Ambiguity-bounded versions of $\mathrm{NP}$, denoted $\mathrm{UP}_{\leq f(n)}$, bound by $f(n)$ the number of accepting paths the nondeterministic polynomial-time Turing machine can have on inputs of length $n$. Such classes range from Valiant's completely unambiguous ($f(n)=1$) class $\mathrm{UP}$ to $\mathrm{NP}$ itself, where there is no bound or, equivalently, there is the toothless exponential bound ($f(n) = 2^{n^{O(1)}}$). This paper seeks to understand which of these classes stand and fall together as to whether they equal deterministic polynomial time. Informally put, what ranges of ambiguities have linked fates? That is, for which pairs of nondecreasing functions, $(f_1 ,f_2)$, satisfying $(\forall n)[f_1(n) \leq f_2(n)]$, does it hold that $\mathrm{P} = \mathrm{UP}_{\leq f_1(n)} \implies \mathrm{P} = \mathrm{UP}_{\leq f_2(n)}$. More particularly, for which pairs does that hold robustly, i.e., it holds in the real world and every relativized world? And for which pairs does that implication fail to hold robustly, i.e., there is an oracle $A$ such that $\mathrm{P}^A = \mathrm{UP}_{\leq f_1(n)}^A \subsetneq \mathrm{UP}_{\leq f_2(n)}^A$? The only previously known positive result is Watanabe's 1988 result that $ \mathrm{P} = \mathrm{UP}_{\leq 1} \implies (\forall k \geq 1)[\mathrm{P} = \mathrm{UP}_{\leq k}]$, which even holds robustly. His result, though lovely, applies only to constant-bounded ambiguities. As our positive result, we present a new class of cases (Theorem 3.8) that apply (and even robustly apply) at greater ambiguity levels. To give our class of cases, we leverage two approaches: a novel path-poisoning approach that works even on superconstant ambiguities (Theorem 3.5) and a new application of the power of padding (Theorems 3.3/3.4). As negative results, we show that for essentially all other cases, no linkage holds robustly.

A formal explanation certifies a prediction with a subset-minimal sufficient reason. Under incremental disclosure, however, evidence arrives field by field, and a normally sufficient reason can be overturned by later information. We study the smallest reason core that remains sufficient under all admissible later disclosures; its size is the robustness radius. We compile a defeasible classifier into an explicit boundary atlas of entry anchors and exit defeaters, and chart the complexity of auditing it (all statements are in the atlas size). Prediction and standing anchors are read by polynomial-time scans of the atlas, without iterative fixpoint computation; a reason's defeater frontier is obtained by scanning and subset-minimizing the defeaters above it. But verifying that a reason core is robust is coNP-complete, and deciding whether a robust core of size at most theta exists is $Σ_2^p$-complete -- a four-cell P / coNP-complete / NP-complete / $Σ_2^p$-complete landscape, with the accepted (A(t)=1) case reaching the second level of the polynomial hierarchy. The decision version of minimal certified disclosure is NP-complete; its optimization version is fixed-parameter tractable in the number of excluded worlds without defeaters, with the general-defeater case open. On exact audits of depth-limited decision trees over standard tabular datasets under a deliberately small Boolean abstraction, the governing parameters fall in a small-parameter regime (robust cores in the low single digits), so exact robust auditing is tractable in these audited cubes; on adversarial instances built from our reductions the hardness bites, with robust cores of size Theta(n). To our knowledge this is the first $Σ_2^p$-complete audit query for disclosure-robust formal explanations.

Twin-width is a graph parameter that has become central to explaining the fixed-parameter tractability of first-order model checking across many graph classes. Despite its algorithmic importance, computing twin-width remains poorly understood: even recognizing graphs of twin-width at most four is NP-hard, and no fixed-parameter approximations parameterized by twin-width itself are known. A recent approach towards breaking this barrier focuses on first developing fixed-parameter algorithms for computing or approximating twin-width under parameterizations distinct from twin-width. Our first result establishes that approximating twin-width is fixed-parameter tractable when parameterized by treedepth, thereby breaking the long-standing barrier that all previous tractable parameterizations were based on deletion distance. The proof proceeds via oriented twin-width, yielding the first constructive evidence that this variant may be easier to handle algorithmically. As our second main result, we show that computing twin-width exactly is fixed-parameter tractable with respect to vertex integrity. This constitutes the first non-trivial parameterized algorithm for computing optimal contraction sequences.

We study the problem of computing a competitive equilibrium with approximately optimal bundles in Fisher markets with separable piecewise-linear concave (SPLC) utility functions, meaning that every buyer receives a $(1-δ)$-optimal bundle, instead of a perfectly optimal one. We establish the first intractability result for the problem by showing that it is PPAD-hard for some constant $δ> 0$, assuming the PCP-for-PPAD conjecture. This hardness result holds even if all buyers have identical budgets (competitive equilibrium with equal incomes), linear capped utilities, and even if we also allow $\varepsilon$-approximate clearing instead of perfect clearing, for any constant $\varepsilon < 1/9$. Importantly, we show that the PCP-for-PPAD conjecture is in fact required to show hardness for constant $δ$: showing PPAD-hardness for finding such approximate market equilibria in a broad class of markets encompassing those generated by our hardness result would prove the conjecture. This is the first natural problem where the conjecture is provably required to establish hardness for it.

Andréka and Maddux classified the relation algebras with at most 3 atoms, and in particular they showed that all of them are representable. Hirsch and Cristiani showed that the network satisfaction problem (NSP) for each of these algebras is in P or NP-hard. The literature contains many results on representations of relation algebras; in particular, some relation algebras with four atoms are not representable. We extend the result of Cristiani and Hirsch to relation algebras with at most 4 atoms: the NSP is always either in P or NP-hard. To this end, we construct universal, fully universal, or even normal representations for these algebras, whenever possible.