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Anytime Cooperative Implicit Hitting Set Solving

Emma Rollón, Javier Larrosa, Aleksandra Petrova

TL;DR

This work addresses efficient WCSP optimization with anytime guarantees by extending the implicit hitting set (HS) framework into a cooperative, parallel scheme. It introduces HS-lb (lower-bound oriented), HS-ub (upper-bound oriented), and notably HS-lub, a shared-memory multithreaded algorithm where both HS-lb and HS-ub run concurrently and exchange cores and bounds, yielding improved convergence and tighter optimality gaps $w^*-lb$ and $w^*-ub$. Empirical results on standard WCSP benchmarks show HS-lub often outperforms its components and, in several cases, competitive with the parallel hbfs/Toulbar2 approach, highlighting the practical viability of a simple, anytime HS strategy. The work also outlines future improvements, including richer HS techniques and solver adaptations, to further enhance performance on large-domain WCSPs.

Abstract

The Implicit Hitting Set (HS) approach has shown to be very effective for MaxSAT, Pseudo-boolean optimization and other boolean frameworks. Very recently, it has also shown its potential in the very similar Weighted CSP framework by means of the so-called cost-function merging. The original formulation of the HS approach focuses on obtaining increasingly better lower bounds (HS-lb). However, and as shown for Pseudo-Boolean Optimization, this approach can also be adapted to compute increasingly better upper bounds (HS-ub). In this paper we consider both HS approaches and show how they can be easily combined in a multithread architecture where cores discovered by either component are available by the other which, interestingly, generates synergy between them. We show that the resulting algorithm (HS-lub) is consistently superior to either HS-lb and HS-ub in isolation. Most importantly, HS-lub has an effective anytime behaviour with which the optimality gap is reduced during the execution. We tested our approach on the Weighted CSP framework and show on three different benchmarks that our very simple implementation sometimes outperforms the parallel hybrid best-first search implementation of the far more developed state-of-the-art Toulbar2.

Anytime Cooperative Implicit Hitting Set Solving

TL;DR

This work addresses efficient WCSP optimization with anytime guarantees by extending the implicit hitting set (HS) framework into a cooperative, parallel scheme. It introduces HS-lb (lower-bound oriented), HS-ub (upper-bound oriented), and notably HS-lub, a shared-memory multithreaded algorithm where both HS-lb and HS-ub run concurrently and exchange cores and bounds, yielding improved convergence and tighter optimality gaps and . Empirical results on standard WCSP benchmarks show HS-lub often outperforms its components and, in several cases, competitive with the parallel hbfs/Toulbar2 approach, highlighting the practical viability of a simple, anytime HS strategy. The work also outlines future improvements, including richer HS techniques and solver adaptations, to further enhance performance on large-domain WCSPs.

Abstract

The Implicit Hitting Set (HS) approach has shown to be very effective for MaxSAT, Pseudo-boolean optimization and other boolean frameworks. Very recently, it has also shown its potential in the very similar Weighted CSP framework by means of the so-called cost-function merging. The original formulation of the HS approach focuses on obtaining increasingly better lower bounds (HS-lb). However, and as shown for Pseudo-Boolean Optimization, this approach can also be adapted to compute increasingly better upper bounds (HS-ub). In this paper we consider both HS approaches and show how they can be easily combined in a multithread architecture where cores discovered by either component are available by the other which, interestingly, generates synergy between them. We show that the resulting algorithm (HS-lub) is consistently superior to either HS-lb and HS-ub in isolation. Most importantly, HS-lub has an effective anytime behaviour with which the optimality gap is reduced during the execution. We tested our approach on the Weighted CSP framework and show on three different benchmarks that our very simple implementation sometimes outperforms the parallel hybrid best-first search implementation of the far more developed state-of-the-art Toulbar2.
Paper Structure (16 sections, 1 theorem, 5 figures, 3 tables, 2 algorithms)

This paper contains 16 sections, 1 theorem, 5 figures, 3 tables, 2 algorithms.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. CSPs and WCSPs
  5. Vectors and Dominance
  6. Cores and Solutions
  7. Two HS-based Schemes
  8. HS-lb
  9. HS-ub
  10. Computing Maximal Cores
  11. Discussion
  12. Any-time HS scheme
  13. Experimental Results
  14. Evaluation setting
  15. Results
  16. ...and 1 more sections

Key Result

Theorem 1

Consider a solution $\vec{h}$ and a set of cores $\mathcal{K}$. Then, $MHV(\mathcal{K})\leq w^* \leq cost(\vec{h})$.

Figures (5)

  • Figure 1: A WCSP with three variables $\{x_1,x_2,x_3\}$ and two cost-functions $F=\{f(\cdot),g(\cdot)\}$ (left). Its vector space, where red and blue dots correspond to cores and solutions, respectively (right).
  • Figure 2: Graphical representation of an arbitrary iteration of HS-lb ($\vec{h}$ and $\vec{k}$) and HS-ub ($\vec{h}'$ and $\vec{k}$). Current $lb$ is $6$, the optimal value has cost $w^*=8$ and $ub>10$. Blue dots represent solutions, red dots cores and the green dot the optimal solution.
  • Figure 3: Two selected executions of HS-lb (solid blue) and HS-ub (dashed red). Plots on the top row (resp. bottom row) show the evolution of bounds (resp. number of accumulated cores) as a function of time.
  • Figure 4: Selected executions of HS-lub and hbfs-16 on instances from SPOT5 (1506 and 414), and Maxclique (brock400-1 and p-hat300-1) benchmarks. Plots show the evolution of the upper and lower bounds as a function of time.
  • Figure 5: Results on EHI-85 (left column) and EHI-90 (right column) benchmarks. Plots on the first row (note the log scale) show, for each instance (each dot), solving time of HS-lub wrt hbfs-16. Black line indicates equal solving times. Plots on the second row show average lower and upper bounds of HS-lub and hbfs-16 as a function of time. Each benchmark has $100$ instances. The average is meaningful because the optimum is $9$ for all of them.

Theorems & Definitions (1)

  • Theorem 1