Tight Bounds for some W[1]-hard Problems Parameterized by Multi-clique-width

Abstract: In this work we contribute to the study of the fine-grained complexity of problems parameterized by multi-clique-width, which was initiated by Fürer [ITCS 2017] and pursued further by Chekan and Kratsch [MFCS 2023]. Multi-clique-width is a parameter defined analogously to clique-width but every vertex is allowed to hold multiple labels simultaneously. This parameter is upper-bounded by both clique-width and treewidth (plus a constant), hence it generalizes both of them without an exponential blow-up. Conversely, graphs of multi-clique-width $k$ have clique-width at most $2^k$, and there exist graphs with clique-width at least $2<sup>{Ω(k)}$. Thus, while the two parameters are functionally equivalent, the fine-grained complexity of problems may differ relative to them. As our first and main result we show that under ETH the Max Cut problem cannot be solved in time $n^{{2^{o(k)}}} \cdot f(k)$ on graphs of multi-clique-width $k$ for any computable function $f$. For clique-width $k$ an $n<sup>{\mathcal{O}(k)}$ algorithm by Fomin et al. [SIAM J. Comput. 2014] is tight under ETH. This makes Max Cut the first known problem for which the tight running times differ for parameterization by clique-width and multi-clique-width and it contributes to the short list of known lower bounds of form $n^{{2^{o(k)}}} \cdot f(k)$. As our second contribution we show that Hamiltonian Cycle and Edge Dominating Set can be solved in time $n<sup>{\mathcal{O}(k)}$ on graphs of multi-clique-width $k$ matching the tight running time for clique-width. These results answer three questions left open by Chekan and Kratsch [MFCS 2023].

• The paper proves that, unless ETH fails, Max Cut cannot be solved in n^{2^{o(k)}·f(k)} time from a multi-k-expression, separating its fine-grained complexity from clique-width.
• The paper develops representative-set and label-choice dynamic programs that solve Hamiltonian Cycle and Edge Dominating Set in n^{O(k)} time on graphs given with a multi-k-expression.
• The results show that multi-label flexibility is costly for problems such as Max Cut, while connectivity and matching-based problems remain tractable when each solution vertex needs only a small number of relevant labels.

Background and motivation

This paper, by Bergougnoux, Chekan, and Kratsch (2604.25841), studies the fine-grained complexity of problems parameterized by multi-clique-width ($\mathrm{mcw}$), a graph width parameter introduced by Fürer that generalizes both treewidth and clique-width without an exponential blow-up. Multi-clique-width uses the same construction operations as clique-width (introduce, union, join, relabel), except that each vertex may hold multiple labels simultaneously. The key structural relations are $\mathrm{mcw} \leq \mathrm{cw}$ and $\mathrm{mcw} \leq \mathrm{tw} + 2$, while graphs of multi-clique-width $k$ have clique-width at most $2^k$, and this is tight up to constants in the exponent. Thus the parameters are functionally equivalent, but their fine-grained behavior may differ.

The paper answers three open questions of Chekan and Kratsch (MFCS 2023), who had shown matching tight bounds for clique-width and multi-clique-width for several problems (e.g., $q$-Coloring, Chromatic Number, Connected Vertex Cover) but could only achieve such results for the weaker parameter fusion-width for Hamiltonian Cycle, Max Cut, and Edge Dominating Set.

Main result: an ETH lower bound for Max Cut

The central contribution is a lower bound separating clique-width from multi-clique-width. Unless ETH fails, Max Cut cannot be solved in time $n^{2^{o(k)} \cdot f(k)}$ on $n$-vertex graphs given with a multi-$k$-expression, for any computable function $f$. Since Max Cut admits an $\mathrm{mcw} \leq \mathrm{cw}$0 algorithm for clique-width [Fomin et al., SIAM J. Comput. 2014], and since a multi-$\mathrm{mcw} \leq \mathrm{cw}$1-expression trivially converts into a $\mathrm{mcw} \leq \mathrm{cw}$2-expression yielding an $\mathrm{mcw} \leq \mathrm{cw}$3 algorithm, this makes Max Cut the first known problem whose tight running times provably differ under the two parameterizations. It also joins the short list of ETH-based lower bounds of the form $\mathrm{mcw} \leq \mathrm{cw}$4 relative to structural parameters; previously such bounds were known only for Chromatic Number, $\mathrm{mcw} \leq \mathrm{cw}$5-Coloring, and Fall Coloring parameterized by clique-width.

