Quasi-Hamiltonian Path Problem Overview

• Quasi-Hamiltonian path problem is defined on semicomplete multipartite digraphs, ensuring each color-class (maximal independent set) is visited at least once.
• It extends the Hamiltonian path concept by enforcing coverage rules on vertex partitions rather than requiring every vertex to be included.
• Although every such digraph admits a quasi-Hamiltonian path, the endpoint-prescribed variant becomes NP-complete for independence number ≥3.

Searching arXiv for papers on the quasi-Hamiltonian path problem and closely related formulations. The quasi-Hamiltonian path problem concerns directed paths that cover every color-class of a semicomplete multipartite digraph at least once. For a semicomplete multipartite digraph $D=(V,A)$ with unique partition $C(D)=\{V_1,\dots,V_c\}$ into maximal independent sets and independence number $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$, a directed path $P$ is quasi-Hamiltonian if, for every color-class $V_i\in C(D)$, $1\le |V(P)\cap V_i|\le |V_i|$. When $\alpha(D)=1$, this notion coincides with the usual Hamiltonian-path notion. Every semicomplete multipartite digraph contains a quasi-Hamiltonian path, but deciding whether there exists such a path with prescribed start and end vertices is NP-complete even for semicomplete multipartite digraphs with independence number exactly $3$ (Brinkmann, 21 Jul 2025).

1. Formal setting and basic notions

In the semicomplete multipartite setting, the vertex set is partitioned into maximal independent sets, also called color-classes or parts. The partition is unique and is denoted

$C(D)=\{V_1,\dots,V_c\},$

with

$\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$

A quasi-Hamiltonian path is therefore not required to visit every vertex; it is required to meet every color-class at least once. Equivalently, for each $C(D)=\{V_1,\dots,V_c\}$0,

$C(D)=\{V_1,\dots,V_c\}$1

This distinction is central: in a multipartite digraph, quasi-Hamiltonicity is a coverage condition over parts rather than a vertex-exhaustion condition over $C(D)=\{V_1,\dots,V_c\}$2 (Brinkmann, 21 Jul 2025).

The principal decision version is the $C(D)=\{V_1,\dots,V_c\}$3-QHP problem. Given an SMD $C(D)=\{V_1,\dots,V_c\}$4 and specified vertices $C(D)=\{V_1,\dots,V_c\}$5, the question is whether there exists an $C(D)=\{V_1,\dots,V_c\}$6-path $C(D)=\{V_1,\dots,V_c\}$7 such that for every color-class $C(D)=\{V_1,\dots,V_c\}$8,

$C(D)=\{V_1,\dots,V_c\}$9

Because every semicomplete multipartite digraph contains a quasi-Hamiltonian path, the endpoint-constrained formulation is the nontrivial algorithmic variant. A common misconception is to treat QHP as merely a relaxed Hamiltonian-path problem; the definition shows that the object being covered is the set of color-classes, not necessarily the whole vertex set.

2. The $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$0-color-constrained framework

A broader formulation introduced for semicomplete multipartite digraphs is the $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$1-color-constrained path problem, abbreviated $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$2-CCP. An instance consists of an SMD $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$3 in which all color-classes have the same size $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$4, two vertices $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$5, and the question whether there exists an $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$6-path $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$7 satisfying

$\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$8

for every color-class $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}$9 (Brinkmann, 21 Jul 2025).

This framework subsumes several standard and nonstandard path problems. Choosing $P$0 recovers the Hamiltonian-path problem in tournaments. Choosing $P$1 yields the quasi-Hamiltonian path problem for general SMDs. Choosing $P$2 enforces that each part loses at least one vertex in the path. The framework also captures the path-version of cycle problems discussed in the same line of work. A plausible implication is that the main combinatorial source of difficulty is not multipartiteness alone, but the interaction between path routing and lower/upper coverage bounds imposed on each color-class.

3. Complexity landscape

The main dichotomy for $P$3-CCP is nearly complete. If either $P$4 or $P$5, then $P$6-CCP is solvable in polynomial time. Otherwise, if $P$7 and $P$8, then $P$9-CCP is NP-complete. For every fixed $V_i\in C(D)$0, all pairs $V_i\in C(D)$1 with $V_i\in C(D)$2, except the trivial reachability case $V_i\in C(D)$3 and the pure Hamiltonian case $V_i\in C(D)$4, yield NP-complete $V_i\in C(D)$5-CCP (Brinkmann, 21 Jul 2025).

