Vertex Subset Problems: An Overview

• Vertex Subset Problems are defined as selecting a subset S of vertices that meet a given combinatorial property, including covering, separation, and distance constraints.
• Recent studies reveal intricate parameterized and kernelization complexity landscapes, with tight ETH lower bounds and FPT algorithms for metrics like vertex cover number and treewidth.
• Algorithmic advances leverage dynamic programming and extended LP formulations to bridge theoretical insights with practical solutions in graph optimization.

A Vertex Subset Problem (VSP) asks, given a graph $G$ and potentially other parameters, for a subset of vertices $S\subseteq V(G)$ satisfying some combinatorial property dictated by $G$ (optionally along with weights, constraints, and target values). Canonical examples encompass dominating set, vertex cover, feedback vertex set, and many fixed-cardinality or distance-based selection problems. VSPs unify broad swathes of combinatorial and algorithmic graph theory, encompassing both classical NP-complete problems and problems with deep parameterized and structural algorithmic complexity. Recent work on VSPs covers their parameterized complexity, kernelization, polyhedral properties, and algorithmic methods—including tight connections to dynamic programming and extension complexity.

1. Formal Definition and General Features

A vertex subset problem is specified by a predicate or property $\varphi(G,S)$ such that $S \subseteq V(G)$ is a feasible solution if and only if $\varphi(G,S)$ holds. The objective can be to minimize (or maximize) $|S|$, the induced subgraph’s measure (e.g., weight, cut-value), or the satisfaction of parameterized constraints. Many important families of VSPs fit the template:

• Covering Problems: Seek $S$ such that every member of a set of substructures (edges, cycles, paths) is “hit” (e.g., vertex cover, feedback vertex set, subset feedback vertex set).
• Separation and Partitioning Problems: Require $S$ whose removal separates the graph in prescribed ways (e.g., vertex separator problems).
• Distance and Identification Problems: Use the metric structure (e.g., metric dimension, geodetic set, outer multiset dimension).
• Cardinality Constraints: Fix $|S|=k$ and optimize a function (subgraph density, edge cuts, etc.).

VSPs are closed under isomorphism and admit characteristic vector encodings, supporting formulations as 0/1-polytope optimization (Oliveira et al., 11 Jan 2026).

2. Parameterized Complexity and Kernelization

The parameterized complexity landscape for VSPs is intricate, with tractability often governed by solution size, natural graph parameters, or structural decompositions:

• Vertex Cover Number $\mathbf{vc}$: For some metric-based VSPs (metric dimension, geodetic set), both admit $2^{\mathcal{O}(\mathsf{vc}^2)} n^{\mathcal{O}(1)}$-time FPT algorithms and kernelizations with $2^{\mathcal{O}(\mathsf{vc})}$ vertices. ETH lower bounds are matching: $2^{o(\mathsf{vc}^2)}$-time FPT algorithms and $2^{o(\mathsf{vc})}$-size kernels are impossible unless ETH fails (Foucaud et al., 2024).
• Structure-Based Parameters: For instance, feedback vertex set number, treewidth, and leafage distinguish between tractable and intractable cases, as shown for the subset feedback vertex set (SFVS) on graphs of bounded leafage and on $H$-free classes, with concrete $n^{\mathcal{O}(\ell)}$ algorithms and corresponding W[1]-hardness for the parameter (Papadopoulos et al., 2021, Paesani et al., 2022).
• Bounded Independent Set Number: For SFVS, polynomial-time solvability exists for $\alpha(G) \le 3$ but hardness appears at $\alpha(G)=4$ (Papadopoulos et al., 2018).
• Fixed-Cardinality VSPs: Generalizations such as $\alpha$-valued fixed-cardinality partitioning unify problems like Densest $k$-Subgraph, Partial Vertex Cover, and Max $(k,n-k)$-Cut, and their kernelization tightly depends on maximum degree, degeneracy, or vertex cover number, with explicit polynomial or infeasibility bounds (Koana et al., 2022).
• Kernelization Lower Bounds: Optimality is established for exponential-size kernels in metric-based problems, and polynomial kernels (e.g., Subset Feedback Vertex Set, Subset Vertex Cover) are shown to be best possible absent complexity collapses (Philip et al., 2019, Brettell et al., 2023).

Problem family	Kernel upper bound	Kernel lower bound (unless assumption fails)
Metric Dimension, Geodetic Set	$2^{\mathcal{O}(\mathsf{vc})}$	No $2^{o(\mathsf{vc})}$ [ETH]
Subset Feedback Vertex Set (split)	$\mathcal{O}(k^2)$	No $k^{2-\epsilon}$ [NP $\nsubseteq$ coNP/poly]

Kernelization and preprocessing also include essential vertices and search-space reductions, identified via packing/covering duality, which tighten FPT algorithms to the subset of nonessential vertices (Bumpus et al., 2022).

