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#Post#: 26--------------------------------------------------
hard randomized small HCP benchmarks
By: gsgs Date: November 12, 2023, 4:24 am
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hard randomized small HCP benchmarks
So, what is the smallest graph of unknown Hamiltonicity ?
--------------------------------------------------------
I uploaded hard small randomized Hamiltonian cycle (HC)
benchmarks to
HTML http://magictour.free.fr/x9.7z
1MB compressed , 11*100 graphs, k=20,30,..,120 ; n=180..1080 in
G&G-format.
Isn't this, what Vandegriend/Culberson,Gent/Connelly,Haythorpe,
Seegers/van Den Berg etc. had been looking for invain ?
They consist of k "xor9"s randomly,balanced connected at their
4 outer vertices.
A xor9 is a P9 with additional edges (1,6) and (4,9) ;
the outer vertices are 1,3,7,9.
Here a picture :
HTML http://magictour.free.fr/xor9.GIF
The edge-density is 2.5 outer edges per outer vertex,
which is close to the phase-transition.
This is almost independent of k in the range that I tested.
It's even harder for balanced graphs, as seen in similar
problems,
e.g. in 3SAT.
So here I also required all outer degrees to be 2 or 3 and
20 edges, 10 outer ones for each xor9 , n=k*9 vertices
and k*15 edges in total where k is the number of xor9s.
There is also at most one edge between different xor9s
and no outer edges go from one xor9 to itself.
These balanced xor9-HCPs near the phase transition typically
have few HCs, if any. 50% are Hamiltonian,30% are uniquely
Hamiltonian.
They typically have also few additional cycle partitions ,
in the exact-cover and SAT encodings corresponding to
cycles in the outer-edges graph of the xor9-graph.
so the naive conversion to exact-cover problems and
SAT-instances
works well and gives only few solutions which are not HCs,
i.e. which have more than one cycle.
I checked this with
"vacul" (Vandegriend,Culberson 1999) ,
"chalat" (Chalaturnyk,2009) ,
"reo100" (chalat with pre-vertex-reordering)
"dlx" (dancing links, Knuth,1996- , version dlx1.w,2016) ,
"kissat" (Biere et.al , versions 3.1 and 4.0)
"clasp" (Kaufmann, on 1 thread and on 4 threads)
"clingo" (Potassco , 2007-2023 , versions 5.2.2 and 5.6.2 from
2023)
"slh" (Flinders Uni,2013)
"minisat"
"cryptominisat" (Soos, -2025 versions 5.0,5.14.1,5.
"glucose"
The complexity is about 1.02^n = 1.2^k nodes with chalat and
1.016^n with kissat.
That's 9000s for k=100,n=900 for a complete search with dlx or
chalat
on a 1.5 GHz Laptop.
kissat is 10 times faster for k=100 (stops at first solution),
but slower for small k<50, presumably kissat has asymptotically
better complexity than chalat,DLX,vacul.
vacul is about 10 times slower than chalat and DLX.
The "triangle-bug" plays no big role here, since there are only
few small cycles.
The treewidth and max. bordersize are large, unbounded;
these are basically just random graphs with vertices replaced by
xor9s,
so border-reorder or treewidth algos won't help much.
In the xor9 we could even delete the middle vertex to make it
harder(n)
but chalat,vacul are slower then, they do not easily detect that
the
middle edge in a xor8 is forced for HCs, this changes when it is
replaced by a degree=2 vertex in a xor9. So I did not use xor8s
here.
Instead of xor9s we can also take any other "in-out" subgraph
(as defined in the change-ringing paper by Haythorpe) with
simiar results. The hard "Stedman" graphs from that paper
or from the FHCP-challenge set are multi 6-in-out graphs,
while xor9s are multi 2-in-out graphs.
The Stedman density is below the phase transition but they are
somehow regular,structured - so there are still solutions.
Randomized multi-6-in-out graphs near the phase-transition
would make even harder HCPs.
#Post#: 29--------------------------------------------------
Re: hard randomized small HCP benchmarks
By: gsgs Date: November 14, 2023, 11:37 pm
---------------------------------------------------------
you can also view this as "railway-graphs",
as I called them.
Rails(paths) can do no acute angles.
Tracks split by track-switches(vertices) and we have terminals
(vertices of degree=2) where people and cargo is exchanged.
And several terminals are grouped into stations.
A train enters a station on a track at one end
trough the switch, proceeds to the terminal of that track
and leaves the station on the same track's other end's switch,
where the rail forks towards the other stations.
