TSP is a classic and extensively studied problem with numerous real-world applications in artificial intelligence and operations research. It is well-known that TSP admits a constant approximation ratio on metric graphs but becomes NP-hard to approximate within any computable function f(n) on general graphs. This disparity highlights a significant gap between the results on metric graphs and general graphs. Recent research has introduced some parameters to measure the ``distance'' of general graphs from being metric and explored FPT approximation algorithms parameterized by these parameters. Two commonly studied parameters are p, the number of vertices in triangles violating the triangle inequality, and q, the minimum number of vertices whose removal results in a metric graph. In this paper, we present improved FPT approximation algorithms with respect to these two parameters. For p, we propose an FPT algorithm with a 1.5-approximation ratio, improving upon the previous ratio of 2.5. For q, we significantly enhance the approximation ratio from 11 to 3, advancing the state of the art in both cases.

In some professional sports leagues, inter-league games are scheduled among multiple divisions or conferences. This inspired us to study the p-partite Traveling Tournament Problem (p-partite TTP), where teams are partitioned into p leagues, and each team plays games against teams from different leagues. Previously, only the case of p=2, known as the Bipartite TTP or BTTP, has been introduced and studied. In this paper, we show that the p-partite TTP is NP-hard for any fixed p≥3, and we propose an efficient algorithm based on a solution to the Traveling Salesman Problem. Furthermore, we prove that the algorithm achieves a notable approximation ratio of 8/3+O(1/n) when p=3. We also conduct experiments demonstrating that the algorithm produces practical schedules with significantly reduced total travel distances, highlighting its effectiveness in generating high-quality multipartite tournament schedules.

The Ride-Sharing Assignment Problem (AAAI 2018) is a fundamental problem in intelligent transportation systems, urban mobility, and algorithmic decision-making. Given a set of m vehicles with initial locations and n requests (n≤mk), each with a specified origin and destination, the goal is to assign at most k requests to each vehicle and compute corresponding routes that minimize the total travel distance. The algorithmic approach depends on whether n=mk or n

We consider a novel vertex cover problem on temporal graphs, where the edges in the graph may change over time, and a vertex selected into the solution has a lifespan d. Specifically, a vertex selected at time t can cover all incident edges in graphs from time slot t to t+d-1. This model effectively captures the scenario of monitoring communication links via secure nodes (monitors) with limited lifespan in a dynamic network. We provide a systematic study of this problem from both theoretical and practical perspectives. We analyze its computational complexity, develop approximation and online algorithms with tight ratios, and present a parameterized algorithm and a tight quadratic kernel under fixed d. Experimental results on random and real-world temporal networks demonstrate the effectiveness of our algorithms. We believe that our systematic study not only reveals the nature of the problem itself, but also paves the way for investigating the ''sustained'' version of other problems on temporal graphs.

In the Clustered TSP (CTSP), we are given an edge-weighted graph satisfying the triangle inequality property, and a family of pairwise disjoint vertex groups. The goal is to find a minimum weight tour that includes all vertices, ensuring that the vertices within each group appear consecutively on the tour. The subgroup planning problem (SGPP) is an extension of CTSP by relaxing some triangle inequality requirements on edge weights. CTSP and SGPP have plentiful applications in AI and robotics. In this paper, we design three improved approximation algorithms for SGPP and CTSP. First, we propose a polynomial-time 2.167-approximation algorithm for SGPP, improving the previous ratio of 3 (IJCAI 2017). Second, we give an FPT 2.072-approximation algorithm for SGPP parameterized by the maximum group size, improving the previous ratio of 2.5 (IJCAI 2017). Third, we prove an FPT (β

The Traveling Tournament Problem (TTP-k) is a well-known benchmark problem in tournament timetabling. It involves designing a feasible double round-robin tournament for a sports league of n teams under several feasibility requirements, while minimizing the total traveling costs of the teams. The parameter k requires that in the tournament at most k consecutive home games or away games for each team are allowed. TTP-k with a small k, especially for k=2,3 and 4, have been extensively studied in the literature. In this paper, we focus on TTP-4 and design an efficient algorithm for it based on minimum weight matching. In theory, we prove that our algorithm has an approximation ratio of 1.625+ε for any constant ε>0, improving the best-known approximation ratio of 1.7+ε. In practice, our experimental results indicate an average improvement of 6.65% over the best-known solutions on 9 benchmark instances.

The Capacitated Vehicle Routing Problem (CVRP) is one of the most extensively studied problems in combinatorial optimization. Based on customer demand, we distinguish three variants of CVRP: unit-demand, splittable, and unsplittable. In this paper, we consider k-CVRP in general metrics and on general graphs, where k is the vehicle capacity. All three versions are APX-hard for any fixed k ≥ 3. Assume that the approximation ratio of metric TSP is 3/2. We present a (5/2 - Θ(√{1/k}))-approximation algorithm for the splittable and unit-demand cases, and a (5/2 + ln 2 - Θ(√{1/k}))-approximation algorithm for the unsplittable case. Our approximation ratio is better than the previous results when k is less than a sufficiently large value, approximately 1.7 x 10⁷. For small values of k, we design independent and elegant algorithms with further improvements. For the splittable and unit-demand cases, we improve the approximation ratio from 1.792 to 1.500 for k = 3, and from 1.750 to 1.500 for k = 4. For the unsplittable case, we improve the approximation ratio from 1.792 to 1.500 for k = 3, from 2.051 to 1.750 for k = 4, and from 2.249 to 2.157 for k = 5. The approximation ratio for k = 3 surprisingly achieves the same value as in the splittable case. Our techniques, such as EX-ITP - an extension of the classic ITP method, have the potential to improve algorithms for other routing problems as well.

