Threshold Minimum Cut with Terminal Quotas: Logarithmic and Planar Approximation Algorithms

Abstract: We study threshold minimum cut problems with a distinguished root vertex, a set of terminals, and a quota. In the threshold minimum edge cut problem (\TMEC), the goal is to find a minimum-cost edge cut that disconnects at least $k$ terminals from the root. In the threshold minimum node cut problem (\TMNC), the goal is to delete a minimum-cost set of nonterminal, nonroot vertices so that at least $k$ terminals become disconnected from the root. We prove three approximation guarantees. First, undirected general-graph \TMEC{} admits a randomized polynomial-time expected $O(\log n)$ approximation via a Räcke-style cut-dominating tree decomposition and an exact dynamic program on trees. A standard repetition argument gives the same asymptotic ratio with high probability. Second, planar \TMEC{} admits a factor-$2$ approximation by reducing the threshold condition to planar weighted balanced cut. Third, bounded-degree planar \TMNC{} admits a $2Δ$-approximation, where $Δ$ is the maximum degree of a deletable vertex, by reducing the node-cost problem to the planar edge-cut problem on the same graph. The results separate exact-quota guarantees from bicriteria small-set-expansion-type guarantees and identify the unbounded-degree planar node-cut case as the main remaining obstacle.

• The paper achieves an expected O(log n) approximation for threshold minimum edge cut on general graphs by combining Räcke-style cut-dominating trees with an exact tree dynamic program that preserves the terminal quota.
• The paper gives a factor-2 approximation for planar threshold edge cuts by reducing the problem exactly to weighted balanced cut, ensuring every solution disconnects at least the required number of terminals.
• The paper obtains a 2Δ-approximation for bounded-degree planar threshold node cuts through a min-endpoint edge-cost transformation, while identifying removal of the degree dependence as a key open challenge.

Problem definition and motivation

The paper studies rooted threshold cut problems. The input is an undirected graph $G=(V,E)$ with nonnegative costs, a root $s$, a terminal set $T\subseteq V\setminus\{s\}$, and a quota $k$. In threshold minimum edge cut (TMEC), one seeks a minimum-cost edge boundary $\delta(X)$ of a set $X$ with $s\notin X$ and $|X\cap T|\ge k$. In threshold minimum node cut (TMNC), one deletes a minimum-cost set of nonterminal, nonroot vertices so that at least $k$ terminals are disconnected from $s$; root and terminals are undeletable, modeled by infinite deletion costs.

These formulations differ from classical minimum $s$0–$s$1 cut in that feasibility is an aggregated quota rather than separation of a single terminal pair. The paper is explicit that its guarantees preserve the quota exactly — every returned solution disconnects at least $s$2 terminals — as opposed to bicriteria guarantees of small-set-expansion type, which typically separate only $s$3 terminals (2606.15324). This exact-feasibility distinction is the paper's central framing device: stronger-looking bounds from min–max partitioning techniques do not directly yield feasible solutions for this problem.

Motivating applications include network reliability (isolating a threshold number of vulnerable hosts from a protected source), image segmentation on planar grid graphs where quotas encode region-size constraints, and cyber-security attack graphs where node cuts represent mitigations blocking harmful terminal states from an initial compromise state.

General-graph TMEC via cut-dominating trees

The first result is a randomized polynomial-time expected $s$4 approximation for TMEC on undirected general graphs. The algorithm uses two components:

Cut-dominating tree decomposition. The paper invokes, as a black box, a Räcke-style hierarchical decomposition theorem [Räcke 2008]: there is a randomized polynomial-time construction of a weighted tree $s$5 whose leaves biject with $s$6, such that for every $s$7, the tree cut value $s$8 dominates the graph cut cost $s$9, while $T\subseteq V\setminus\{s\}$0.

Exact dynamic program on trees. On any weighted tree whose leaves correspond to original vertices, TMEC is solvable exactly. Rooting the tree at the leaf for $T\subseteq V\setminus\{s\}$1, the DP computes $T\subseteq V\setminus\{s\}$2: the minimum cost of cutting edges strictly inside subtree $T\subseteq V\setminus\{s\}$3 so that exactly $T\subseteq V\setminus\{s\}$4 terminals below $T\subseteq V\setminus\{s\}$5 are disconnected from $T\subseteq V\setminus\{s\}$6, with terminal counts truncated at $T\subseteq V\setminus\{s\}$7. For each child, either the parent edge is kept (inheriting the child's DP table) or cut (contributing $T\subseteq V\setminus\{s\}$8 terminals at edge cost), and child tables are combined by knapsack convolution. A straightforward implementation runs in $T\subseteq V\setminus\{s\}$9 time.

