Maximum Coverage $k$-Antichains and Chains: A Greedy Approach

TL;DR

This work studies Maximum Coverage k-antichains (MA-$k$) and related problems on DAGs by leveraging Greene-Kleitman theorems and minimum-cost flow reductions on a carefully constructed network $G'$. The authors obtain near-optimal and exact results: (i) an MA-$k$ algorithm running in $|E|^{1+o(1)}$ time independent of $k$; (ii) a randomized near-linear-time algorithm with runtime $ ilde{O}(oldsymbol{ abla}_k|E|)$; and (iii) a $(1-1/e)$-approximation for MA-$k$ with time $O(oldsymbol{ abla}_1^2|V| + (oldsymbol{ abla}_1+k)|E|)$. They also extend these approaches to MAP-$k$, MC-$k$, MCP-$k$, and related path-cover variants using a parameterized cycle-canceling framework and recent advances in negative-cost cycles. A greedy-set-cover-based approach yields a $(1-1/e)$-approximation, with efficient implementations; however, the paper also proves lower bounds showing greedy can be provably suboptimal on specially constructed DAGs, highlighting intrinsic limits of greedy strategies for these problems. Overall, the work advances both exact and approximate algorithmic techniques for coverage via antichains in DAGs and clarifies the trade-offs between exactness, parameterization, and greedy methods in this domain.

Abstract

Given an input acyclic digraph $G = (V,E)$ and a positive integer $k$, the problem of Maximum Coverage $k$-Antichains (resp., Chains) denoted as MA-$k$ (resp., MC-$k$) asks to find $k$ sets of pairwise unreachable vertices, known as antichains (resp., $k$ subsequences of paths, known as chains), maximizing the number of vertices covered by these antichains (resp. chains). While MC-$k$ has been recently solved in (almost) optimal $O(|E|^{1+o(1)})$ time [Kogan and Parter, ICALP 2022], the fastest known algorithm for MA-$k$ is a recent $(k|E|)^{1+o(1)}$-time solution [Kogan and Parter, ESA 2024] as well as a $1/2$ approximation running in $|E|^{1+o(1)}$ time in the same paper. In this paper, we leverage a paths-based proof of the Greene-Kleitmann (GK) theorem with the help of the greedy algorithm for set cover and recent advances on fast algorithms for flows and shortest paths to obtain the following results for MA-$k$: - The first (exact) algorithm running in $|E|^{1+o(1)}$ time, hence independent in $k$. - A randomized algorithm running in $\tilde{O}(α_k|E|)$ time, where $α_k$ is the size of the optimal solution. That is, a near-linear parameterized running time, generalizing the result of [Mäkinen et al., ACM TALG] obtained for $k=1$. - An approximation algorithm running in time $O(α_1^2|V| + (α_1+k)|E|)$ with approximation ratio of $(1-1/e) > 0.63 > 1/2$. Our last two solutions rely on the use of greedy set cover, first exploited in [Felsner et al., Order 2003] for chains, which we now apply to antichains. We complement these results with two examples (one for chains and one for antichains) showing that, for every $k \ge 2$, greedy misses a $1/4$ portion of the optimal coverage. We also show that greedy is a $Ω(\log{|V|})$ factor away from minimality when required to cover all vertices: previously unknown for sets of chains or antichains.

Figures (5)

• Figure 1: A DAG $G$ and its corresponding $\alpha_k$ and $\beta_k$ networks $G'$, with their difference on edge $(t,s)$ shown. Namely, in the $\alpha_k$ network, the edge $(t,s)$ gets cost $k$, while in the $\beta_k$ network, it gets capacity $k$. For clarity, most edges of the form $(s,v^{in})$ and $(v^{out},t)$ are omitted. The edges of type $e_v^1$ are drawn thicker, and have cost $c=-1$ and capacity $u=1$.
• Figure 2: Illustrations of the graphs used in \ref{['lem:upperBoundCoverage-I']}. The greedy chains are obtained in the order blue, red, yellow.
• Figure 3: Illustrations of the graphs used in \ref{['lem:upperBoundCoverage-II']}. The greedy antichains are obtained in the order blue, red, yellow.
• Figure 4: Illustration of the instance $G^C_6$, with $6$ alternating paths shows as different colors. Each vertex is labeled by its weight $w$ (which can be reduced to $w$ vertices as shown in the bottom of the figure). The green path of a single vertex is $P_1$, and the blue path of length $6$ is $P_6$. The vertices are ordered by $P_6[j-1] \to P_5[j-1] \to \dots \to P_j[j-1]$ for every $j = 1,\dots,6$. Moreover, we ensure that $P_j[j-1] \to P_6[j+1]$ for all $j = 1, \dots, 4=6-2$.
• Figure 5: Illustration of the graph $G^A_2$ from \ref{['lem:lowerBoundPartition-partII']}.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Theorem 5
• Theorem 6: Greene-Kleitman (1976)
• Theorem 7: Schrijver (2003)
• Lemma 1
• proof
• Lemma 2: \ref{['thm:almostLinear']}, part I