Faster Parameterized Vertex Multicut

Abstract: In the {\sc Vertex Multicut} problem the input consists of a graph $G$, integer $k$, and a set $\mathbf{T} = {(s_1, t_1), \ldots, (s_p, t_p)}$ of pairs of vertices of $G$. The task is to find a set $X$ of at most $k$ vertices such that, for every $(s_i, t_i) \in \mathbf{T}$, there is no path from $s_i$ to $t_i$ in $G - X$. Marx and Razgon [STOC 2011 and SICOMP 2014] and Bousquet, Daligault, and Thomassé [STOC 2011 and SICOMP 2018] independently and simultaneously gave the first algorithms for {\sc Vertex Multicut} with running time $f(k)n<sup>{O(1)}$. The running time of their algorithms is $2<sup>{O(k<sup>3)}n<sup>{O(1)}$ and $2<sup>{O(k<sup>{O(1)})}n<sup>{O(1)}$, respectively. As part of their result, Marx and Razgon introduce the {\em shadow removal} technique, which was subsequently applied in algorithms for several parameterized cut and separation problems. The shadow removal step is the only step of the algorithm of Marx and Razgon which requires $2<sup>{O(k<sup>3)}n<sup>{O(1)}$ time. Chitnis et al. [TALG 2015] gave an improved version of the shadow removal step, which, among other results, led to a $k<sup>{O(k<sup>2)}n<sup>{O(1)}$ time algorithm for {\sc Vertex Multicut}. We give a faster algorithm for the {\sc Vertex Multicut} problem with running time $k<sup>{O(k)}n<sup>{O(1)}$. Our main technical contribution is a refined shadow removal step for vertex separation problems that only introduces an overhead of $k^{O(k)}\log n$ time. The new shadow removal step implies a $k<sup>{O(k<sup>2)}n<sup>{O(1)}$ time algorithm for {\sc Directed Subset Feedback Vertex Set} and a $k<sup>{O(k)}n<sup>{O(1)}$ time algorithm for {\sc Directed Multiway Cut}, improving over the previously best known algorithms of Chitnis et al. [TALG 2015].

• The paper gives the fastest known Vertex Multicut algorithm, solving instances in k^{O(k)}n^{O(1)} time through iterative compression, refined shadow removal, and bipedal-instance reduction.
• Its randomized shadow-covering method, derandomized with splitters, succeeds with probability at least k^{-2k}/e and improves the prior k^{O(k^2)} overhead.
• The LP-guided branching framework also improves Directed Multiway Cut to O*(k^{O(k)}) and Directed Subset Feedback Vertex Set to O*(k^{O(k^2)}), while leaving single-exponential dependence as an open goal.

Problem and context

In the Vertex Multicut problem, the input is a graph $G$, a set $\mathbf{T}$ of terminal pairs, and an integer $k$; the task is to find a vertex set of size at most $k$ that separates every terminal pair. The parameterized complexity of this problem with respect to the cut size $k$ was a long-standing open question until Marx and Razgon [(2602.13981)'s cited literature] and Bousquet, Daligault, and Thomassé independently resolved it in 2011, both via algorithms running in time $2^{O(k^{O(1)})}n^{O(1)}$. The subsequent improvement by Chitnis et al., based on a faster shadow removal step, yielded a $k^{O(k^2)}n^{O(1)}$ algorithm. This paper presents the current fastest algorithm for Vertex Multicut, running in time $O^*(k^{O(k)})$, where $O^*(\cdot)$ suppresses polynomial factors in the input size.

The paper also observes a subtle point in the prior literature: the "reduction to the bipedal case" step of Marx and Razgon was analyzed as contributing a $2^{O(k^3)}$ factor, but a tighter reading of their bound gives only $\mathbf{T}$0, i.e., $\mathbf{T}$1. Consequently, Chitnis et al.'s improved shadow removal already implied a $\mathbf{T}$2 bound for Vertex Multicut, though this was never stated explicitly.

Main results

The central theorem is:

Theorem (Main). Vertex Multicut can be solved in time $\mathbf{T}$3.

Since Edge Multicut reduces to Vertex Multicut by a solution-size-preserving reduction, the same bound applies to Edge Multicut. Two further corollaries follow from the paper's refined shadow removal procedure applied within existing frameworks:

Problem	New bound	Previous best
Edge/Vertex Multicut	$\mathbf{T}$4	$\mathbf{T}$5
Directed Subset Feedback Vertex Set	$\mathbf{T}$6	$\mathbf{T}$7
Directed Multiway Cut	$\mathbf{T}$8	$\mathbf{T}$9

Algorithmic framework

The algorithm follows the four-stage architecture of Marx and Razgon: iterative compression, shadow removal, reduction to bipedal instances, and solving bipedal instances. Iterative compression reduces Vertex Multicut to its compression variant at a cost of $k$0: given a known solution $k$1 of size at most $k$2, one seeks a disjoint solution $k$3 of size at most $k$4 that is additionally a multiway cut for $k$5. The final stage reuses the $k$6-time algorithm for bipedal instances via a reduction to Almost 2-SAT. The two intermediate stages are completely redesigned; each is implemented as a partial branching algorithm — producing a family of sub-instances such that no-instances map to no-instances, while yes-instances preserve some desired structural property in at least one branch.