The reduction starts from Multicolored Independent Set, which has no $\mathrm{mcw} \leq \mathrm{cw}$6 algorithm under ETH [Lokshtanov, Marx, Saurabh 2011]. The construction builds on gadgets of Fomin et al.—$\mathrm{mcw} \leq \mathrm{cw}$7-gadgets ($\mathrm{mcw} \leq \mathrm{cw}$8 disjoint length-2 paths forcing endpoints apart in optimal cuts), $\mathrm{mcw} \leq \mathrm{cw}$9-gadgets (length-3 paths forcing endpoints together), and $\mathrm{mcw} \leq \mathrm{tw} + 2$0-gadgets (triangles of $\mathrm{mcw} \leq \mathrm{tw} + 2$1-gadgets)—and introduces a generalized conditional gadget $\mathrm{mcw} \leq \mathrm{tw} + 2$2 built around a complete $\mathrm{mcw} \leq \mathrm{tw} + 2$3-partite "core" $\mathrm{mcw} \leq \mathrm{tw} + 2$4. Choosing $\mathrm{mcw} \leq \mathrm{tw} + 2$5 ensures any partition violating a gadget loses at least $\mathrm{mcw} \leq \mathrm{tw} + 2$6 crossed edges, so every optimal partition must satisfy all gadgets.

Each edge $\mathrm{mcw} \leq \mathrm{tw} + 2$7 of the source graph gets its own copy of a selection gadget consisting of cliques $\mathrm{mcw} \leq \mathrm{tw} + 2$8 and $\mathrm{mcw} \leq \mathrm{tw} + 2$9 for sets $k$0, where $k$1 are complementary families of size $k$2 with $k$3. Selecting vertex $k$4 corresponds to placing exactly $k$5 vertices of $k$6 on one side; five attached conditional gadgets per copy enforce that not both endpoints of the corresponding edge are selected. Copies are chained by $k$7-gadgets so all copies select consistently.

Two design choices are crucial for keeping the multi-clique-width logarithmic:

• Extending from $k$8 to $k$9 and adding $2^k$0-gadgets between $2^k$1 and $2^k$2 makes partitions of complementary sets behave complementarily, enabling "at least" constraints via "at most" gadgets.
• Making $2^k$3 a clique unless $2^k$4 turns what would be a thickened matching between $2^k$5 and $2^k$6 into a thickened anti-matching: two vertices $2^k$7 and $2^k$8 are adjacent iff $2^k$9. This allows edges to be created with only $q$0 labels: vertices of $q$1 and $q$2 receive label sets $q$3 and $q$4, and joining $q$5 with $q$6 for each $q$7 creates exactly the required adjacencies. The resulting expression is linear, and the whole instance is computable in time polynomial in $q$8 and $q$9.

Correctness follows because the budget forces optimality on every gadget, which pins down the counts $n^{2^{o(k)} \cdot f(k)}$0 across all copies, and violating an edge constraint would contradict the conditional-gadget guarantees. Plugging the construction into a hypothetical $n^{2^{o(k)} \cdot f(k)}$1 algorithm yields an $n^{2^{o(k)} \cdot f(k)}$2 algorithm for Multicolored Independent Set, contradicting ETH.

Algorithm for Hamiltonian Cycle

The paper shows Hamiltonian Cycle can be decided in time $n^{2^{o(k)} \cdot f(k)}$3 given a multi-$n^{2^{o(k)} \cdot f(k)}$4-expression, matching the tight bound for clique-width (the $n^{2^{o(k)} \cdot f(k)}$5 lower bound of Fomin et al. transfers trivially since $n^{2^{o(k)} \cdot f(k)}$6-expressions are multi-$n^{2^{o(k)} \cdot f(k)}$7-expressions).

The algorithm adapts the representative-set machinery of Bergougnoux, Kanté, and Kwon (Algorithmica 2020). Partial solutions are path packings of the graph constructed so far, equipped with a label choice mapping each path endpoint to one label that may later create the missing incident edge via a join. Each partial solution induces an auxiliary multigraph on the label set $n^{2^{o(k)} \cdot f(k)}$8, with one edge per path connecting the chosen labels. Although a path packing now admits many label choices (up to $n^{2^{o(k)} \cdot f(k)}$9 options per endpoint), the relevant equivalence—identifying multigraphs with equal degree sequences and connected-component partitions—still bounds the number of representatives by $n$0.

