w$ are bounded by constants, the problems are solvable in polynomial time. Assume vertices have weight~1. One would like algorithms whose running times are of the form $O(f(k,w) n^c)$ for some constant $c$ independent of $k$ and $w$, in which case the problems are said to be fixed-parameter tractable with respect to $k$ and $w$). We show that, under a complexity-theoretic assumption, no such algorithms exist. However, we do give algorithms with running time $O(c^wn^{k+1})$. Thus if the diameter of the graph is moderately small and the number of districts is very small, our algorithm is useable

The Coming Divorce Decline

This article analyzes U.S. divorce trends over the past decade and considers their implications for future divorce rates. Modeling women’s odds of divorce from 2008 to 2017 using marital events data from the American Community Survey, I find falling divorce rates with or without adjustment for demographic covariates. Age-specific divorce rates show that the trend is driven by younger women, which is consistent with longer term trends showing uniquely high divorce rates among people born in the Baby Boom period. Finally, I analyze the characteristics of newly married women and estimate the trend in their likelihood of divorcing based on the divorce models. The results show falling divorce risks for more recent marriages. The accumulated evidence thus points toward continued decline in divorce rates. The United States is progressing toward a system in which marriage is rarer and more stable than it was in the past

Parental Age and Cognitive Disability among Children in the United States

Funding for Open Access provided by the UMD Libraries Open Access Publishing Fund.Some risks of having children at older ages are widely documented, and the “biological clock” is a popular media concern, but the association between cognitive disability generally and both mothers’ and fathers’ age is not well known. This article assesses descriptively the relationship between children’s cognitive disability and parents’ age at birth, using a sample of 353,119 children aged five to eleven living with two married parents from the 2009-2011 American Community Survey. Cognitive disability varied by parental age categories from 1.8 percent to 5.4 percent, with overall rates of 2.2 percent. Odds of disability were much more strongly associated with mothers’ age at birth than with fathers’ age at birth, with the highest odds for children whose mothers were age 45 or higher at the time of their birth (adjusted odds ratio 2.7 relative to age 30 to 34) and the lowest for those born to mothers in their early 30s. These results demonstrate that the risk is strongly associated with the mother’s age at birth—but not the father’s. This is consistent with previous research showing that it is the mother’s health, rather than age per se, that is most important for the health of their children

Headed Toward Equality? Housework Change in Comparative Perspective

This paper examined gendered housework in the larger context of comparative social change, asking specifically whether cross-national differences in domestic labor patterns converge over time. Our analysis of data from 13 countries (N =11,065) from the 1994 and 2002 International Social Survey Program (ISSP), confirmed that social context matters in shaping couples’ division of labor at home, but also showed that context affects patterns of change. Our results suggested that, compared to the most egalitarian countries, the shift in housework patterns was greatest among the most traditional countries. This provides support for the thesis of cultural convergence, but the evidence did not suggest that such convergence will lead to complete equality in the foreseeable future

The Opioid Epidemic Has Disrupted Children’s Living Arrangements

The contemporary drug overdose crisis has had profound impacts on children and families in the United States. This brief summarizes how children’s living arrangements have changed during the opioid epidemic. The authors find that opioid overdose deaths are associated with decreasing shares of children living with two married parents and increases in shares of children living with unmarried but cohabiting parents, single fathers, and adults other than their parents. These changes have been most pronounced among White children

Parental Age and Cognitive Disability among Children in the United States

Abstract: Some risks of having children at older ages are widely documented, and the "biological clock" is a popular media concern, but the association between cognitive disability generally and both mothers' and fathers' age is not well known. This article assesses descriptively the relationship between children's cognitive disability and parents' age at birth, using a sample of 353,119 children aged five to eleven living with two married parents from the 2009-2011 American Community Survey. Cognitive disability varied by parental age categories from 1.8 percent to 5.4 percent, with overall rates of 2.2 percent. Odds of disability were much more strongly associated with mothers' age at birth than with fathers' age at birth, with the highest odds for children whose mothers were age 45 or higher at the time of their birth (adjusted odds ratio 2.7 relative to age 30 to 34) and the lowest for those born to mothers in their early 30s. These results demonstrate that the risk is strongly associated with the mother's age at birth-but not the father's. This is consistent with previous research showing that it is the mother's health, rather than age per se, that is most important for the health of their children

The Heated Core of the Radio-Quiet Galaxy Cluster A644

We present an analysis of a Chandra ACIS-I observation of the massive galaxy cluster A644. This cluster was previously classified as a cooling flow, but no radio emission has been detected from its cD galaxy. Outside the core (R ~75 kpc ~0.03R_vir) the hot ICM has properties consistent with a (relaxed) cool-core cluster out to the largest radii investigated (R ~415 kpc ~0.14 R_vir). Over this region the gravitating mass profile is described well by a Navarro-Frenk-White profile with concentration parameter, c = 6.1 +/- 1.2, and virial radius, R_vir = 2.9 +/- 0.4 Mpc. However, inside the core the temperature and entropy profiles reverse their inward radial decline and rise at the center; the inner temperature profile is inconsistent with a constant at the 2.3 sigma level. Although the core region does not display X-ray cavities or filamentary structures characteristic of radio-loud, cool-core clusters, the peak of the X-ray emission is offset from that of the centroid of the global X-ray halo by ~60 kpc. The position of the cD galaxy lies approximately between the X-ray peak and centroid, further testifying to a merger origin for the properties of the X-ray emission in the core. We discuss the implications of A644 and the small number of radio-quiet, cool-core clusters for the AGN feedback paradigm to suppress cooling flows in clusters.Comment: 10 pages, 4 figures, Accepted for Publication in The Astrophysical Journa