Shaowei Cai

Local search with edge weighting and configuration checking heuristics for minimum vertex cover

The Minimum Vertex Cover (MVC) problem is a well-known combinatorial optimization problem of great importance in theory and applications. In recent years, local search has been shown to be an effective and promising approach to solve hard problems, such as MVC. In this paper, we introduce two new local search algorithms for MVC, called EWLS (Edge Weighting Local Search) and EWCC (Edge Weighting Configuration Checking). The first algorithm EWLS is an iterated local search algorithm that works with a partial vertex cover, and utilizes an edge weighting scheme which updates edge weights when getting stuck in local optima. Nevertheless, EWLS has an instance-dependent parameter. Further, we propose a strategy called Configuration Checking for handling the cycling problem in local search. This is used in designing a more efficient algorithm that has no instance-dependent parameters, which is referred to as EWCC. Unlike previous vertex-based heuristics, the configuration checking strategy considers the induced subgraph configurations when selecting a vertex to add into the current candidate solution. A detailed experimental study is carried out using the well-known DIMACS and BHOSLIB benchmarks. The experimental results conclude that EWLS and EWCC are largely competitive on DIMACS benchmarks, where they outperform other current best heuristic algorithms on most hard instances, and dominate on the hard random BHOSLIB benchmarks. Moreover, EWCC makes a significant improvement over EWLS, while both EWLS and EWCC set a new record on a twenty-year challenge instance. Further, EWCC performs quite well even on structured instances in comparison to the best exact algorithm we know. We also study the run-time behavior of EWLS and EWCC which shows interesting properties of both algorithms.

NuMVC: an efficient local search algorithm for minimum vertex cover

The Minimum Vertex Cover (MVC) problem is a prominent NP-hard combinatorial optimization problem of great importance in both theory and application. Local search has proved successful for this problem. However, there are two main drawbacks in state-of-the-art MVC local search algorithms. First, they select a pair of vertices to exchange simultaneously, which is timeconsuming. Secondly, although using edge weighting techniques to diversify the search, these algorithms lack mechanisms for decreasing the weights. To address these issues, we propose two new strategies: two-stage exchange and edge weighting with forgetting. The two-stage exchange strategy selects two vertices to exchange separately and performs the exchange in two stages. The strategy of edge weighting with forgetting not only increases weights of uncovered edges, but also decreases some weights for each edge periodically. These two strategies are used in designing a new MVC local search algorithm, which is referred to as NuMVC. We conduct extensive experimental studies on the standard benchmarks, namely DIMACS and BHOSLIB. The experiment comparing NuMVC with state-of-the-art heuristic algorithms show that NuMVC is at least competitive with the nearest competitor namely PLS on the DIMACS benchmark, and clearly dominates all competitors on the BHOSLIB benchmark. Also, experimental results indicate that NuMVC finds an optimal solution much faster than the current best exact algorithm for Maximum Clique on random instances as well as some structured ones. Moreover, we study the effectiveness of the two strategies and the run-time behaviour through experimental analysis.

Local search for Boolean Satisfiability with configuration checking and subscore

This paper presents and analyzes two new efficient local search strategies for the Boolean Satisfiability (SAT) problem. We start by proposing a local search strategy called configuration checking (CC) for SAT. The CC strategy results in a simple local search algorithm for SAT called Swcc, which shows promising experimental results on random 3-SAT instances, and outperforms TNM, the winner of SAT Competition 2009. However, the CC strategy for SAT is still in a nascent stage, and Swcc cannot yet compete with Sparrow2011, which won SAT Competition 2011 just after Swcc had been designed. The CC strategy seems too strict in that it forbids flipping those variables even with great scores, if they do not satisfy the CC criterion. We improve the CC strategy by adopting an aspiration mechanism, and get a new variable selection heuristic called configuration checking with aspiration (CCA). The CCA heuristic leads to an improved algorithm called Swcca, which exhibits state-of-the-art performance on random 3-SAT instances and crafted ones. The third contribution concerns improving local search algorithms for random k-SAT instances with k>3. Although the SAT community has made great achievements in solving random 3-SAT instances, the progress lags far behind on random k-SAT instances with k>3. This work proposes a new variable property called subscore, which is utilized to break ties in the CCA heuristic when candidate variables for flipping have the same score. The resulting algorithm CCAsubscore is very efficient for solving random k-SAT instances with k>3, and significantly outperforms other state-of-the-art ones. Combining Swcca and CCAsubscore, we obtain a local search SAT solver called CCASat, which was ranked first in the random track of SAT Challenge 2012. Additionally, we perform theoretical analyses on the CC strategy and the subscore property, and show interesting results on these two heuristics. Particularly, our analysis indicates that the CC strategy is more effective for k-SAT with smaller k, while the subscore notion is not suitable for solving random 3-SAT.