The paper verifies that representation is maintained across all node types of a simplified multi-expression (where relabels either forget a label or add one label to another, and introduces carry a single label):

• Forget-label nodes: discard multigraphs with nonzero degree at the forgotten label.
• Add-label nodes ($n$1): each endpoint mapped to $n$2 may independently switch to $n$3; the family $n$4 captures all switch patterns and is enumerable in $n$5 time by branching on multiplicities.
• Union nodes: pairwise edge-disjoint unions of child multigraphs.
• Join nodes ($n$6): applying the combine operation (replacing two red edges incident to $n$7 and $n$8 by one edge) up to $n$9 times, interleaved with the reduction operator $k$0 after every step, preserves representation.

At the root, checking for a single-edge auxiliary multigraph with endpoints pinned to two fresh labels decides Hamiltonicity between a fixed adjacent pair; iterating over all edges gives the final algorithm. The main technical contribution is showing the representation argument survives non-unique label choices inherent to multi-labelings, particularly in the add-label and join cases.

Algorithm for Edge Dominating Set

Edge Dominating Set is also solvable in $k$1 time given a multi-$k$2-expression, again matching the transferred clique-width lower bound. The algorithm rests on the folklore equivalence: $k$3 has an edge dominating set of size at most $k$4 iff there exist a vertex cover $k$5 and a matching $k$6 with $k$7 and $k$8.

Partial solutions are pairs $k$9; a label choice maps each vertex of $f$0 to one of its labels—the single label expected to later create its matching edge, if any. A footprint is a triple $f$1 recording the union $f$2 of labels of uncovered vertices (to keep $f$3 independent at joins), the profile $f$4 of the label choice, and the matching size. There are at most $f$5 footprints per node, and the paper gives closed-form transition rules for all four node types: forgetting discards footprints with $f$6; add-label shifts $f$7 units of mass from $f$8 to $f$9 for each $\mathrm{mcw} \leq \mathrm{cw}$00; unions sum components; joins decrease both $\mathrm{mcw} \leq \mathrm{cw}$01 and $\mathrm{mcw} \leq \mathrm{cw}$02 by $\mathrm{mcw} \leq \mathrm{cw}$03 while increasing $\mathrm{mcw} \leq \mathrm{cw}$04 by $\mathrm{mcw} \leq \mathrm{cw}$05, subject to $\mathrm{mcw} \leq \mathrm{cw}$06. A sentinel label added to every introduced vertex handles unmatched cover vertices at the root check.

The contrast with Max Cut is instructive: for Hamiltonian Cycle and Edge Dominating Set, only a constant number of labels per vertex participate in solution-relevant edge creation, so guessing one label suffices. For Max Cut, every edge may be cut, so in principle all labels of a vertex matter—which is precisely why the lower bound construction exploits anti-matchings to force large multi-clique-width behavior.

Limitations and open questions

All algorithms assume a multi-$\mathrm{mcw} \leq \mathrm{cw}$07-expression is part of the input; this is standard for such results but substantive, since the best known FPT approximation of multi-clique-width goes through rank-width with a double-exponential ratio. The lower bound is stated under ETH and concerns XP-type running times; whether the $\mathrm{mcw} \leq \mathrm{cw}$08 upper bound for Max Cut via conversion to clique-width is optimal in a finer sense (e.g., under SETH) remains unaddressed. The separation established is between clique-width and multi-clique-width only; no problem is yet known to separate fusion-width from multi-clique-width, or all three parameters simultaneously, for either XP or FPT problems. Finally, whether the double-exponential approximation of multi-clique-width can be improved is left open.

Conclusion

The paper resolves three questions of Chekan and Kratsch concerning fine-grained complexity under multi-clique-width. For Max Cut it establishes an ETH-based lower bound of $\mathrm{mcw} \leq \mathrm{cw}$09, providing the first known separation between clique-width and multi-clique-width and extending the sparse list of double-exponential-in-the-exponent lower bounds for structural parameters. For Hamiltonian Cycle and Edge Dominating Set it provides $\mathrm{mcw} \leq \mathrm{cw}$10 algorithms matching the tight clique-width bounds, demonstrating that connectivity-style problems whose solutions touch few labels per vertex retain their fine-grained complexity under the more permissive multi-labeling model.