The Traveling Tournament Problem (TTP-k) is a well-known benchmark problem in tournament timetabling and has been extensively studied in the field of AI. In this problem, we are going to design a double round-robin schedule such that each pair of teams plays one game in each other's home venue, minimizing the total distance traveled by all n teams (n is even) under the constraint that each team can have at most k-consecutive home games or away games. The Linear Distance Traveling Tournament Problem (LDTTP-k), where all teams are located on a line, was introduced by Hoshino and Kawarabayashi (AAAI 2012). For LDTTP-3, they gave a 4/3-approximation algorithm for n≡4 (mod 6) teams. In this paper, we show that for any 3≤k=o(∛n), LDTTP-k allows an efficient polynomial-time approximation scheme (EPTAS).

Network protection has drawn a certain interest in network optimization. One of the most effective and widely used methods to protect networks from failures is to establish backup paths for working paths. For example, we find${k}$node-disjoint paths between a source and a sink with one working path and${k}\,\,-$1 backup paths. However, the demand for full protection of a network is somewhat too restrictive and there may not exist${k}$node-disjoint paths in the network due to the limitation of geographical environments. On the other hand, the occurrence probability of node failures is usually much less than that of link failures in real-world models. To save network resources, we turn to establish link-disjoint paths allowing a few shared nodes. We study the problem of finding${k}$link-disjoint paths between a source and a sink under the constraint that the number of nodes shared by at least${r}$paths is at most$\delta $, minimizing the total link weight. First, we systematically study the computational complexity of the problem with respect to three parameters${k}$,${r}$, and$\delta $. Then, we build an integer linear programming for the general model and design a polynomial-time algorithm for the case that${k}\,\,=\,\,{r}$by using the techniques of augmenting paths and splitting nodes. Finally, we carry out experimentations on synthetic and real networks that show the effectiveness of our algorithms in practice.

The Traveling Tournament Problem (TTP) is a well-known benchmark problem in the field of tournament timetabling, which asks us to design a double round-robin schedule such that each pair of teams plays one game in each other’s home venue, minimizing the total distance traveled by all n teams (n is even). TTP-k is the problem with one more constraint that each team can have at most k consecutive home games or away games. The case where k = 3, TTP-3, is one of the most investigated cases. In this paper, we improve the approximation ratio of TTP-3 from (1.667+ε) to (1.598+ε), for any ε > 0. Previous schedules were constructed based on a Hamiltonian cycle of the graph. We propose a novel construction based on triangle packing. Then, by combining our triangle packing schedule with the Hamiltonian cycle schedule, we obtain the improved approximation ratio. The idea of our construction can also be extended to k ≥ 4. We demonstrate that the approximation ratio of TTP-4 can be improved from (1.750+ε) to (1.700+ε) by the same method. As an additional product, we also improve the approximation ratio of LDTTP-3 (TTP-3 where all teams are allocated on a straight line) from 4/3 to (6/5+ε).

The Traveling Tournament Problem is a well-known benchmark problem in tournament timetabling, which asks us to design a schedule of home/away games of n teams (n is even) under some feasibility requirements such that the total traveling distance of all the n teams is minimized. In this paper, we study TTP-2, the traveling tournament problem where at most two consecutive home games or away games are allowed, and give an effective algorithm for n/2 being odd. Experiments on the well-known benchmark sets show that we can beat previously known solutions for all instances with n/2 being odd by an average improvement of 2.66%. Furthermore, we improve the theoretical approximation ratio from 3/2+O(1/n) to 1+O(1/n) for n/2 being odd, answering a challenging open problem in this area.

Network survivability has drawn certain interest in network optimization. However, the demand for full protection of a network is usually too restrictive. To overcome the limitation of geographical environments and to save network resources, we turn to establish backup networks allowing a few common nodes. It comes out the problem of finding k link-disjoint paths between a given pair of source and sink in a network such that the number of common nodes shared by at least two paths is bounded by a constant and the total link weight of all paths is minimized under the above constraints. For the case k = 2, where we have only one backup path, several fast algorithms have been developed in the literature. For the case k > 2, little results are known. In this paper, we first establish the NP-hardness of the problem with general k. Motivated by the situation that each node in a network may have a capability of multicasting, we also study a restricted version with one more requirement that each node can be shared by at most two paths. For the restricted version, we build an ILP model and design a fast algorithm by using the techniques of augmenting paths and splitting nodes. Furthermore, experimental results on synthetic and real networks show that our algorithm is effective in practice.