CCLS: An Efficient Local Search Algorithm for Weighted Maximum Satisfiability

The maximum satisfiability (MAX-SAT) problem, especially the weighted version, has extensive applications. Weighted MAX-SAT instances encoded from real-world applications may be very large, which calls for efficient approximate methods, mainly stochastic local search (SLS) ones. However, few works exist on SLS algorithms for weighted MAX-SAT. In this paper, we propose a new heuristic called CCM for weighted MAX-SAT. The CCM heuristic prefers to select a CCMP variable. By combining CCM with random walk, we design a simple SLS algorithm dubbed CCLS for weighted MAX-SAT. The CCLS algorithm is evaluated against a state-of-the-art SLS solver IRoTS and two state-of-the-art complete solvers namely akmaxsat_ls and New WPM2, on a broad range of weighted MAX-SAT instances. Experimental results illustrate that the quality of solution found by CCLS is much better than that found by IRoTS, akmaxsat_ls and New WPM2 on most industrial, crafted and random instances, indicating the efficiency and the robustness of the CCLS algorithm. Furthermore, CCLS is evaluated in the weighted and unweighted MAX-SAT tracks of incomplete solvers in the Eighth Max-SAT Evaluation (Max-SAT 2013), and wins four tracks in this evaluation, illustrating that the performance of CCLS exceeds the current state-of-the-art performance of SLS algorithms on solving MAX-SAT instances.

TCA: An Efficient Two-Mode Meta-Heuristic Algorithm for Combinatorial Test Generation (T)

Covering arrays (CAs) are often used as test suites for combinatorial interaction testing to discover interaction faults of real-world systems. Most real-world systems involve constraints, so improving algorithms for covering array generation (CAG) with constraints is beneficial. Two popular methods for constrained CAG are greedy construction and meta-heuristic search. Recently, a meta-heuristic framework called two-mode local search has shown great success in solving classic NPhard problems. We are interested whether this method is also powerful in solving the constrained CAG problem. This work proposes a two-mode meta-heuristic framework for constrained CAG efficiently and presents a new meta-heuristic algorithm called TCA. Experiments show that TCA significantly outperforms state-of-the-art solvers on 3-way constrained CAG. Further experiments demonstrate that TCA also performs much better than its competitors on 2-way constrained CAG.

Local Search with Configuration Checking for SAT

Local Search is an appealing method for solving the Boolean Satisfiability problem (SAT). However, this method suffers from the cycling problem which severely limits its power. Recently, a new strategy called configuration checking (CC) was proposed, for handling the cycling problem in local search. The CC strategy was used to improve a state-of the-art local search algorithm for Minimum Vertex Cover. In this paper, we propose a novel local search strategy for the satisfiability problem, i.e., the CC strategy for SAT. The CC strategy for SAT takes into account the circumstances of the variables when selecting a variable to flip, where the circumstance of a variable refers to truth values of all its neighboring variables. We then apply it to design a local search algorithm for SAT called SWcc (Smoothed Weighting with Configuration Checking). Experimental results show that the CC strategy for SAT is more efficient than the previous strategy for handling the cycling problem called tabu. Moreover, SWcc significantly outperforms the best local search SAT solver in SAT Competition 2009 called TNM on large random 3-SAT instances.

