COCOON 2013: Computing and Combinatorics pp 434-445

# Maximum Balanced Subgraph Problem Parameterized above Lower Bound

• Robert Crowston
• Gregory Gutin
• Mark Jones
• Gabriele Muciaccia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7936)

## Abstract

We consider graphs without loops or parallel edges in which every edge is assigned + or −. Such a signed graph is balanced if its vertex set can be partitioned into parts V 1 and V 2 such that all edges between vertices in the same part have sign + and all edges between vertices of different parts have sign − (one of the parts may be empty). It is well-known that every connected signed graph with n vertices and m edges has a balanced subgraph with at least $$\frac{m}{2} + \frac{n-1}{4}$$ edges and this bound is tight. We consider the following parameterized problem: given a connected signed graph G with n vertices and m edges, decide whether G has a balanced subgraph with at least $$\frac{m}{2} + \frac{n-1}{4}+\frac{k}{4}$$ edges, where k is the parameter.

We obtain an algorithm for the problem of runtime 8 k (kn) O(1). We also prove that for each instance (G,k) of the problem, in polynomial time, we can either solve (G,k) or produce an equivalent instance (G′,k′) such that k′ ≤ k and |V(G′)| = O(k 3). Our first result generalizes a result of Crowston, Jones and Mnich (ICALP 2012) on the corresponding parameterization of Max Cut (when every edge of G has sign −). Our second result generalizes and significantly improves the corresponding result of Crowston, Jones and Mnich for MaxCut: they showed that |V(G′)| = O(k 5).

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## Authors and Affiliations

• Robert Crowston
• 1
• Gregory Gutin
• 1
• Mark Jones
• 1
• Gabriele Muciaccia
• 1
1. 1.Royal Holloway, University of LondonUK