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Maximum Balanced Subgraph Problem Parameterized above Lower Bound

  • Robert Crowston
  • Gregory Gutin
  • Mark Jones
  • Gabriele Muciaccia
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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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

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

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