A fuel reformer is configured in an exhaust line upstream of a LNT. To desulfate the LNT, over a first period, fuel is provided to the exhaust to make the exhaust rich. While producing reformate, the reformer is allowed to heat to a relatively large extent, typically at least about 75° C. Before the reformer overheats, the provision of fuel is reduced and typically stopped entirely for a second period, over which the reformer cools. The heating and cooling periods are repeated to complete the desulfation process. The reduced efficiency of operating the reformer over an extended temperature range is more than compensated for by the benefits of extending the rich pulse.