From: AAAI-00 Proceedings. Copyright © 2000, AAAI (www.aaai.org). All rights reserved. Depth-First Branch-and-Bound versus Local Search: A Case Study Weixiong Zhang Information Sciences Institute and Computer Science Department University of Southern California 4676 Admiralty Way, Marina del Rey, CA 90292 Email: [email protected] Abstract 1983). DFBnB explores a state space in a depth-first order, and finds many suboptimal solutions with in- Depth-first branch-and-bound (DFBnB) is a complete creasingly better qualities. These solutions are approx- algorithm that is typically used to find optimal solu- imations to the optimal solution if DFBnB is termi- tions of difficult combinatorial optimization problems. It can also be adapted to an approximation algorithm nated prematurely. Furthermore, DFBnB is also an and run as an anytime algorithm, which are the sub- anytime algorithm. An anytime algorithm (Dean & jects of this paper. We compare DFBnB against the Boddy 1988) can provide a solution at any time dur- Kanellakis-Papadimitriou local search algorithm, the ing its execution, and is able to improve the qual- best known approximation algorithm, on the asymmet- ity of the current best solution with more computa- ric Traveling Salesman Problem (ATSP), an important tion. DFBnB finds better suboptimal solutions with NP-hard problem. Our experimental results show that more computation and eventually reaches the opti- DFBnB significantly outperforms the local search on mal solution. In contrast to the importance of any- large ATSP and various ATSP structures, finding bet- time problem solving and the effort of developing ter solutions faster than the local search; and the qual- new anytime algorithms (Hoebel & Zilberstein 1997; ity of approximate solutions from a prematurely ter- minated DFBnB, called truncated DFBnB, is several Horvitz & Zilberstein 1996), DFBnB has not been stud- times better than that from the local search. ied as an approximation or anytime algorithm so far. We study DFBnB as an approximation and any- time algorithm in this paper. We compare it Introduction and Overview against the Kanellakis-Papadimitriou local search algo- Depth-first branch-and-bound (DFBnB) (Balas & Toth rithm on the asymmetric Traveling Salesman Problem 1985; Papadimitriou & Steiglitz 1982) and local (ATSP) (Kanellakis & Papadimitriou 1980). This local search (Lin & Kernighan 1973; Kanellakis & Papadim- search algorithm is an adaptation and extension of the itriou 1980; Johnson 1990; Johnson & McGeoch 1997) well-known Lin-Kernighan local search algorithm (Lin are the two most applied search methods for solving & Kernighan 1973), and the only local search algo- combinatorial optimization problems, such as planning rithm for the ATSP which we found in the literature. and scheduling. DFBnB is usually the algorithm for We choose the ATSP due to the following two reasons. finding optimal solutions of large problems, due to the First, the ATSP is an important problem in the NP- virtue of its linear-space requirement. Local search, hard class (Papadimitriou & Steiglitz 1982) and has on the other hand, is a method for high-quality ap- many practical applications. Many difficult combina- proximate solutions, and has been shown to be effec- torial optimization problems, such as vehicle routing, tive and efficient on many combinatorial optimization workshop scheduling and computer wiring, can be for- problems, such as the symmetric Traveling Salesman mulated and solved as the ATSP (Lawler et al. 1985). Problem (TSP) (Lin & Kernighan 1973; Johnson 1990; Second, despite its importance, little work has been Johnson & McGeoch 1997). done on the ATSP, which is disproportional to that on Besides that DFBnB is an efficient complete algo- the symmetric TSP (see (Johnson & McGeoch 1997) rithm for optimal solutions, it can be used as an ap- for an excellent survey and references cited). It will be proximation algorithm1, as suggested in (Ibaraki et al. very helpful if information regarding which algorithm should be used for a particular type of ATSP is avail- Copyright c 2000, American Association for Artificial In- able to guide algorithm selection in practice. telligence (www.aaai.org). All rights reserved. 