TY - GEN

T1 - Multiway k-Cut in Static and Dynamic Graphs

T2 - 58th IEEE Conference on Decision and Control, CDC 2019

AU - Baranwal, Mayank

AU - Srivastava, Amber

AU - Salapaka, Srinivasa M.

N1 - Funding Information:
1Department of Electrical and Computer Engineering at the University of Michigan, Ann-Arbor, 48109 MI, USA. mayankb@umich.edu 2Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, 61801 IL, USA. aasrvstv6@illinois.edu,bsalapaka@illinois.edu The authors would like to acknowledge NSF grants ECCS 15-09302, CMMI 14-63239 and CNS 15-44635 for supporting this work.

PY - 2019/12

Y1 - 2019/12

N2 - This work presents a maximum entropy principle based algorithm for solving minimum multiway k-cut problem defined over static and dynamic digraphs. A multiway k-cut problem requires partitioning the set of nodes in a graph into k subsets, such that each subset contains one prespecified node, and the corresponding total cut weight is minimized. These problems arise in many applications and are computationally complex (NP-hard). In the static setting this article presents an approach that uses a relaxed multiway k-cut cost function; we show that the resulting algorithm converges to a local minimum. This iterative algorithm is designed to avoid poor local minima with its run-time complexity as ~ O(kIN3), where N is the number of vertices and I is the number of iterations. In the dynamic setting, the edge-weight matrix has an associated dynamics with some of the edges in the graph capable of being influenced by an external input. The objective is to design the dynamics of the controllable edges so that multiway k-cut value remains small (or decreases) as the graph evolves under the dynamics. Also it is required to determine the time-varying partition that defines the minimum multiway k-cut value. Our approach is to choose a relaxation of multiway k-cut value, derived using maximum entropy principle, and treat it as a control Lyapunov function to design control laws that affect the weight dynamics. Simulations on practical examples of interactive foreground-background segmentation, minimum multiway k-cut optimization for non-planar graphs and dynamically evolving graphs that demonstrate the efficacy of the algorithm, are presented.

AB - This work presents a maximum entropy principle based algorithm for solving minimum multiway k-cut problem defined over static and dynamic digraphs. A multiway k-cut problem requires partitioning the set of nodes in a graph into k subsets, such that each subset contains one prespecified node, and the corresponding total cut weight is minimized. These problems arise in many applications and are computationally complex (NP-hard). In the static setting this article presents an approach that uses a relaxed multiway k-cut cost function; we show that the resulting algorithm converges to a local minimum. This iterative algorithm is designed to avoid poor local minima with its run-time complexity as ~ O(kIN3), where N is the number of vertices and I is the number of iterations. In the dynamic setting, the edge-weight matrix has an associated dynamics with some of the edges in the graph capable of being influenced by an external input. The objective is to design the dynamics of the controllable edges so that multiway k-cut value remains small (or decreases) as the graph evolves under the dynamics. Also it is required to determine the time-varying partition that defines the minimum multiway k-cut value. Our approach is to choose a relaxation of multiway k-cut value, derived using maximum entropy principle, and treat it as a control Lyapunov function to design control laws that affect the weight dynamics. Simulations on practical examples of interactive foreground-background segmentation, minimum multiway k-cut optimization for non-planar graphs and dynamically evolving graphs that demonstrate the efficacy of the algorithm, are presented.

UR - http://www.scopus.com/inward/record.url?scp=85082503957&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85082503957&partnerID=8YFLogxK

U2 - 10.1109/CDC40024.2019.9029914

DO - 10.1109/CDC40024.2019.9029914

M3 - Conference contribution

AN - SCOPUS:85082503957

T3 - Proceedings of the IEEE Conference on Decision and Control

SP - 3948

EP - 3953

BT - 2019 IEEE 58th Conference on Decision and Control, CDC 2019

PB - Institute of Electrical and Electronics Engineers Inc.

Y2 - 11 December 2019 through 13 December 2019

ER -