Minimum Cycle Cover Problem Research Articles

ILP formulations for monitoring-cycle design based on segment routing

The minimum cycle cover problem in segment routing is to find a set of monitoring cycles such that (a) they originate from the same vantage point, (b) cover every directed link in the network, (c) each cycle can be encoded by a segment list smaller than a given maximum size, and (d) the total length of all cycles is minimized. To solve the problem using Integer Linear Programming (ILP), the key challenge is how to capture all the subtle features of segment routing using linear constraints. In this paper, five original ILP formulations, numbered from ILP1 to ILP5, are designed. In ILP1, the classic flow-based approach is adopted for constructing cycles. In ILP2, a more efficient voltage-based approach is adopted. We then extend ILP2 to ILP3 to jointly minimize the total segment list size. Since the time required to detect a network failure is determined by the longest cycle, we also extend ILP2 to ILP4 to jointly minimize the length of the longest cycle. As finding optimal solutions for large-sized topologies is time-consuming, we propose to limit the set of candidate segments for constructing cycles. Although the resulting ILP5 cannot guarantee optimal solutions, we show that it always outperforms the best heuristic algorithm. Finally, the performance gain of using a dedicated monitoring topology for probe packets, as adopted by all existing heuristic algorithms, is studied.

New LP relaxations for Minimum Cycle/Path/Tree Cover Problems

Given an undirected complete graph G=(V,E) with nonnegative edge weight function obeying the triangle inequality, a set {C1,C2, …,Ck} of cycles is called a cycle cover if V⊆⋃i=1kV(Ci), where V(Ci) represents the set of vertices in Ci, and its cost is given by the maximum weight of the cycles. The Minimum Cycle Cover Problem (MCCP) aims to find a cycle cover of cost at most λ with the minimum number of cycles. We propose new LP relaxations for MCCP as well as its variants, called the Minimum Path Cover Problem (MPCP) and the Minimum Tree Cover Problem, where the cycles are replaced by paths or trees. Moreover, we give new LP relaxations for a special case of the rooted version of MCCP/MPCP. We show that these LP relaxations have significantly better integrality gaps than the previous relaxations.

New approximation algorithms for the minimum cycle cover problem

Given an undirected weighted graph G=(V,E) with nonnegative weight function obeying the triangle inequality, a set {C1,C2, …,Ck} of cycles is called a cycle cover if V⊆⋃i=1kV(Ci) and its cost is given by the maximum weight of the cycles. The Minimum Cycle Cover Problem aims to find a cycle cover of cost at most λ with the minimum number of cycles. An O(n2) 24/5-approximation algorithm and an O(n5) 14/3-approximation algorithm are given by Yu and Liu (Improved approximation algorithms for some min-max cycle cover problems. Theoretical Computer Science 654 (2016) 45–58). However, the original proofs for approximation ratios are incomplete. In this paper we first present a corrected simplified analysis on the 24/5-approximation algorithm. Based on the simplified approach of analysis and some new observations, we present a new 14/3-approximation algorithm that runs in O(n3) and give an improved 32/7-approximation algorithm that runs in O(n5).

Improved approximation algorithms for some min-max and minimum cycle cover problems

Given an undirected weighted graph G=(V,E), a set {C1,C2,…,Ck} of cycles is called a cycle cover of the vertex subset V′ if V′⊆∪i=1kV(Ci) and its cost is given by the maximum weight of the cycles. The Min-Max Cycle Cover Problem (MMCCP) is to find a minimum cost cycle cover of V with at most k cycles. The Rooted Min-Max Cycle Cover Problem (RMMCCP) is to find a minimum cost cycle cover of V∖D with at most k cycles, each of which contains one vertex in D. The Minimum Cycle Cover Problem (MCCP) aims to find a cycle cover of V of cost at most λ with the minimum number of cycles. We propose approximation algorithms for MMCCP and RMMCCP with performance ratios 5 and 6, respectively. These results improve the previous algorithms in term of both approximation ratios and running times. For MCCP we obtain a 143-approximation algorithm that has the same time complexity as the previous best 5-approximation algorithm. Moreover, we transform a ρ-approximation algorithm for TSP into approximation algorithms for MMCCP, RMMCCP and MCCP with ratios 4ρ, 4ρ+1 and 4ρ, respectively.

Minimum cycle cover and Chinese postman problems on mixed graphs with bounded tree-width

Let M = ( V , E , A ) be a mixed graph with vertex set V , edge set E and arc set A . A cycle cover of M is a family C = { C 1 , … , C k } of cycles of M such that each edge/arc of M belongs to at least one cycle in C . The weight of C is ∑ i = 1 k | C i | . The minimum cycle cover problem is the following: given a strongly connected mixed graph M without bridges, find a cycle cover of M with weight as small as possible. The Chinese postman problem is: given a strongly connected mixed graph M , find a minimum length closed walk using all edges and arcs of M . These problems are NP-hard. We show that they can be solved in polynomial time if M has bounded tree-width.