solver: replace cell DFS with skyline search

Represent partial placements as column heights and branch on the narrowest local valley. This removes the board-area cell state and makes first-solution search substantially smaller for feasible orders.

Expose direct-search and candidate-order benchmark controls so the skyline core can be measured independently of odd-order construction. Document the completeness argument and the 10/11 test-tier decision.

Tests: Release, Debug, ASan, and UBSan CTest (10 passed each)

Refs: #4

benchmarks/run.py

@@ -74,16 +74,16 @@ def environment(binary):
             "logical_cpus": os.cpu_count(),
         },
         "workers": 1,
-        "search_policy": "single-threaded, deterministic, largest-fitting-first",
+        "search_policy": "single-threaded, deterministic, smallest-width valley",
         "seed": None,
     }
 
 
-def run_once(binary, order, mode, timeout):
+def run_once(binary, order, mode, candidate_order, search_route, timeout):
     started = time.monotonic()
     try:
         process = subprocess.run(
-            [str(binary), str(order), mode],
+            [str(binary), str(order), mode, candidate_order, search_route],
             check=False,
             capture_output=True,
             text=True,
@@ -216,6 +216,12 @@ def main():
     parser.add_argument("--warmup", type=int, default=1)
     parser.add_argument("--repetitions", type=int, default=5)
     parser.add_argument("--timeout", type=float, default=600.0)
+    parser.add_argument(
+        "--candidate-order",
+        choices=("ascending", "descending"),
+        default="ascending",
+    )
+    parser.add_argument("--direct-search", action="store_true")
     parser.add_argument(
         "--measure-overhead",
         action="store_true",
@@ -241,11 +247,25 @@ def main():
         mode_results = {mode: {"runs": []} for mode in modes}
         for trial in range(args.warmup):
             for mode in trial_modes(modes, trial):
-                run_once(args.binary, order, mode, args.timeout)
+                run_once(
+                    args.binary,
+                    order,
+                    mode,
+                    args.candidate_order,
+                    "direct" if args.direct_search else "public",
+                    args.timeout,
+                )
         for trial in range(args.repetitions):
             for mode in trial_modes(modes, trial):
                 mode_results[mode]["runs"].append(
-                    run_once(args.binary, order, mode, args.timeout)
+                    run_once(
+                        args.binary,
+                        order,
+                        mode,
+                        args.candidate_order,
+                        "direct" if args.direct_search else "public",
+                        args.timeout,
+                    )
                 )
         for result in mode_results.values():
             result["summary"] = summarize(result["runs"])
@@ -270,6 +290,8 @@ def main():
             "warmup_runs": args.warmup,
             "measured_repetitions": args.repetitions,
             "per_run_timeout_seconds": args.timeout,
+            "candidate_order": args.candidate_order,
+            "search_route": "direct" if args.direct_search else "public",
             "stdout": "captured; rendered grid suppressed by probe",
         },
         "cases": cases,