Setting	Complexity status	Remarks
$V_i\in C(D)$6-CCP	Polynomial time	Reachability
$V_i\in C(D)$7	Polynomial time	Tournaments; $V_i\in C(D)$8 solved in $V_i\in C(D)$9
Fixed $1\le |V(P)\cap V_i|\le |V_i|$0, $1\le |V(P)\cap V_i|\le |V_i|$1	NP-complete	Nearly complete dichotomy

Specializing to quasi-Hamiltonian paths gives a standalone hardness result: deciding whether an SMD $1\le |V(P)\cap V_i|\le |V_i|$2 with $1\le |V(P)\cap V_i|\le |V_i|$3 admits an $1\le |V(P)\cap V_i|\le |V_i|$4-quasi-Hamiltonian path is NP-complete. This clarifies an otherwise potentially misleading point: the unrestricted existence of a quasi-Hamiltonian path in an SMD is guaranteed, but the endpoint-prescribed version remains computationally difficult already at independence number $1\le |V(P)\cap V_i|\le |V_i|$5. Notable open problems are the Hamiltonian path problem on semicomplete multipartite digraphs and the quasi-Hamiltonian path problem restricted to semicomplete multipartite digraphs with independence number $1\le |V(P)\cap V_i|\le |V_i|$6.

4. Reduction mechanisms for NP-completeness

For $1\le |V(P)\cap V_i|\le |V_i|$7, hardness is obtained by reductions from $1\le |V(P)\cap V_i|\le |V_i|$8-in-$1\le |V(P)\cap V_i|\le |V_i|$9-SAT or more general $\alpha(D)=1$0-$\alpha(D)=1$1-SAT variants. The core construction uses variable gadgets that force a binary routing choice and clause color-classes whose bounded-coverage constraints enforce the corresponding satisfiability condition (Brinkmann, 21 Jul 2025).

When $\alpha(D)=1$2, the reduction uses a $\alpha(D)=1$3-gadget $\alpha(D)=1$4. This gadget has exactly two internally vertex-disjoint paths from $\alpha(D)=1$5 to $\alpha(D)=1$6, one visiting a $\alpha(D)=1$7-chain of length $\alpha(D)=1$8 and one visiting a $\alpha(D)=1$9-chain of length $3$0, arranged so that no $3$1-path can mix $3$2- and $3$3-vertices. By coloring these vertex-sets together with an extra dummy vertex, the construction keeps color-classes of size at most $3$4. For the case $3$5, the reduction uses the classical $3$6-gadget of Bang-Jensen, Maddaloni and Simonsen, which again has exactly two natural $3$7-subpaths, through the $3$8-clique or the $3$9-clique.

Clause enforcement is handled by turning each clause $C(D)=\{V_1,\dots,V_c\},$0 into a color-class whose three vertices correspond to three literal appearances in appropriate gadget positions. The covering bounds then force that each clause-class contributes between $C(D)=\{V_1,\dots,V_c\},$1 and $C(D)=\{V_1,\dots,V_c\},$2 hits. Chaining is achieved by identifying $C(D)=\{V_1,\dots,V_c\},$3 with $C(D)=\{V_1,\dots,V_c\},$4, or by adding an explicit arc, so that any global $C(D)=\{V_1,\dots,V_c\},$5-path must traverse the variable gadgets in sequence. The analysis shows that any valid $C(D)=\{V_1,\dots,V_c\},$6-path induces a truth assignment by selecting exactly one route through each gadget, and conversely any assignment with the desired clause-counts yields a path.

The extension from $C(D)=\{V_1,\dots,V_c\},$7 to $C(D)=\{V_1,\dots,V_c\},$8 is obtained by padding with new dummy vertices and a small inductive gadget that increases every color-class by one while incrementing both $C(D)=\{V_1,\dots,V_c\},$9 and $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$0 by $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$1. This gives the lifting step required for the full dichotomy.

5. Structural results for independence number at most $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$2

For $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$3, the known NP-hardness reductions do not apply; the constructions inherently need parts of size $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$4. In this regime, the literature develops sufficient conditions for the existence of quasi-Hamiltonian $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$5-paths in $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$6-strong SMDs, closely paralleling classical tournament arguments (Brinkmann, 21 Jul 2025).

One theorem states that if $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$7 is a $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$8-strong SMD, $\alpha(D)=\max\{|V_1|,\dots,|V_c|\}.$9 are distinct vertices, neither $C(D)=\{V_1,\dots,V_c\}$00 nor $C(D)=\{V_1,\dots,V_c\}$01 lies in any $C(D)=\{V_1,\dots,V_c\}$02-cycle, both $C(D)=\{V_1,\dots,V_c\}$03 and $C(D)=\{V_1,\dots,V_c\}$04 remain $C(D)=\{V_1,\dots,V_c\}$05-strong, $C(D)=\{V_1,\dots,V_c\}$06 is not $C(D)=\{V_1,\dots,V_c\}$07-strong, and $C(D)=\{V_1,\dots,V_c\}$08, then $C(D)=\{V_1,\dots,V_c\}$09 contains an $C(D)=\{V_1,\dots,V_c\}$10-quasi-Hamiltonian path. Another theorem states that if $C(D)=\{V_1,\dots,V_c\}$11 is a $C(D)=\{V_1,\dots,V_c\}$12-strong SMD with $C(D)=\{V_1,\dots,V_c\}$13 and $C(D)=\{V_1,\dots,V_c\}$14, $C(D)=\{V_1,\dots,V_c\}$15 are distinct vertices, neither lies in a $C(D)=\{V_1,\dots,V_c\}$16-cycle, $C(D)=\{V_1,\dots,V_c\}$17, $C(D)=\{V_1,\dots,V_c\}$18 and $C(D)=\{V_1,\dots,V_c\}$19 remain $C(D)=\{V_1,\dots,V_c\}$20-strong, and every $C(D)=\{V_1,\dots,V_c\}$21-separator of $C(D)=\{V_1,\dots,V_c\}$22 and $C(D)=\{V_1,\dots,V_c\}$23 is trivial, then $C(D)=\{V_1,\dots,V_c\}$24 has an $C(D)=\{V_1,\dots,V_c\}$25-quasi-Hamiltonian path.