Refined shadow removal

Shadow removal is applicable to any vertex-deletion problem satisfying two axioms: monotonicity under adding vertices to a solution, and the property that if $k$7 is a solution with $k$8 in the shadow of $k$9, then $k$0 is itself a solution. The core is a randomized shadow covering procedure: color each vertex red with probability $k$1 and blue otherwise, then output the set $k$2 of vertices having some important $k$3–$k$4 separator of size at most $k$5 that is entirely red. For any solution $k$6 satisfying the standard "pushing" conditions, $k$7 covers the shadow of $k$8 and avoids $k$9 with probability at least $k$0 — improving on success probabilities of $k$1 (Marx–Razgon) and $k$2 (Chitnis et al.).

Two ingredients make this work. First, a strengthened hitting-set lemma for important separators: for any vertex set $k$3 not fully containing any important separator of size at most $k$4, there exists a hitting set of size at most $k$5 disjoint from $k$6. This is proved constructively by iteratively marking vertices on farthest minimum separators, exploiting their uniqueness and the fact that every important separator contains the farthest minimum separator. Second, the classical pushing argument shows that every yes-instance admits a $k$7-shadow-removable solution — one whose shadow vertices are separated from $k$8 by important separators contained in the solution, while no important separator separating a solution vertex from $k$9 lies entirely inside it. Combining these, the red/blue coloring succeeds when the (unknown) solution is red and the union of the per-vertex hitting sets (size at most $2^{O(k^{O(1)})}n^{O(1)}$0) is blue, giving probability $2^{O(k^{O(1)})}n^{O(1)}$1. Derandomization via splitters yields a deterministic family of $2^{O(k^{O(1)})}n^{O(1)}$2 candidate sets, each processed by the torso operation to enforce shadowlessness.

Reduction to bipedal instances

The second redesigned step uses the half-integral LP relaxation of Vertex Multiway Cut as a measure. Branching Rule 1 exploits non-zero vertices: forcing such a vertex out of the solution raises $2^{O(k^{O(1)})}n^{O(1)}$3 by at least $2^{O(k^{O(1)})}n^{O(1)}$4, so branching between deletion and torso decreases the measure $2^{O(k^{O(1)})}n^{O(1)}$5 by at least $2^{O(k^{O(1)})}n^{O(1)}$6 per branch.

When no non-zero vertex exists, the paper proves the key structural identity $2^{O(k^{O(1)})}n^{O(1)}$7, relating the LP optimum to isolating min-cuts. The proof analyzes complementary slackness: choosing a dual optimum maximizing untight vertices, the neighborhoods $2^{O(k^{O(1)})}n^{O(1)}$8 of untight regions are pairwise disjoint (otherwise complementary slackness would force a non-zero vertex), and every positive dual path crosses exactly two such boundaries — established via an exchange argument showing that otherwise the untight set could be enlarged or another non-zero vertex would exist. This identity licenses Branching Rule 2 over farthest isolating min-cut boundaries: deleting a boundary vertex decreases the measure by at least 1, while contracting it into $2^{O(k^{O(1)})}n^{O(1)}$9 strictly increases $k^{O(k^2)}n^{O(1)}$0, which forces a non-zero vertex to appear and hence progress in the next step. The search tree has depth $k^{O(k^2)}n^{O(1)}$1 and branching factor $k^{O(k^2)}n^{O(1)}$2, giving $k^{O(k^2)}n^{O(1)}$3 leaves.

On leaves assumed to admit a contractible shadowless solution, the region decomposition induced by the farthest isolating min-cuts satisfies strong properties: the region $k^{O(k^2)}n^{O(1)}$4 is empty, regions with $k^{O(k^2)}n^{O(1)}$5 are empty, and edges exist only between single-terminal regions or between a singleton region and a two-terminal region containing it. Contracting the inner boundaries of singleton regions into their terminals then produces a bipedal instance preserving shadowless solutions, completing the pipeline.

Limitations and open questions

The paper concedes several points. The shadow covering analysis is existential and probabilistic; whether the $k^{O(k^2)}n^{O(1)}$6 overhead in shadow removal is optimal, or can be improved toward single-exponential, is left open. The overall running time remains $k^{O(k^2)}n^{O(1)}$7, dominated jointly by iterative compression, shadow removal, and the bipedalization branching; the authors explicitly pose whether a $k^{O(k^2)}n^{O(1)}$8 (single-exponential) algorithm for Edge Multicut exists as an interesting direction. Additionally, the correctness argument for Lemma on contractible solutions notes an edge case — components entirely contained in the solution — handled by assuming optimality is preserved through the earlier procedures, a dependency the authors flag in a footnote. Finally, the corollaries for Directed Subset Feedback Vertex Set and Directed Multiway Cut rely on substituting the new shadow removal into the frameworks of Chitnis et al.; no tighter analysis specific to those problems' other bottlenecks is provided.

Conclusion

This paper improves the parameterized complexity of Vertex Multicut from $k^{O(k^2)}n^{O(1)}$9 to $O^*(k^{O(k)})$0 through two technical contributions: a shadow removal procedure with $O^*(k^{O(k)})$1 overhead built on a strengthened important-separator hitting-set lemma, and an LP-guided branching scheme grounded in the identity equating twice the multiway cut LP optimum with the sum of isolating min-cuts. The same machinery yields improved algorithms for Directed Subset Feedback Vertex Set and Directed Multiway Cut, and the techniques are candidates for application to other parameterized separation problems amenable to shadowless-solution and bipedal-compression formulations.