For simplicity and convenience I assume here that the only
switches are at the station exits/entries , one on each end
of a track. Also my switches allows forking into multiple
directions,
while in practice this would probably by realised by several
consecutive simple bi-switches.
So, we have a normal,undirected,simple graph G=(V,E)
with an additional 1-factor (perfect matching) of "tracks".
(omitting the terminals)
A tour is a cycle where every second edge
is from the 1-factor.
The tracks can be partitioned into stations and a station-tour
is a tour which visits every station exactly once.
If every station has 1 track, then a station-tour is
a perfect 1-factorization.
If every station has 2 tracks , then
these station-tours are in 1-1 correspondence with
Hamiltonian cycles in the corresponding xor9-graph ,
The hard Stedman problems from the change-ringing paper and
the 1001-FHCP-challenge correspond then to station-tours
in a railway graph, where each station has 6 tracks.
Other than cycles in most normal graphs these tours have
typically
very few subcycles (chords).
At least in my hard instances with edge-density near the
Hamiltonian phase transition.
Instead of 1-factor you can use triangles (2-factor)
with similar results. (xor13-graph)
#Post#: 40--------------------------------------------------
Re: hard randomized small HCP benchmarks
By: gsgs Date: September 8, 2025, 8:50 am
---------------------------------------------------------
these 1100 SAT-instances
HTML http://magictour.free.fr/x9-1100.7z
were submitted to the SAT competition 2024
and 40 of them were included in their 400 competition
benchmarks.
---------------------------------------
The x9-s* instances from the SAT-2024-competition are instances
with 5s variables and 4s EO-clauses (exactly-one-clauses) which
then are encoded as cnf-instances with the "pairwise" method.
The EO-clauses are ordered and partitioned into s "stations"
with 4 EO-clauses each - called "gates" N,E,S,W.
Each variable is in 4 EO-clauses : 2 gates from 2 station .
2 EO-clauses in the same station with a common variable have
gates {N,E} or {E,S} or {S,W} or {W,N} but not {N,S} or {E,W}.
It is also required that each EO-clause has 4,5,or 6
EO-literals.
Besides that the procedure is fully random.
-----------------
The motivation was to find hard Hamiltonian-Cycle (HCP)
benchmarks
from which these x9-instances are derived and why the name is
"x9".
There are other, similar constructions , but this one gave the
hardest(n=8s) HCPs, requiring approximately 1.17^s/100000
seconds.
The phase transition of variables/clauses is at (approximately
?) 1.25
and seems to be contant(s).
The SAT-instances do not have the difficult "reachability"
constraint, so they do not exactly encode HCPs, but it turns out
that a good proportion of SAT-solutions correspond to HSs.
This is usually not the case for normal HCPs.
So we can encode these instances as HCPs or ECPs or SATs
and compare the solvers for these problems.
For the larger instances, as those selected for SAT-2024,
it turned out that kissat4.0 and clasp4 were the best
while for the smaller ones chalat,reo100,DLX were also equally
good.
vacul,clingo,SLH were worse , also
minisat,cryptominisat,glucose,
kissat 3.1, clasp1
The time needed is mostly random and can vary on different
random-seeds
almost much (well, maybe 70% as wild guess) as different
instances
with the same s.
Here is a C-program to convert the x9-SAT-instances into graphs
for Hamiltonian-Cycle-Solvers:
HTML http://magictour.free.fr/sat2cha.c
---------------------------------------------
Given a simple undirected graph G=(V,E) and a subset W of VxVxV,
let an W-cycle be a cycle where (u,v,w) is in W for each pair of
adjacent edges (u,v),(v,w) in the cycle .