3. Key Structural and Algorithmic Results

• Metric-based distance problems: The metric dimension and geodetic set exhibit $2^{\Theta(\mathsf{vc}^2)}$-FPT algorithms and nearly matching kernel and lower bounds (Foucaud et al., 2024). The outer multiset dimension generalizes metric dimension with multiset representations, is always defined, and is strictly larger for many graphs, but is NP-complete and lacks strong parameterized or approximation results (Gil-Pons et al., 2019).
• Feedback problems: SFVS generalizes the feedback vertex set and Multiway Cut. SFVS is FPT in $k$, but NP-complete on chordal or split graphs. For split graphs, a reduction to 3-Hitting Set yields optimal quadratic-size kernels and a tight $2^k$-time FPT algorithm (Philip et al., 2019). On interval and permutation graphs, and for co-bipartite graphs, polynomial-time algorithms are established via dynamic programming or enumeration (Papadopoulos et al., 2017). The complexity is sharply determined by forbidden induced subgraph $H$; dichotomies chart the exact frontier between polynomial and NP-complete cases (Paesani et al., 2022).
• Separation and partitioning: Vertex separator problems are addressed via continuous bilinear programming with multilevel refinement, establishing equivalence to the discrete optimum, and yielding practical competitiveness with state-of-the-art heuristics (Hager et al., 2013).
• Optimization and approximation: For the broad subclass of “subset” problems (where solutions are vertex sets with a size objective), strong FPT-inapproximability holds for most W[1]-hard or W[2]-hard cases. However, dual-parameterization (using $n-k$) frequently enables FPT-approximation schemes, even for otherwise intractable problems (Bonnet et al., 2013).

4. Polyhedral and Extended Formulation Theory

A key advance is the formal correspondence between dynamic programming for VSPs on tree-like decompositions and small extended LP formulations of the associated solution polytopes. For any VSP admitting a DP algorithm with table size $\alpha(k,n)$ on treewidth-$k$ graphs, the extension complexity of its polytope is $O(\alpha(k,n)\cdot n)$, and this bound is ETH-tight: no smaller extended formulations exist under ETH for classic problems such as Independent Set, Vertex Cover, Dominating Set, or $d$-Coloring (Oliveira et al., 11 Jan 2026).

This equivalence unifies combinatorial and polyhedral algorithmics, shows optimality of known DPs and LP formulations, and motivates further study into other decompositions (e.g., clique-width, pathwidth). Meta-theorems confirm the generality for all MSO-definable VSPs, yielding linear-parameterized LPs on bounded-treewidth inputs.

5. Notable Specializations and Variations

• Subset feedback problems: Subset Feedback Vertex Set, Subset Vertex Cover, Node Multiway Cut, and others, extend the feedback paradigm; their tractability is highly sensitive to hereditary graph classes, forbidden substructures, and dual-parameterizations (Cygan et al., 2010, Brettell et al., 2023).
• Cut and partitioning generalizations: The Bounded Generalized Min-Cut problem, involving superadditive set functions and cardinality constraints, admits polynomial-time enumeration of all near-optimal solutions for fixed bounds. This yields new tractability for surjective and lower-bounded valued CSPs (Matl et al., 2019).
• Vertex isoperimetric and extremal problems: Minimizing boundary size for fixed-size vertex subsets leads to canonical nested extremal sets, proved via compression and centralization methods in grids, tori, and cubes (Veomett et al., 2012).

6. Open Problems and Directions

Numerous questions remain open in the complexity theory and algorithmic design of VSPs:

• Is there FPT (or even XP) tractability for the outer multiset dimension parameterized by known structural measures?
• Can the gaps between kernelization and parameterized lower bounds for leafage, mim-width, or other advanced parameters be closed for major VSPs (Papadopoulos et al., 2021, Paesani et al., 2022)?
• Do efficient LP extended formulations exist for VSPs on decompositions beyond treewidth—such as clique-width or hybrid measures—and how do these relate to the table sizes of solution-preserving DPs (Oliveira et al., 11 Jan 2026)?
• Which other CSPs, or more general surjective/lower-bounded constraint problems, can be described and efficiently solved via the bounded generalized min-cut paradigm (Matl et al., 2019)?

The literature on vertex subset problems continues to bridge deeply theoretical and practical concerns, with tight connections to parameterized complexity, combinatorial optimization, structural graph theory, and polyhedral combinatorics. The landscape is marked by crystallized tractability frontiers, optimal algorithm/kernel lower bounds, and ongoing questions at the intersection of discrete optimization, structure, and efficient computation.