The approximation analysis chains three inequalities: the optimal graph solution induces a feasible tree solution costing $k$0, which has expectation at most $k$1; the DP returns a tree solution no more expensive; and by domination, mapping back to the graph yields a feasible cut of cost at most the tree cost. Hence $k$2. A Markov-inequality repetition argument gives the same asymptotic ratio with probability $k$3 using $k$4 independent samples. Disconnected graphs are handled by counting already-disconnected terminals at zero cost and reducing the quota.

An implication worth noting: because the reduction goes through exact optimization on trees, the returned cut always satisfies the quota exactly — the logarithmic loss comes entirely from the expected distortion of the decomposition, not from rounding of feasibility.

Planar TMEC: factor-2 via balanced cut

For planar graphs, the paper obtains a polynomial-time factor-$k$5 approximation by reducing to the planar balanced-cut theorem of Garg, Saran, and Vazirani [Garg et al. 1999], which gives a factor-$k$6 approximation for minimum-cost $k$7-balanced edge cuts when $k$8 under unary vertex weights.

The reduction assigns weight $k$9 to the root, weight $\delta(X)$0 to each terminal, and weight $\delta(X)$1 elsewhere, giving total weight $\delta(X)$2 where $\delta(X)$3, and sets the balance parameter $\delta(X)$4. Two directions make this work exactly:

• Every feasible TMEC solution induces a $\delta(X)$5-balanced cut: the root side has weight at most $\delta(X)$6, and the terminal side has weight at most $\delta(X)$7. Thus $\delta(X)$8.
• Conversely, if $\delta(X)$9 is a feasible $X$0-balanced cut and $X$1 is the root component of $X$2, then $X$3 forces $X$4, so $X$5 contains at least $X$6 terminals. Since $X$7, the cost is preserved.

Combining both directions yields $X$8. The paper notes the weights are bounded by $X$9 and hence unary-expandable in polynomial size, so the unary-weight assumption of the balanced-cut theorem is satisfied without a pseudoapproximation caveat.

Bounded-degree planar TMNC: $s\notin X$0-approximation

For planar TMNC with maximum degree $s\notin X$1 over deletable vertices $s\notin X$2, the paper proves a $s\notin X$3-approximation by reducing node costs to edge costs on the same graph. Edges receive derived cost $s\notin X$4, with root and terminals assigned a large sentinel cost $s\notin X$5, where $s\notin X$6.

Two lemmas connect the optima:

• Node-to-edge: an optimal node separator $s\notin X$7 induces a feasible edge cut $s\notin X$8 whose crossing edges all touch $s\notin X$9, so $|X\cap T|\ge k$0.
• Edge-to-node: if $|X\cap T|\ge k$1, no crossing edge has both endpoints undeletable (its cost would be $|X\cap T|\ge k$2), so covering each crossing edge with a cheapest deletable endpoint yields a separator $|X\cap T|\ge k$3 with $|X\cap T|\ge k$4 that disconnects at least $|X\cap T|\ge k$5 terminals.

Since a finite feasible solution exists, $|X\cap T|\ge k$6, guaranteeing the returned edge cut satisfies $|X\cap T|\ge k$7, so the cover lemma applies. Chaining gives $|X\cap T|\ge k$8.

The degree dependence enters only through the node-to-edge direction, where a separator vertex may be charged once per incident crossing edge. The paper identifies removing this factor $|X\cap T|\ge k$9 — likely requiring either a direct planar balanced vertex separator approximation or a more faithful vertex-to-edge transformation — as the main remaining obstacle. It also notes the result assumes finiteness of some feasible solution; instances with no finite feasible solution are declared infeasible under the undeletability constraints.

Limitations and open questions

Three limitations are stated or implicit in the analysis. First, the general-graph guarantee is randomized and expected; derandomization is not addressed, though high-probability success follows from independent repetition at the cost of additional running time. Second, the planar TMEC result inherits the unary-weight assumption of the Garg–Saran–Vazirani theorem; the paper sidesteps it via polynomial-size expansion but does not address whether binary-weight balanced cuts would improve constants. Third, the TMNC bound degrades linearly with $k$0, and the paper leaves open whether a constant-factor planar node-cut approximation is achievable. A second open question is improving the exact-quota general-graph ratio below $k$1: existing small-set-expansion methods [Bansal et al. 2014] suggest better bicriteria bounds, but preserving $k$2 exactly remains, per the paper, the key technical barrier.

Conclusion

The paper establishes three approximation guarantees for rooted threshold cut problems: an expected $k$3 approximation for TMEC on general undirected graphs via Räcke decompositions plus exact tree dynamic programming, a factor-$k$4 approximation for planar TMEC via an exact reduction to planar weighted balanced cut, and a $k$5-approximation for bounded-degree planar TMNC via a min-endpoint edge-cost transformation. All guarantees are exact-quota, distinguishing them from bicriteria alternatives, and the degree-bounded restriction on the planar node-cut case is identified as the principal open obstacle.