1We use the term approximation algorithm loosely to re- The paper is structured as follows. We discuss the fer to an algorithm that is able to find a suboptimal solution. ATSP, DFBnB, truncated DFBnB and the Kanellakis- Such an algorithm does not provide a quality guarantee un- Papadimitriou local search algorithm in Section 2. In der the definition of -approximation algorithm (Papadim- Section 3, we describe various ATSP structures used itriou 1994). in our experiments. In Section 4, we investigate initial tour-construction heuristics. In Section 5, we compare selects a recently generated node n to examine next. If DFBnB with the local search algorithm. We discuss the AP solution of n is a complete tour, meaning that n the features of DFBnB and the weakness of the local is a leaf node in the search tree, and its cost is less than search algorithm on the ATSP in Section 6. Finally, we the current upper bound α, α is revised to the cost of conclude and discuss future work in Section 7. n.Ifn’s AP solution is not a complete tour and its cost is greater than or equal to α, n is pruned, because node The Problem and the algorithms costs are non-decreasing along a path from the root so that no descendent of n will have a cost smaller than { ··· } Given n cities, 1, 2, ,n , and a matrix (ci,j) that de- n’s cost. Otherwise, n is expanded, generating all its fines the costs of pairs of cities, the Traveling Salesman child nodes. To find an optimal goal node quickly, the Problem (TSP) is to find a minimum-cost tour that vis- children n should be searched in an increasing order of its each city once and returns to the starting city. When their costs. This is called node ordering. We use node the cost matrix is asymmetric, i.e., the cost from city i ordering in this study. to city j is not necessarily equal to the cost from j to DFBnB can be used as an anytime algorithm. Dur- i, the problem is the asymmetric TSP (ATSP). ing the depth-first exploration of a state space, DF- BnB may encounter many leaf nodes and improve the DFBnB and truncated DFBnB best solution at hand continuously. Furthermore, DF- Branch-and-bound (BnB) (Balas & Toth 1985; Pa- BnB can be stopped at any time during its execution. padimitriou & Steiglitz 1982) solves an ATSP as a This is an extension to the M-Cut strategy suggested state-space search and uses the assignment problem as in (Ibaraki et al. 1983), which terminates BnB with a a lower-bound cost function. The assignment problem fixed amount of computation. We call DFBnB with an (AP) (Martello & Toth 1987; Papadimitriou & Stei- early termination truncated DFBnB. Among all possi- glitz 1982) is to assign to each city i another city j, ble stopping points, of particular interest is where the with ci,j as the cost of the assignment, such that the first leaf node is reached. When no initial tour is used, total cost of all assignments is minimized. The AP is this simple, special truncated DFBnB is a greedy search a relaxation of the ATSP, since the assignments need in a search space, which always chooses to explore next not form a complete tour. Therefore, the AP cost is the minimum-cost child node of the current state until a lower bound on the ATSP tour cost. If the AP so- it reaches a leaf node. When a high-quality initial tour lution happens to be a complete tour, it is also the is employed, DFBnB may not necessarily encounter a solution to the ATSP. The BnB search takes the origi- leaf node before the total allocated computation is ex- nal ATSP as the root of the state space and repeats the hausted. Without confusion, we call DFBnB that ter- following two steps. First, solve the AP of the current minates when it reaches a leaf node or consumes all problem. If the AP solution is not a complete tour, allowed computation truncated DFBnB in the rest of decompose it into subproblems by subtour elimination. this paper. Specifically, select a subtour from the AP solution, and generate subproblems by excluding some edges from Kanellakis-Papadimitriou local search the assignments, so as to eliminate the subtour. There Local search is based on a fundamental concept called are many subtour-elimination heuristics (Balas & Toth neighborhood structure. If two TSP tours differ by λ 1985), and we use the Carpaneto-Toth scheme in our ex- edges, and one can be changed to the other by swap- periments (Carpaneto & Toth 1980), which generates ping the different edges, one tour is a λ-change neighbor no duplicate subproblem. Next, select as the current of the other. A neighborhood structure is established problem a new subproblem that has been generated but by defining the legitimate changes. Within a neighbor- not yet expanded. This process continues until there is hood, a tour is a local optimum if it is the best among no unexpanded problem, or until all unexpanded prob- its neighbors. Given a neighborhood structure, a local lems have costs greater than or equal to the cost of the search moves from a tour to a neighboring tour that has best complete tour found so far. Note that a subprob- a smaller cost until a local optimum is reached. lem is more constrained than its parent problem, there- The Kanellakis-Papadimitriou local search algorithm fore the AP cost to the subproblem must be as much as for the ATSP (Kanellakis & Papadimitriou 1980) fol- that to the parent. This means that the AP cost func- lows the Lin-Kernighan local search algorithm for the tion is monotonically nondecreasing with search depth.