Distance-guided local search

Abstract

We present several techniques that use distances between candidate solutions to achieve intensification in Local Search (LS) algorithms. An important drawback of classical LS is that after visiting a very high-quality solution the search process can “forget about it” and continue towards very different areas. We propose a method that works on top of a given LS to equip it with a form of memory so as to record the highest-quality visited areas (spheres). More exactly, this new method uses distances between candidate solutions to perform a coarse–grained recording of the LS trajectory, i.e., it records a number of discovered spheres. The (centers of the) spheres are kept sorted in a priority queue in which new centers are continually inserted as in insertion-sort algorithms. After thoroughly investigating a sphere, the proposed method resumes the search from the first best sphere center in the priority queue. The resulting LS trajectory is no longer a continuous path, but a tree-like structure, with closed branches (already investigated spheres) and open branches (as-yet-unexplored spheres). We also explore several other techniques relying on distances, e.g., in Section 2.3, we show how to use distance information to prevent the search from looping indefinitely on large (quasi-)plateaus. Experiments on three problems based on different encodings (partitions, vectors and permutations) confirm the intensification potential of the proposed ideas.

Appendices

A The underlying local searches and their streamlined calculations

1.1 A.1 Graph k-coloring

The underlying LS for graph k-coloring is the Tabu Search (TS) from Porumbel et al. (2010). A solution s is represented as an array of length n such that $s_v$ is the color of vertex v. A neighboring solution can be obtained by simply changing the color $s_v$ of any conflicting vertex v to some $s'_v$. By focusing on conflicting vertices, this neighborhood helps the search process to concentrate on influential moves and to avoid irrelevant ones, because changing the color of a non-conflicting vertex would not directly improve the objective function.

After executing a move and assigning a new color to a vertex v, v can not receive again the lost color for the next $T_\ell$ iterations. The value of $T_\ell$ is set at $\texttt {random}(10)+0.6\cdot obj(s)+ \Big \lfloor \frac{iters_{\texttt {plat}}}{1000}\Big \rfloor$, where $iters_{\texttt {plat}}$ is the number of last moves with no objective function variation. The last term is only introduced to change $T_\ell$ when the algorithm is blocked looping on a plateau and the objective value does not change for 1000 moves. Each series of consecutive 1000 moves with no objective function variation triggers the increment of all subsequent tabu list lengths, which encourages TS to choose more and more moves that have not been executed in the past, until the objective changes again and TS leaves the plateau. This additional term prevents the search process from getting blocked looping on a plateau while not affecting its behavior outside plateaus.

To rapidly choose the best neighbor of s, this TS uses a $n \times k$ table $\Gamma$ such that $\Gamma _{v,s'_v}$ indicates the number of conflicts of v if v received color $s'_v$. As such, $\Gamma _{v,s'_v}-\Gamma _{v,s_v}$ represents the objective function variation associated to the move that changes the color of v from $s_v$ into $s'_v$. After performing a move, $\Gamma$ can be updated in O(n) time (because only columns $s_v$ and $s'_v$ might require updating).

1.2 A.2 k-cluster: incremental calculations of objective value and distance

The main ideas of the k-clique Tabu Search (TS) algorithm were presented in the first paragraphs of Sect. 4.2. We here describe how it uses incremental calculations to rapidly find the best swap of vertices at each iteration. For this, the TS uses a table that associates to each non-selected vertex $v^{\text {out}}$ the number of edges that it can bring to the current solution. For a selected vertex $v^{\text {in}}$, this table records the number of edges linked to $v^{\text {in}}$ in the current solution. To find the best swap, it is enough to consider each selected vertex $v^{\text {in}}$ and each non-selected one $v^{\text {out}}$ and to calculate (in constant time using the above table!) the objective function variation of swapping $v^{\text {in}}$ with $v^{\text {out}}$. After executing the move, the table values of $v^{\text {in}}$ and $v^{\text {out}}$ are quite easily updated, by scanning their neighbors modified by the last move. For a more complex and faster calculation streamlining scheme, we refer the reader to Qinghua and Hao (2013). However, using a slower (and more pedagogical) algorithm poses no problem for the empirical evaluations needed in this paper.

The calculation of the distance from the current solution s to the current center c is also incremental. If $s'$ is obtained from s by swapping a and b, then

$$\begin{aligned} d(s',c)= & {} d(s,c)- \underbrace{\Big ([s_a\ne c_a] + [s_b\ne c_b]\Big )}_{ old \,contribution \,to \,the \,Hamming \,distance} \\&+ \underbrace{\Big ([s_b\ne c_a]+[s_a\ne c_b]\Big )}_{new\, contribution \,to \,the \,Hamming \,distance\, }, \end{aligned}$$

where [S] is the Iverson bracket, i.e., [S] is 1 when the statement S is true and 0 otherwise. If the move consists of deselecting a selected vertex $a=v^{\text {in}}$ and of selecting a non-selected vertex $b=v^{\text {out}}$, the above formula becomes

$$\begin{aligned} d(s',c)= d(s,c)- \Big ([1 \ne c_a] + [0\ne c_b]\Big ) + \Big ([0\ne c_a] +[1\ne c_b]\Big ). \end{aligned}$$

One can check all possible cases of $c_a$ and $c_b$ to see this leads to the following simpler formula:

$$\begin{aligned} d(s',c)=d(s,c)+2\cdot c_a-2\cdot c_b. \end{aligned}$$

1.3 A.3 Capacitated arc-routing (CARP)

The underlying LS for CARP is based on a simplification of the Iterated Local Search (ILS) from Porumbel et al. (2017). The original ILS considers a search space of permutations that are decoded into explicit routes using a decoder (see below). The main simplifications are the following. First, all Column Generation (CG) components of the algorithm from Porumbel et al. (2017) are removed, allowing one to more easily compare LS with DGLS, using less external components. Secondly, the neighborhood is restricted to only use adjacent transpositions (swaps), i.e., a neighbor permutation is constructed by swapping consecutive elements of the current permutation. This allows one to achieve a correlation between a distance $d(s_a, s_b)$ and the number of LS moves needed to reach $s_a$ from $s_b$. We do not use any post-decoder as in Porumbel et al. (2017).

The perturbation operator of this ILS consists of inserting in the current solution a route (sequence) discovered earlier during the search (Porumbel et al. 2017, §2.1). More exactly, to perturb the current permutation s, we extract a route r from a pool, we inject r at the beginning of s and we remove from s any duplicate element of r. The pool is continually updated throughout the search, by adding routes discovered by the ILS at different moments of the search.

Finally, the decoder consists of a dynamic programming routine of linear complexity in terms of the number of clients $|E_R|$, i.e., the complexity is $O(|E_R|)$. More precisely, given input permutation $s = (s_1 , s_2 , \dots s_m )$, the decoder determines a set of routes of minimum total cost that service all required edges in the order $s_1,~s_2,\dots s_m$. Since the decoder is relatively computationally intensive, the distance calculations do not introduce an important slowdown in the search.

B MDS plots of other DGLS trajectories for graph k-coloring

C The success rate of DGLS for different values of runsPerSphere and maxRadius

Here, we analyze the effectiveness of DGLS over several values of runsPerSphere and maxRadius on the graph coloring and the k-clique problem (in Table 7 and resp. Table 8). The caption of Tables 7 and 8 is self explanatory.

Table 7 The success rate of DGLS for different value of $\texttt {maxRadius}$ (above table) or $\texttt {runsPerSphere}$ (below table) on graph coloring

Table 8 The success rate of DGLS for different value of $\texttt {maxRadius}$ (above table) or $\texttt {runsPerSphere}$ (below table) on the k-clique problem