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Edge expansion of a graph: SDP-based computational strategies

Akshay Gupte, Melanie Siebenhofer, Angelika Wiegele

TL;DR

This work tackles the NP-hard problem of computing the edge expansion $h(G)$ (the Cheeger constant) of graphs by introducing two SDP-based exact algorithms. The first splits the problem by fixing the smaller side size $k$ (the $k$-bisection problem), obtaining strong SDP lower bounds and transforming fixed-$k$ instances into max-cut problems solvable by SDP-based solvers, while the second applies Dinkelbach's fractional programming framework to solve the parametrized problems with SDP subproblems. The authors show robust performance up to graphs with about 400 vertices, outperforming standard branch-and-cut solvers on many instances and providing the first SDP-based solvers for this metric. They also provide a practical capability to verify lower bounds on $h(G)$, which is relevant for problems like Mihail-Vazirani on 0/1-polytopes. The results demonstrate the complementary strengths of split-and-bound and Dinkelbach-based strategies across graph families, and point to future directions in convexification and submodularity exploitation to further scale these SDP-based approaches.

Abstract

Computing the edge expansion of a graph is a famously hard combinatorial problem for which there have been many approximation studies. We present two variants of exact algorithms using semidefinite programming (SDP) to compute this constant for any graph. The first variant uses the SDP relaxation first to reduce the search space considerably. The problem is then transformed into instances of max-cut problems, which are solved with an SDP-based state-of-the-art solver. Our second variant to compute the edge expansion uses Dinkelbach's algorithm for fractional programming. This is, we have to solve a parametrized optimization problem and again we use semidefinite programming to obtain solutions of the parametrized problems. Numerical results demonstrate that with our algorithms one can compute the edge expansion on graphs up to 400 vertices in a routine way, including instances where standard branch-and-cut solvers fail. To the best of our knowledge, these are the first SDP-based solvers for computing the edge expansion of a graph.

Edge expansion of a graph: SDP-based computational strategies

TL;DR

This work tackles the NP-hard problem of computing the edge expansion (the Cheeger constant) of graphs by introducing two SDP-based exact algorithms. The first splits the problem by fixing the smaller side size (the -bisection problem), obtaining strong SDP lower bounds and transforming fixed- instances into max-cut problems solvable by SDP-based solvers, while the second applies Dinkelbach's fractional programming framework to solve the parametrized problems with SDP subproblems. The authors show robust performance up to graphs with about 400 vertices, outperforming standard branch-and-cut solvers on many instances and providing the first SDP-based solvers for this metric. They also provide a practical capability to verify lower bounds on , which is relevant for problems like Mihail-Vazirani on 0/1-polytopes. The results demonstrate the complementary strengths of split-and-bound and Dinkelbach-based strategies across graph families, and point to future directions in convexification and submodularity exploitation to further scale these SDP-based approaches.

Abstract

Computing the edge expansion of a graph is a famously hard combinatorial problem for which there have been many approximation studies. We present two variants of exact algorithms using semidefinite programming (SDP) to compute this constant for any graph. The first variant uses the SDP relaxation first to reduce the search space considerably. The problem is then transformed into instances of max-cut problems, which are solved with an SDP-based state-of-the-art solver. Our second variant to compute the edge expansion uses Dinkelbach's algorithm for fractional programming. This is, we have to solve a parametrized optimization problem and again we use semidefinite programming to obtain solutions of the parametrized problems. Numerical results demonstrate that with our algorithms one can compute the edge expansion on graphs up to 400 vertices in a routine way, including instances where standard branch-and-cut solvers fail. To the best of our knowledge, these are the first SDP-based solvers for computing the edge expansion of a graph.
Paper Structure (32 sections, 11 theorems, 48 equations, 3 figures, 1 table, 3 algorithms)

This paper contains 32 sections, 11 theorems, 48 equations, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Introduction
  2. Contribution and outline
  3. Notation
  4. Formulations and SDP relaxations
  5. Semidefinite relaxations
  6. Illustrative examples for motivation
  7. Fixing the size $k$: Bisection problem
  8. SDP lower bounds for the bisection problem
  9. A heuristic for the bisection problem
  10. Transformation to a max-cut problem
  11. Split & bound
  12. Pre-elimination
  13. Impact of pre-elimination on sample instances
  14. Stopping exact computations early and updating $u^*$
  15. Order of selecting values $k$ from $\mathcal{I}$
  16. ...and 17 more sections

Key Result

Proposition 2.1

The optimal solution of the second SDP in eq:SDP-expansion is $\lambda_{2}(L)/2$.

Figures (3)

  • Figure 1: Lower and upper bounds for each $k$.
  • Figure 2: Lower and upper bounds for each $k$.
  • Figure 3: Performance comparison of the exact algorithms. Note the different scale on the $x$-axis: the plot on the left displays the time range from 0 to 500 seconds, the plot on the right from 500 seconds to 3 hours.

Theorems & Definitions (20)

  • Proposition 2.1
  • proof
  • Lemma 3.1
  • proof
  • Corollary 3.2
  • Proposition 4.1
  • proof
  • Proposition 5.1
  • proof
  • Theorem 5.2
  • ...and 10 more