The associated proof sketches use longest quasi-Hamiltonian paths, linear-decomposition, arc-shortcut arguments, merging lemmas, and the “$C(D)=\{V_1,\dots,V_c\}$26-internally-disjoint-paths $C(D)=\{V_1,\dots,V_c\}$27 QHP” condition. Although no full polynomial-time algorithm is given for QHP when $C(D)=\{V_1,\dots,V_c\}$28, the structural theorems mirror the key steps in the $C(D)=\{V_1,\dots,V_c\}$29 algorithm of Bang-Jensen–Manoussakis–Thomassen for tournaments. The stated algorithmic consequence is therefore suggestive rather than definitive: QHP in SMDs with $C(D)=\{V_1,\dots,V_c\}$30 may be decidable in polynomial time by recursively reducing to smaller digraphs, checking $C(D)=\{V_1,\dots,V_c\}$31-strong connectivity, enumerating $C(D)=\{V_1,\dots,V_c\}$32-separators, and applying merging and linear-decomposition in $C(D)=\{V_1,\dots,V_c\}$33 or $C(D)=\{V_1,\dots,V_c\}$34 time.

6. Relations to Hamiltonian path and adjacent formulations

For independence number at most $C(D)=\{V_1,\dots,V_c\}$35, Hamiltonian-path structure is substantially sharper than the current quasi-Hamiltonian picture. A biorientation of the complete graph minus a disjoint union of paths of length at most $C(D)=\{V_1,\dots,V_c\}$36 is unilaterally connected if and only if it has a Hamiltonian path. Applied to SMDs with $C(D)=\{V_1,\dots,V_c\}$37, this yields the corollary that a semicomplete multipartite digraph $C(D)=\{V_1,\dots,V_c\}$38 with $C(D)=\{V_1,\dots,V_c\}$39 is unilaterally connected if and only if it has a Hamiltonian path. Hence the Hamiltonian-path problem in this regime can be solved in $C(D)=\{V_1,\dots,V_c\}$40 time by checking unilateral connectivity via linear-decomposition (Brinkmann, 21 Jul 2025).

An adjacent but distinct line of work studies Hamiltonian paths and cycles with precedence constraints given by a partial order on the vertex set. In that model, one asks for a Hamiltonian path or cycle whose vertex order extends a partial order $C(D)=\{V_1,\dots,V_c\}$41. For the path problem, a permutation $C(D)=\{V_1,\dots,V_c\}$42 is a $C(D)=\{V_1,\dots,V_c\}$43-extension of $C(D)=\{V_1,\dots,V_c\}$44 if $C(D)=\{V_1,\dots,V_c\}$45, and the path must visit each vertex exactly once along graph edges while respecting this extension. This is not the same as quasi-Hamiltonian color-class coverage. The width-theoretic results are correspondingly different: POHPP is NP-complete for graphs of pathwidth $C(D)=\{V_1,\dots,V_c\}$46, polynomial-time solvable on graphs of pathwidth $C(D)=\{V_1,\dots,V_c\}$47 and treewidth $C(D)=\{V_1,\dots,V_c\}$48, and NP-complete on rectangular grid graphs of height at least $C(D)=\{V_1,\dots,V_c\}$49 (Beisegel et al., 30 Jun 2025).

7. Open questions

The boundary cases left unresolved are structurally narrow but conceptually important. The $C(D)=\{V_1,\dots,V_c\}$50-Hamiltonian-path problem in semicomplete multipartite digraphs of unbounded $C(D)=\{V_1,\dots,V_c\}$51, corresponding to the case $C(D)=\{V_1,\dots,V_c\}$52, is not known to be in $C(D)=\{V_1,\dots,V_c\}$53 nor shown NP-complete. The pure quasi-Hamiltonian path problem on SMDs with $C(D)=\{V_1,\dots,V_c\}$54 also remains open: neither hardness nor a complete polynomial-time algorithm is currently available (Brinkmann, 21 Jul 2025).

Within the Schaefer-style classification, the small cases $C(D)=\{V_1,\dots,V_c\}$55 and $C(D)=\{V_1,\dots,V_c\}$56 remain unclassified beyond the Hamiltonian-path case. Closing these gaps would complete the complexity map of $C(D)=\{V_1,\dots,V_c\}$57-CCP. The present structural evidence for $C(D)=\{V_1,\dots,V_c\}$58 suggests a possible polynomial-time theory, but that implication remains provisional rather than established.