Focused random walk with configuration checking and break minimum for satisfiability

Stochastic local search SLS algorithms, especially those adopting the focused random walk FRW framework, have exhibited great effectiveness in solving satisfiable random 3-satisfiability 3-SAT instances. However, they are still unsatisfactory in dealing with huge instances, and are usually sensitive to the clause-to-variable ratio of the instance. In this paper, we present a new FRW algorithm dubbed FrwCB, which behaves more satisfying in the above two aspects. The main idea is a new heuristic called CCBM, which combines a recent diversification strategy named configuration checking CC with the common break minimum BM variable-picking strategy. By combining CC and BM in a subtle way, CCBM significantly improves the performance of FrwCB, making FrwCB achieve state-of-the-art performance on a wide range of benchmarks. The experiments show that FrwCB significantly outperforms state-of-the-art SLS solvers on random 3-SAT instances, and competes well on random 5-SAT, random 7-SAT and structured instances.

Clause States Based Configuration Checking in Local Search for Satisfiability

Two-mode stochastic local search (SLS) and focused random walk (FRW) are the two most influential paradigms of SLS algorithms for the propositional satisfiability (SAT) problem. Recently, an interesting idea called configuration checking (CC) was proposed to handle the cycling problem in SLS. The CC idea has been successfully used to improve SLS algorithms for SAT, resulting in state-of-the-art solvers. Previous CC strategies for SAT are based on neighboring variables, and prove successful in two-mode SLS algorithms. However, this kind of neighboring variables based CC strategy is not suitable for improving FRW algorithms. In this paper, we propose a new CC strategy which is based on clause states. We apply this clause states based CC (CSCC) strategy to both two-mode SLS and FRW paradigms. Our experiments show that the CSCC strategy is effective on both paradigms. Furthermore, our developed FRW algorithms based on CSCC achieve state-of-the-art performance on a broad range of random SAT benchmarks.

Local Search for Minimum Weight Dominating Set with Two-Level Configuration Checking and Frequency Based Scoring Function

The Minimum Weight Dominating Set (MWDS) problem is an important generalization of the Minimum Dominating Set (MDS) problem with extensive applications. This paper proposes a new local search algorithm for the MWDS problem, which is based on two new ideas. The first idea is a heuristic called two-level configuration checking (CC2), which is a new variant of a recent powerful configuration checking strategy (CC) for effectively avoiding the recent search paths. The second idea is a novel scoring function based on the frequency of being uncovered of vertices. Our algorithm is called CC2FS, according to the names of the two ideas. The experimental results show that, CC2FS performs much better than some state-of-the-art algorithms in terms of solution quality on a broad range of MWDS benchmarks.

CCEHC: An efficient local search algorithm for weighted partial maximum satisfiability

Weighted maximum satisfiability and (unweighted) partial maximum satisfiability (PMS) are two significant generalizations of maximum satisfiability (MAX-SAT), and weighted partial maximum satisfiability (WPMS) is the combination of the two, with more important applications in practice. Recently, great breakthroughs have been made on stochastic local search (SLS) for weighted MAX-SAT and PMS, resulting in several state-of-the-art SLS algorithms CCLS, Dist and DistUP. However, compared to the great progress of SLS on weighted MAX-SAT and PMS, the performance of SLS on WPMS lags far behind. In this paper, we present a new SLS algorithm named CCEHC for WPMS. CCEHC employs an extended framework of CCLS with a heuristic emphasizing hard clauses, called EHC. With strong accents on hard clauses, EHC has three components: a variable selection mechanism focusing on configuration checking based only on hard clauses, a weighting scheme for hard clauses, and a biased random walk component. Extensive experiments demonstrate that CCEHC significantly outperforms its state-of-the-art SLS competitors. Further experimental results on comparing CCEHC with a state-of-the-art complete solver show the effectiveness of CCEHC on a number of application WPMS instances, and indicate that CCEHC might be beneficial in practice. Also, empirical analyses confirm the effectiveness of each component underlying the EHC heuristic.