[Paper Review] Approximating multicut and the demand graph

Chandra Chekuri, Vivek Madan|arXiv (Cornell University)|Jan 16, 2017

Complexity and Algorithms in Graphs11 citations

TL;DR

This paper presents a 2-approximation algorithm for the undirected Multicut problem in n^O(t) time when the demand graph excludes an induced matching of size t, using a reduction to uniform metric labeling rather than flow-cut gaps. In contrast, under the Unique Games Conjecture, it proves that for directed Multicut with fixed demand graphs, no better approximation exists than the worst-case flow-cut gap, and shows a k-approximation when the demand graph excludes certain induced subgraphs, generalizing the directed multiway cut result.

ABSTRACT

In the minimum Multicut problem, the input is an edge-weighted supply graph G = (V, E) and a demand graph H = (V, F). Either G and H are directed (Dir-MulC) or both are undirected (Undir-MulC). The goal is to remove a minimum weight set of supply edges E' ⊆ E such that in G − E' there is no path from s to t for any demand edge (s, t) ∈ F. Undir-MulC admits O(log k)-approximation where k is the number of edges in H while the best known approximation for Dir-MulC is min{k, O(|V|11/23)}. These approximations are obtained by proving corresponding results on the multicommodity flow-cut gap. In this paper we consider the role that the structure of the demand graph plays in determining the approximability of Multicut. We obtain several new positive and negative results.In undirected graphs our main result is a 2-approximation in nO(t) time when the demand graph excludes an induced matching of size t. This gives a constant factor approximation for a specific demand graph that motivated this work, and is based on a reduction to uniform metric labeling and not via the flow-cut gap.In contrast to the positive result for undirected graphs, we prove that in directed graphs such approximation algorithms can not exist. We prove that, assuming the Unique Games Conjecture (UGC), that for a large class of fixed demand graphs Dir-MulC cannot be approximated to a factor better than the worst-case flow-cut gap. As a consequence we prove that for any fixed k, assuming UGC, Dir-MulC with k demand pairs is hard to approximate to within a factor better than k. On the positive side, we obtain a k approximation when the demand graph excludes certain graphs as an induced subgraph. This generalizes the known 2 approximation for directed Multiway Cut to a larger class of demand graphs.

Motivation & Objective

• To investigate how the structural properties of the demand graph influence the approximability of the Multicut problem.
• To develop improved approximation algorithms for Multicut by exploiting specific structural constraints on the demand graph.
• To establish tight inapproximability bounds for directed Multicut under the Unique Games Conjecture.
• To generalize known results, such as the 2-approximation for directed multiway cut, to broader classes of demand graphs.

Proposed method

• Reduces the undirected Multicut problem to uniform metric labeling when the demand graph excludes an induced matching of size t.
• Employs a dynamic programming approach over the structure of the demand graph to achieve the 2-approximation in n^O(t) time.
• Uses the Unique Games Conjecture to prove inapproximability lower bounds for directed Multicut with fixed demand graphs.
• Identifies forbidden induced subgraphs that allow a k-approximation for directed Multicut, extending the multiway cut result.
• Analyzes the multicommodity flow-cut gap in relation to demand graph structure, but avoids relying on it for the main positive result.
• Applies structural graph theory to characterize demand graphs that permit constant-factor approximations.

Experimental results

Research questions

• RQ1 Can the approximability of Multicut be improved by restricting the structure of the demand graph?
• RQ2 What structural properties of the demand graph enable constant-factor approximation algorithms for undirected Multicut?
• RQ3 Under the Unique Games Conjecture, what are the tight inapproximability bounds for directed Multicut with fixed demand graphs?
• RQ4 Can the 2-approximation for directed multiway cut be generalized to a broader class of demand graphs?
• RQ5 How does the presence of induced matchings or other subgraphs affect the flow-cut gap and approximation hardness?

Key findings

• A 2-approximation algorithm for Undir-MulC is achieved in n^O(t) time when the demand graph excludes an induced matching of size t.
• The approximation is obtained via reduction to uniform metric labeling, not through flow-cut gap analysis.
• Under the Unique Games Conjecture, no better approximation exists than the worst-case flow-cut gap for a large class of fixed directed demand graphs.
• For any fixed k, Dir-MulC with k demand pairs cannot be approximated to a factor better than k under the UGC.
• A k-approximation is achieved for Dir-MulC when the demand graph excludes certain induced subgraphs, generalizing the multiway cut result.
• The structural constraints on the demand graph are shown to be decisive in determining the approximability of Multicut in both undirected and directed settings.