Unsigned linear unichromosomal genomes with genes labeled 1, . . . , n are represented by permutations of these labels, whose adjacencies are the unordered pairs of genes occurring consecutively. The breakpoint distance counts the adjacencies of one genome that are absent from the other, and a breakpoint median minimizes the total breakpoint distance to a given set of genomes. Deciding whether a permutation is a median often requires a minimization over the whole symmetric group, which makes the median set resistant to exact counting. We replace this global condition by a local one: a c-approximate median is a genome having at most c adjacencies absent from all the input genomes. For a fixed genome we introduce its approximate median inverse, the set of ordered k-tuples of genomes ad- mitting it as a c-approximate median, and we construct and count this set explicitly. For k genomes chosen uniformly and independently at random, this gives the probability that a given genome is a c-approximate median and the expected number of such approximate medians. The relaxation is not only technical: approximate medians and near-optimal genomes co- incide up to a shift of the tolerance that remains bounded in probability as the number of genes grows. Finally, we derive an explicit upper bound on the probability that a given genome is an exact median, and on the expected number of exact breakpoint medians.
Phylogenetic distance deconvolution represents mixture-to-reference distances by weighted sources placed on a reference tree. We study the minimum number of strictly interior sources that explains a distance profile, and what incomplete references cost. On a positive-rational reduced tree we give an exact algorithm with 2^O(K) poly(B) bit complexity for a free positive mean pendant and any nonempty set of observed leaves; the single-exponential dependence on K is tight under ETH, already with every leaf observed. With arbitrary missing references and a prescribed pendant the problem is W[1]-hard in the source budget K, even on positive binary trees supplied with a strict explanation using at most K+1 sources. We then use the algorithm as an instrument. Removing references can only lower the minimum source count, never raise it, at every error tolerance. Which references are missing then matters far more than how many. Under nearest-to-source deletion a median of two references can be erased before the generating support stops being the unique smallest explanation, and that budget stays nearly flat as the panel grows (Spearman 0.10 over 20 to 361 references, against 0.82 for random loss). Discarding 7,931 of 10,575 references from a real 16S panel at random costs 1.8% of the tolerance at which one source suffices. Discarding comparably many as whole clades costs 90%. That tolerance needs neither ground truth nor an error model, which makes the model measurable on real data. The generating configuration's worst-leaf residual exceeds the one-source optimum in all 296 real ANI cases, and in 248 after refitting the common pendant. The phenomenon is not unobservable in practice. A point estimate of the count alone does not quantify it. Methods for this problem should report the tolerance at which an explanation of a given size exists.
Organisms must constantly weigh exploiting the resources they know against exploring for better ones. When foragers compete, the best behavior for each forager also depends on what its competitors do. How does competition change the strategic balance between exploration and exploitation? Here we address this question in a minimal model: two $ε$-greedy temporal-difference learners on a one-dimensional lattice, who start at a small reward at one end and are unaware of a larger reward at the other. For a lone forager, the optimal exploration rate increases monotonically with the value of the distant reward, and with its distance up to a critical distance. When two foragers share the lattice and split rewards, the choice of exploration rate becomes a game whose Nash equilibria have both foragers exploring, both staying, or one exploring while the other stays (anti-coordination). Under local best-response dynamics, pairs of foragers often converge on both exploring or both staying, even when anti-coordination would be socially optimal. This social dilemma arises for a wide range of parameters, and it persists even in a simplified model without explicit spatial structure. A minimal, naturalistic signal, in which one forager informs the other that it has found the distant reward, can remove many of the dilemmas in which both foragers explore, and can produce a Nash equilibrium whose collective payoff exceeds that of every outcome available without signaling.