--------------------------------------
The x9-instances above can also be encoded as
"W-disjoint-cycle-cover problems"
generalizing W-Hamiltonian-cycle problems , which gives longer,
harder
encodings. However, in the general case a nice encoding
as for the x9s is not possible, so it makes sense to study these
encodings too. I called them the corresponding y9-encodings and
uploaded them to
HTML http://magictour.free.fr/y9-1100.7z
Below are the solution counts and timings with kissat 4.0 :
--------------------------
average time in seconds needed by kissat 4.0
100 instances x9 and y9 submitted / 1000 instances x9t and r9t
st,x9:e(s/e) ,y9:e(s/e) , SAT x9:e(s/e) , y9:e(s/e) , SAT
,corr(X9,y9)
----------------------------------------------------------------
---x
20,.006(.20) , .028(.16) , 80 ,.006(.17) , .028( ) , 825 , 07
30,.010(.17) , .070(.42) , 84 ,.010(.15) , .068( ) , 813 , 38
40,.019(.36) , .138(.32) , 83 ,.018(.35) , .130( ) , 836 , 57
50,.053(.66) , .276(.34) , 88 ,.058(.68) , .249(.36) , 841 , 36
60,.184(.57) , .521(.40) , 93 ,.202(.71) , .532(.49) , 842 , 40
70,.713(.82) , 1.32(.62) , 89 ,.723(.84) , 1.33( ) , 875 , 57
80,2.45(.86) , 4.70(.73) , 90 ,2.58(.82) , 4.16(.79) , 886 , 52
90,6.57(.84) , 12.5(.85) , 91 ,8.00(.76) , 12.9 , 897 , 53
100,24.3(.97) , 39.9(.86) , 91 ,24.8 , 40.2 , 904 ,
72
110,99.6(1.11), 144(.94) , 91 ,96.0 , 152 , 912 , 61
120,326(1.45) , 476(1.23) , 93 ,331 , , 921 , 45
st:stations = number of x9-subgraphs in the corresponding graph
e:expectation value , mean time per instance
s:standard deviation
todo : split by sat and unsat !
file with all individual timings and solution counts :
HTML http://magictour.free.fr/x9y9.csv
#Post#: 41--------------------------------------------------
Re: hard randomized small HCP benchmarks
By: gsgs Date: December 30, 2025, 12:39 am
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here is a file with empirical timings for x9 instances
HTML http://magictour.free.fr/blaetter.txt
and here a file with solution-counts :
HTML http://magictour.free.fr/solz.txt
and here a file with empirical tests (mainly kissat 4.0)
in random regular balanced wedge-graphs
HTML http://magictour.free.fr/wed.txt
#Post#: 42--------------------------------------------------
Re: hard randomized small HCP benchmarks
By: gsgs Date: December 31, 2025, 2:49 am
---------------------------------------------------------
here is what chatgpt 5.1 wrote on 2025-12-30
after some chat about the topic and reading
the previous posts from this thread :
================================================================
=======
Summary: Hard Randomized Hamiltonian-Cycle Benchmarks via WHC
and x9 Gadgets
Motivation
Purely random or balanced random graphs are no longer good
benchmarks
for the Hamiltonian Cycle Problem (HCP): modern solvers handle
them easily.
However, balanced Wedge-Hamiltonian Cycle (WHC) instances still
exhibit a
sharp phase transition and remain hard near the 50%
satisfiability point.
This hardness can be compiled back into ordinary HCP instances
using
small gadgets, yielding new families of hard, randomized
benchmarks.
Core construction: x9 stations
The basic building block is an x9 gadget (station):
Start with a path P9
Add chords (1,6) and (4,9)
{1,3,7,9}.
A graph is built from many such stations, randomly and
balancedly
connected through their ports. Each station has exactly 10
incident
external edges (10-regular at the station level), with outer
degrees
constrained to 2 or 3, keeping the construction close to the
Hamiltonian
phase transition.
Intuitively, each station behaves like a constrained in-out
component: a Hamiltonian cycle can only traverse it via certain
compatible port pairs. Incompatible arrivals/departures are
forbidden.
Exact-cover (x9) vs WHC (y9) encodings
There are two main encodings of the same underlying constraint
system:
x9 instances (Exact Cover / EO-SAT)
Each station offers a small number of valid traversal patterns.
One traversal is chosen per station.
Global consistency is enforced by exact-cover constraints on
inter-station connections.
This encoding is compact and fast.
These were the primary instances submitted to SAT Competition
2024.
y9 instances (WHC encoding)
The same constraints are expressed as a Wedge-Hamiltonian-Cycle
problem.
Larger encodings with more variables/constraints.
Empirically slower (x2 for Kissat, often worse for others).
Useful to study WHC directly, but less efficient as SAT
benchmarks.
Relation to WHC and hardness transfer
The x9 construction can be viewed as a reduction from WHC to HC:
Balanced random WHC instances are hard near their phase
transition.
x9 gadgets preserve this hardness when translated into ordinary
HCP instances.
Unlike naive HCP encodings, these instances have few spurious
cycle
covers:a good fraction of SAT solutions correspond to genuine
Hamiltonian cycles.
