doc(roadmap): expand algorithms and complexity coverage

ROADMAP.md

@@ -148,6 +148,19 @@ In order:
 - VOLE as an ideal functionality, and VOLE-in-the-head;
 - libzk's compositional structure as a composition of these components, without ciphersuites.
 
+### Concurrent and quantum security
+
+- **Concurrent composition:** extend the protocol semantics with environments, scheduling,
+  session identifiers and concurrent executions; prove a composition theorem with efficient
+  simulators and explicit setup, corruption and communication assumptions. Then develop
+  selected adaptive-corruption and non-malleability applications, alongside the static-malicious
+  OT target.
+- **Quantum-secure cryptography:** uniform quantum-efficient distinguishers, computational
+  indistinguishability with quantum side information, and security reductions for concrete
+  constructions. Connect the quantum-computation and lattice tracks through these adversary
+  models; justify each reduction in that model. Later targets include quantum oracle access,
+  the quantum random-oracle model, and extraction or rewinding under their precise hypotheses.
+
 ## Complexity classes and reductions
 
 ### Coverage catalogue
@@ -213,6 +226,17 @@ index links to canonical definitions rather than keeping a second set.
 - **Immunity:** immune and bi-immune sets, P-bi-immune sets in EXP (Balcázar and Schöning), and
   almost-everywhere hardness.
 
+### Average-case complexity
+
+- Distribution ensembles with computability and samplability certificates; distributional
+  reductions with explicit domination and resource bounds; DistNP, errorless average-case
+  classes and heuristic classes, keeping runtime tails and permitted error distinct. Reuse the
+  specialized Liu–Pass hardness notions through proved bridges.
+- **Levin completeness:** a distributional bounded-halting problem complete for the chosen
+  polynomial-time computable-distribution class, followed by a separately scoped extension to
+  samplable distributions. Connect the framework to one-way functions, learning and the
+  metacomplexity track ([#798]), preserving the distribution and reduction conventions.
+
 ### Counting and modular classes
 
 - The Another Solution Problem (ASP) framework: certify polynomial-time forward and inverse
@@ -271,6 +295,14 @@ universe size, not supplied as unverified input tables. In dependency order:
   structures and algorithms.
 - A witness-carrying reduction structure once a client needs one, for example size bounds for
   Cook–Levin.
+- **Logical lower-bound methods:** Ehrenfeucht–Fraïssé and pebble games with their logical
+  equivalence theorems, locality and concrete nondefinability results; then counting and
+  fixed-point expressiveness, including Cai–Fürer–Immerman-style separations. Reuse upstream
+  model-theoretic foundations and connect the methods to the ordered capture theorems.
+- **Dynamic descriptive complexity:** dynamic programs and update semantics with explicit
+  initialization, auxiliary relations and domain conventions; invariant-preservation rules and
+  reachability in DynFO as a target. Relate logical update restrictions to the stateful-algorithm
+  interfaces through explicit translations.
 
 ### Uniform circuit families
 
@@ -283,7 +315,8 @@ DLOGTIME-uniform AC⁰ under explicit conventions.
 
 - **TFNP:** CLS, further local search, fixed-point and equilibrium problems.
 - **Parameterized:** strongly uniform variants (computable `f`), further explicit polynomial
-  kernels, the W and A hierarchies, and parameterized lower bounds.
+  kernels, the W and A hierarchies, and parameterized lower bounds; connect these to
+  [structural graph algorithms](#structural-graph-algorithms) and their decomposition costs.
 - **Exponential-time SAT** ([#894]): amplify Schöning's random walk, with a time bound, then
   PPSZ. Further fine-grained reductions and consequences of ETH, SETH and other named hypotheses.
 
@@ -305,8 +338,15 @@ DLOGTIME-uniform AC⁰ under explicit conventions.
 
 ### Proof complexity
 
-Proof-space lower bounds, Frege, cutting planes, polynomial calculus and sum-of-squares, and
-certified DPLL solver runtimes.
+- Proof-space lower bounds, Frege, cutting planes, polynomial calculus and sum-of-squares, and
+  certified DPLL solver runtimes.
+- **Feasible interpolation:** from a Resolution refutation of `A(x,y) ∧ B(x,z)`, construct a
+  circuit separating the `x` assignments extendible to `A` from those extendible to `B`, with
+  proof-size and extraction-time bounds. Prove the monotone variant under its polarity
+  hypotheses and transfer circuit and communication lower bounds to proof-size lower bounds.
+- **Proof search:** automatizability and weak automatizability with explicit encodings and
+  runtime dependence on the shortest proof; proof-search reductions built on the Cook–Reckhow
+  and p-simulation APIs; then reflection principles and bounded-arithmetic witnessing theorems.
 
