Persist Map of Lists

See com.intellij.openapi.updateSettings.impl.pluginsAdvertisement.PluginsAdvertiser.KnownExtensions for a similar setup and solution

I managed to do it based on the code you suggested, though I'm not sure if the method I save the file is correct.

import com.intellij.configurationStore.XmlSerializer;
import com.intellij.openapi.components.ServiceManager;
import com.intellij.openapi.project.Project;
import com.intellij.openapi.util.JDOMUtil;
import com.intellij.openapi.util.io.FileUtil;
import com.intellij.util.xmlb.annotations.OptionTag;
import com.intellij.util.xmlb.annotations.XMap;
import org.jetbrains.annotations.NotNull;

import java.io.File;
import java.io.IOException;
import java.util.*;

public class ProjectCacheService {

    private static final String CACHE_XML = "cache.xml";

    public ProjectCacheService(Project project) {
    }

    public static ProjectCacheService getInstance(@NotNull Project project) {
        return ServiceManager.getService(project, ProjectCacheService.class);
    }

    public static class CachedMap {
        @OptionTag
        @XMap
        public Map<String, CacheStringList> cacheMap = new HashMap();

        @SuppressWarnings("unused")
        public CachedMap() {
        }

        public CachedMap(Map<String, List<String>> map) {
            Iterator iterator = map.keySet().iterator();

            while (iterator.hasNext()) {
                String key = (String) iterator.next();
                this.cacheMap.put(key, new CacheStringList(map.get(key)));
            }
        }

        public List<String> find(String key) {
            CacheStringList cacheStringList = this.cacheMap.get(key);
            return cacheStringList != null ? cacheStringList.cacheList : null;
        }
    }

    public static class CacheStringList {
        public List<String> cacheList = new ArrayList<>();

        @SuppressWarnings("unused")
        public CacheStringList() {
        }

        public CacheStringList(List<String> list) {
            this.cacheList.addAll(list);
        }
    }

    public static CachedMap loadExtensions(Project currentProject) {
        try {
            File file = getCacheFile(currentProject);
            if (file.isFile()) {
                return XmlSerializer.deserialize(JDOMUtil.load(file), CachedMap.class);
            }
        } catch (Exception e) {
            e.printStackTrace();
        }
        return null;
    }

    public static void saveMapInCache(Map<String, List<String>> map, Project currentProject) throws IOException {
        File file = getCacheFile(currentProject);
        if (!file.isFile()) {
            FileUtil.ensureCanCreateFile(file);
        }
        JDOMUtil.write(XmlSerializer.serialize(new CachedMap(map)), file);
    }

    private static File getCacheFile(Project currentProject) {
        return new File(currentProject.getBasePath(), CACHE_XML);
    }
}

Thanks!