This explains why the benchmarks remain hard despite lacking
explicit
reachability constraints.
Phase transition and empirical hardness
Empirical testing (hundreds to thousands of instances, many
solvers)
shows:
A clear phase transition near constant density, largely
independent of size.
Around the transition:
50% of instances are Hamiltonian,
30% are uniquely Hamiltonian,
very few alternative cycle partitions exist.
Median runtime peaks close to this transition.
Observed time scaling for HCP solving is approximately:
T(n)=1.016^n/100000 to1.02^n/100000,
depending on solver and encoding, where
n is the number of vertices.
This scaling holds robustly across random seeds, with high
variance
typical of hard instances.
Benchmark relevance
1100 x9 SAT instances were generated; 40 were selected for the
SAT
Competition 2024 main benchmark set.
Larger instances favor modern CDCL solvers (e.g. Kissat 4.0,
Clasp 4),
while smaller ones are also hard for classical HCP solvers (DLX,
Chalat, Vacul).
The construction generalizes: other in-out gadgets (e.g.
Stedman-type, multi-track stations) produce similar effects, but
x9 gave the hardest
small instances tested.
Big picture
Balanced random HC is easy.
Balanced random WHC is hard near its phase transition.
x9 gadgets allow this WHC hardness to be transferred back to HC,
producing small, randomized, phase-transition-controlled
benchmarks.
These benchmarks are solver-agnostic, scalable, and empirically
validated.
================================================================
==
#Post#: 44--------------------------------------------------
Re: hard randomized small HCP benchmarks
By: gsgs Date: January 1, 2026, 5:27 am
---------------------------------------------------------
and here is what Grok 4.1 wrote on 2025-12-31
here's a concise, updated summary of the wedge-graph / x9
framework
that incorporates everything we've discussed
(gadget history, randomness insight, phase transition behavior,
empirical hardness, and the key WCC vs. WHC gap).
Feel free to post it directly below ChatGPT's original summary.
Summary: Random Balanced In-Out Gadget Graphs State-of-the-Art
Hard Random Hamiltonian Cycle Benchmarks (2023-2025)
Wedge graphs (V,W) with W<=VxVxV define allowed length-2
transitions (turns).
The associated normal graph has an edge {u,v} iff some wedge
uses it.
A wedge-Hamiltonian cycle (WHC) is a Hamiltonian cycle
respecting all
turn constraints; a wedge-cycle cover (WCC) is a vertex-disjoint
union
of such cycles.
Classic in-out gadgets on paths with chords (Papadimitriou
"diamonds",
Haythorpe S1/X3, S2/x9, ) were originally used in structured
NP-completeness reductions and crafted hard instances
(FHCP Challenge change-ringing graphs). They enforce a small set
of
compatible in–out traversals via ports.
The breakthrough (independent experiments 2023, SAT Comp 2024
benchmarks)
was to connect many copies of the same gadget randomly but
balanced on
ports (regular or near-regular station graph). This yields a
controllable
random model with a sharp phase transition for both WCC and WHC
in the
full graph.
Key properties of the x9 family (P9 + chords 1-6,4-9, ports
{1,3,7,9},
10-regular station graph):
Exact-cover encoding on stations naturally solves WCC.
Near critical density (~constant ~2.01 wedges/possible edges in
associated
graph), ~50% of instances are Hamiltonian in the ordinary
undirected graph
(i.e., admit a WHC).
Hamiltonian instances almost always have a unique Hamiltonian
cycle.
~15-20% are unsat even for WCC; the remaining ~30-35% have WCCs
but no
single cycle (merging fails) the costly gap for solvers.
Empirical hardness: CDCL runtimes peak sharply, scaling
exponentially
~1.016^n/100000 - 1.02^n/100000 .
40 instances selected for SAT Competition 2024 main track; still
among the
hardest systematic HC benchmarks.
Broader balanced regular wedge graphs (r=3 to r=12 in associated
graph)
show similar transitions, with WHC fraction stabilizing at
10-18% near
threshold and rising to >95% supercritical. The small but
critical gap
between WCC and WHC existence - opposite to most crafted hard
instances
where covers are easy but merging impossible - is what produces
the
extreme average-case hardness.
These random in-out gadget graphs are currently the strongest
known
generatable family of hard plain Hamiltonian Cycle instances,
combining unique solutions, sharp thresholds, and resistance to
modern CDCL solvers.
(If you post this, feel free to attribute it to Grok summary,
based on discussions with the researcher or similar.)
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