 ## Circuits and algebraic complexity
 
@@ -377,6 +417,10 @@ each statement against the numbered result. Organize modules by subject, not cha
   list-size/agreement guarantees and decoding runtime as separate obligations; concrete
   list-decodable codes for Nisan–Wigderson.
 - **Boolean analysis:** hypercontractivity beyond `p ≤ 2 ≤ q` and its applications.
+- **Resource-bounded measure and dimension:** martingales and gales with explicit computation
+  budgets, their success sets, and measure and dimension within complexity classes; connect
+  quantitative randomness to the Kolmogorov-complexity development. State the resource bounds
+  and uniformity of the betting strategies separately from ordinary measure-theoretic facts.
 
 ## Communication, queries, learning and other models
 
@@ -390,7 +434,12 @@ each statement against the numbered result. Organize modules by subject, not cha
   The first Count-Min milestone is scoped in [#1201](https://github.com/cslib-community/complexitylib/issues/1201).
 - **Learning:** machine-certified running times for the LMN and Kushilevitz–Mansour learners;
   distributional problems and reductions, samplability, average-to-worst-case connections,
-  PAC/agnostic/SQ learning, learning lower bounds, compression and refutation.
+  PAC/agnostic/SQ learning, learning lower bounds, compression and refutation. Give the
+  computational–statistical gap program concrete clients: an SQ oracle with tolerance and query
+  costs, statistical-dimension lower bounds for a specified planted problem, and corresponding
+  sample and runtime upper bounds. Develop low-degree likelihood-ratio and sum-of-squares
+  comparisons where justified, distinguishing restricted-model bounds from unrestricted
+  computational hardness. Use the [distributional framework](#average-case-complexity).
 - **Property and distribution testing:** access models, one/two-sided and tolerant testing,
   query and sample bounds, canonical testers and lower bounds.
 - **Kolmogorov complexity and metacomplexity** ([#798], [#884]): the near-linear threshold regime
@@ -409,13 +458,102 @@ each statement against the numbered result. Organize modules by subject, not cha
   circuit, channel and measurement semantics with uniformity and approximation costs, reusing
   suitable quantum foundations rather than incompatible models; BQP, QMA and related query,
   communication and interactive classes, with amplification, containments and complete
-  problems.
+  problems. Supply the [quantum adversary models](#concurrent-and-quantum-security) needed by
+  cryptographic reductions.
+
+### Differential privacy and adaptive data analysis
+
+- Dataset adjacency and `(ε, δ)`-differential privacy for probabilistic programs, with
+  postprocessing and adaptive composition. Start with a finite or discrete mechanism proving
+  both privacy and utility, including sampling cost and finite-precision error.
+- The exponential mechanism, private query answering and private learning; privacy–accuracy
+  and sample-complexity lower bounds; then adaptive data-analysis and generalization guarantees.
+  Keep the adjacency relation, interaction model, privacy parameters and computational resources
+  explicit, reusing the probability, oracle-computation and learning APIs.
 
 [#1459]: https://github.com/cslib-community/complexitylib/issues/1459
 [#1463]: https://github.com/cslib-community/complexitylib/issues/1463
 [#1481]: https://github.com/cslib-community/complexitylib/issues/1481
 [#1482]: https://github.com/cslib-community/complexitylib/issues/1482
 
