[Paper Review] Alternating Path and Coloured Clustering

Leizhen Cai, On Yin Leung|arXiv (Cornell University)|Jul 27, 2018

Advanced Graph Theory Research5 references4 citations

TL;DR

This paper reveals that the Colored Clustering problem—maximizing stable edges in an edge-colored graph via vertex coloring—is equivalent to finding a maximum vertex-monochromatic subgraph or removing the fewest edges to eliminate all alternating paths. It presents an O(m^{3/2} log n)-time algorithm for edge-bicolored graphs via minimum cut reduction, proves NP-completeness for edge-tricolored planar bipartite graphs of max degree four, and gives FPT algorithms parameterized by stable and unstable edges with a 4k-vertex kernel and 2k² + k-edge kernel.

ABSTRACT

In the Coloured Clustering problem, we wish to colour vertices of an edge coloured graph to produce as many stable edges as possible, i.e., edges with the same colour as their ends. In this paper, we reveal that the problem is in fact a maximum subgraph problem concerning monochromatic subgraphs and alternating paths, and demonstrate the usefulness of such connection in studying these problems. We obtain a faster algorithm to solve the problem for edge-bicoloured graphs by reducing the problem to a minimum cut problem. On the other hand, we push the NP-completeness of the problem to edge-tricoloured planar bipartite graphs of maximum degree four. Furthermore, we also give FPT algorithms for the problem when we take the numbers of stable edges and unstable edges, respectively, as parameters.

Motivation & Objective

• To reframe Colored Clustering as a maximum subgraph problem involving vertex-monochromatic subgraphs and alternating path removal.
• To improve the time complexity of solving Colored Clustering on edge-bicolored graphs.
• To establish NP-completeness for edge-tricolored planar bipartite graphs with maximum degree four.
• To develop FPT algorithms for Colored Clustering parameterized by the number of stable and unstable edges.
• To derive a polynomial kernel of size O(k) for the unstable edge parameterization.

Proposed method

• Reduce the Colored Clustering problem on edge-bicolored graphs to a minimum cut problem to achieve faster computation.
• Construct the condensed graph G* from the original graph G to simplify edge-conflict analysis and kernelization.
• Use the edge-conflict graph X(G*) to model conflict pairs and reduce Conflict-Pair Removal to vertex cover.
• Apply a 1.2783^k-time algorithm for weighted vertex cover on X(G*) to solve Conflict-Pair Removal.
• Leverage the equivalence between Colored Clustering and Alternating Path Removal to transfer algorithmic techniques.
• Derive a kernel with at most 4k vertices and 2k² + k edges for the unstable edge parameterization using structural bounds on colorful vertices and unstable edges.

Experimental results

Research questions

• RQ1 Can the Colored Clustering problem on edge-bicolored graphs be solved faster than O(m^{3/2}n) time using a minimum cut-based approach?
• RQ2 Is Colored Clustering NP-complete for edge-tricolored planar bipartite graphs of maximum degree four?
• RQ3 Can FPT algorithms be developed for Colored Clustering when parameterized by the number of stable edges or unstable edges?
• RQ4 Does a polynomial kernel exist for the problem of finding a vertex-monochromatic subgraph with at least k edges?
• RQ5 Is there an FPT algorithm for destroying all alternating cycles by deleting at most k edges?

Key findings

• An O(m^{3/2} log n)-time algorithm is developed for edge-bicolored graphs by reducing to minimum cut, improving upon the prior O(m^{3/2}n)-time algorithm based on bipartite independent sets.
• The NP-completeness of Colored Clustering is extended to edge-tricolored planar bipartite graphs with maximum degree four.
• FPT algorithms are designed for both stable and unstable edge parameters, with a kernel of size at most 4k vertices and 2k² + k edges for the unstable edge parameter.
• The Conflict-Pair Removal problem is solved in O(m + n + 1.2783^k) time using kernelization and weighted vertex cover on the edge-conflict graph.
• The problem of finding a vertex-monochromatic subgraph with at least k edges admits a kernel of size O(k), suggesting potential for faster exact algorithms.
• The connection between Colored Clustering, vertex-monochromatic subgraphs, and alternating path removal provides a unifying framework for algorithmic analysis.