+## Algorithms and optimization
+
+### Optimization and duality
+
+- **Linear programming:** rational input and output encodings, primal and dual certificates,
+  weak and strong duality, and infeasible or unbounded outcomes. Reuse Mathlib's convexity and
+  linear-algebra foundations; certify certificate checking and a polynomial-time LP algorithm
+  with bit complexity and intermediate precision accounted for.
+- **Separation and optimization:** a precise oracle interface and a separation–optimization
+  equivalence with its representation, accuracy and well-boundedness hypotheses. Extend to
+  convex optimization through explicit oracle and numerical-error contracts.
+- **Primal–dual approximation and multiplicative weights:** reusable analyses with vertex cover
+  and set cover as first primal–dual clients; multiplicative-weights regret and approximation
+  bounds with finite-precision implementations, then applications to games, learning and
+  optimization.
+
+### Combinatorial optimization
+
+Coordinate native graph-algorithm work with [CSLib's graph-algorithm directions](https://github.com/leanprover/cslib/blob/main/CONTRIBUTING.md#graph-algorithms-and-graph-foundations),
+reusing graph and matroid foundations and the shared data-structure interfaces.
+
+- **Flows and matching:** integral max-flow/min-cut and a native augmenting-path algorithm with
+  charged runtime; then minimum-cost flow and weighted bipartite matching through common
+  residual-network and duality machinery.
+- **Spanning trees and matroids:** minimum spanning trees with sorting and disjoint-set costs;
+  weighted matroid greedy optimality with an independence-oracle interface; then matroid
+  intersection with explicit oracle and implementation costs.
+- **Submodular optimization:** the `1 - 1/e` approximation for nonnegative monotone submodular
+  maximization under a cardinality constraint, charging value-oracle calls separately from their
+  concrete realizations; then further constraints and approximation bounds.
+
+### Online algorithms and competitive analysis
+
+- Request sequences, online strategies, offline optima and competitive ratios with additive
+  constants; distinguish deterministic and randomized strategies and oblivious and adaptive
+  adversaries. Compare algorithms in the same action and cost model, including initialization.
+- **Paging:** an LRU competitive bound and a matching deterministic lower bound as the first
+  clients. Reuse native amortized certificates and make adversarial request generation explicit.
+- **Adaptive search trees:** a BST access-and-rotation model, the splay access lemma, static
+  optimality, working-set and dynamic-finger bounds, and Wilber-style lower bounds. State dynamic
+  optimality as a conjectural research target with explicit strategy and sequence quantifiers.
+
+### Structural graph algorithms
+
+- Tree decompositions, width and normalization into a suitable decomposition form; a reusable
+  dynamic-programming evaluator over a supplied decomposition, with state-space, transition,
+  join and runtime bounds. Start with native independent-set, coloring and SAT/CSP decision and
+  counting clients, specifying the graph whose width is measured.
+- **Algorithmic metatheorems:** MSO-to-automata/evaluator compilation and a Courcelle-style
+  theorem with computable parameter dependence on width and formula size. Keep the supplied
+  decomposition and its encoding explicit; certify decomposition finding separately. Connect
+  the evaluator to descriptive complexity, parameterized reductions and constraint networks.
+
+### Sampling and approximate counting
+
+- **Markov-chain sampling:** finite kernels, stationarity, reversibility and total-variation
+  mixing; coupling, path coupling, conductance and canonical-path arguments. Implement
+  Metropolis sampling for encoded rational target weights, charging initialization, arithmetic
+  and every transition, with support and mixing hypotheses explicit.
+- **FPRAS and almost-uniform sampling:** interfaces separating relative error, failure
+  probability, output-distribution error and runtime, including zero-count instances. Start
+  with a complete DNF-counting FPRAS; prove sampling/counting reductions for suitably
+  self-reducible relations, with the sampling and counting oracle costs included.
+- Extend to randomized approximate counting for matchings and permanents of nonnegative
+  rational matrices through explicit mixing and reduction theorems. Reuse the coupling and
+  expander APIs and connect to the separate Stockmeyer oracle-counting track where their
+  hypotheses agree.
+
+### Constructive probabilistic methods
+
+- **Moser–Tardos resampling:** an actual resampling program, its correctness and expected
+  resampling bound, and native sampling and event-detection costs, building on the variable
+  form of the Local Lemma.
+- **Conditional expectations and pessimistic estimators:** certified choice and update rules,
+  algorithmic alteration, and concrete coloring, satisfiability or combinatorial-construction
+  clients. Preserve the quantitative guarantee when converting an existence argument into a
+  deterministic or randomized algorithm.
+
 ## Machines, RAM and data structures
 
 - **Native distinct-position frontend** ([#1482]): initialized signed input produces exact
@@ -593,6 +731,19 @@ each statement against the numbered result. Organize modules by subject, not cha
   and a `690n²Dg` charged bound in the source's admissible parameter regime. Prime selection,
   solving the prepared instances, product/backend composition and the full #1394 endpoint remain.
 
+### Parallel, external-memory and distributed models
+
+- **Parallel algorithms:** work–span semantics and a processor-scheduling or simulation theorem,
+  followed by prefix scan, sorting and graph primitives with both work and span bounds. State
+  processor access, synchronization and scheduling overhead explicitly.
+- **Memory hierarchies:** external-memory and ideal-cache models with explicit capacity and
+  block-size parameters; scanning, B-tree operations, blocked matrix multiplication and
+  cache-oblivious sorting as clients. Connect arithmetic-step, memory-access and cache-miss
+  bounds through concrete realizations, including replacement and initialization conventions.
+- **Distributed complexity:** LOCAL and CONGEST models with explicit identifiers, round,
+  message-size and randomness conventions; locality lower bounds and communication-to-round
+  reductions with corresponding algorithmic examples.
+
 ## Coverage and reuse
 
 The long-term bar is to support formalizing every paper in the