A New Approximation Algorithm for the Asymmetric TSP with Triangle Inequality By Markus Bläser Presented By: Chris Standish chriss@cs.tamu.edu 23 November 2005 1
Outline Problem Definition Frieze s Generic Algorithm of 1982 Bottleneck in Frieze s Algorithm LP Formulation for Cycle Covers Bläsers Generic Algorithm of 2003 Dealing with Fractional LP Fractional Solutions Discussion and Homework 2
The Asymmetric Minimum TSP The Asymmetric Minimum Traveling Salesman Problem (ATSP) is: Problem 1 Asymmetric Minimum TSP Given: A complete, directed, edge-weighted graph G(V, E), without self-loops, and an edge weight function w which satisfies the triangle inequality, and which is asymmetric, and non-negative. Find: A Hamiltonian tour of G of minimum weight. A weight function w : E Q 0 which satisfies the triangle inequality means that: w(u, v) w(u, x) + w(x, v) for all distinct u, x, v V. An asymmetric weight function w is such that w(u, v) w(v, u). We will abbreviate w(u, v) as w uv If we consider this problem defined with an arbitrary weight function then this problem is NPO-complete. So we don t expect to be able to find a PTAS unless P = NP [1]. 3
Definitions Frieze, et. al. [2] developed an algorithm in 1982 that, until now, has had the best approximation ratio for minimum ATSP. But first we need some definitions: k-cycle - A cycle is called a k-cycle if is has length exactly k. cycle cover - A cycle cover of a directed graph G is a set of directed simple cycles such that each node is part of exactly one cycle, i.e., the set of cycles spans the graph. k-cycle cover - A cycle cover C is called a k-cycle cover if each cycle in C has length at least k. cycle cover weight - Let C = {C 1, C 2,...,C t } be a cycle cover. Then the weight of the cycle cover is w(c) = t i=1 w(c i) where w(c i ) = (u,v) C i w uv partial cycle cover - If C is a collection of node disjoint cycles, but not a spanning one, then C is called a partial cycle cover. 4
Frieze s Algorithm Freize s generic algorithm [3] to find an approximate minimum-weighted tour can be described as: Algorithm 1 Generic ATSP Step 1) Create a minimum weight cycle cover C for the graph G Step 2) For each cycle in C remove a maximum-weighted edge to form a set of vertex disjoint paths P. Step 3) Patch together the resulting set of paths in P to produce a tour T of G The algorithm has a performance ratio of 1 log n. That is, it always delivers a solution which is at most 1 log n times the weight of an optimal solution [2]. The goal of this paper is to show that this threshold is not tight. The paper gives an algorithm with a 0.999 log n performance ratio. 5
An Example A PARTIAL 2 CYCLE COVER A TOUR Figure 1: Constructing a tour. Most edges are not shown for clarity. 6
Bottleneck in the Generic Algorithm Notice that if we could construct a n -cycle cover we can find an optimal 2 tour directly. So why don t we? The problem with the above algorithm is the difficulty of computing a minimum weight cycle cover. In fact, computing a minimum weight 3-cycle cover is APX-hard, even if the weight function satisfies the triangle inequality [4]. This paper adapts Frieze s algorithm to find a good partial cycle cover instead of a cycle cover. 7
Computing Cycle Covers The problem of computing a minimum weight cycle cover can be solved by a relaxed linear program (LP). Let x uv be the variable that represents the edge (u, v). If x uv = 1 then the edge is included in a cycle, if x uv = 0 then it is not. The relaxed linear program can be formulated as: Minimize (u,v) E w uv x uv subject to u x uv = 1 for all v V (in-degree constraints) v x uv = 1 for all u V (out-degree constraints) x uv 0 for all (u, v) E (non-negativity constraints) 8
Relaxed LP formulation In this particular LP formulation we encounter something quite nice. The solution matrix X = (x uv) n n to this relaxed LP happens to be totally unimodular. What this results in, is an X that has entries that are either 0 or 1. So the optimal solution is exact and we don t have to deal with fractional x uv values. However, we have not constrained the length of the cycles in the cycle cover. Notice that the larger the value of k, the better our approximate tour will be. This LP can be solved in polynomial time, i.e., O(n 3 ) 9
Good Partial Cycle Covers In this paper, the generic ATSP algorithm has been modified to use the notion of a good partial cycle cover, see Figure 1. TSP(G) - Let TSP(G) be the weight of an optimal tour in G. b-good partial cycle cover - is a partial cycle cover which satisfies the condition where α log β b 0 < b 1, α 1 is such that the weight of the partial cycle cover is α TSP(G), and β is such that the number of cycles in the partial cycle cover is β V, 0 < β < 1. The generic algorithm of Bläser is also recursive. 10
The b-good Partial Cycle Cover Problem The b-good Partial Cycle Cover Problem is: Problem 2 The b-good Partial Cycle Cover Problem (b-gpcc) Given: A directed graph G(V, E), and an asymmetric edge weight function w that satisfies the triangle inequality. Find: A partial cycle cover C of weight w(c) = α TSP(G), which has β V cycles, and such that α log β b 11
Recursive Generic Algorithm The generic algorithm this paper uses to find an approximate tour can be described as: Algorithm 2 Generic Recursive ATSP Step 1) Create a b-good cycle cover C for the graph G Step 2) For each cycle in C, choose one arbitrary node. Let V be the set of nodes consisting of these nodes, together with all the nodes in V that are not contained in any cycle of C. Step 3) Recursively compute a TSP tour T of the graph G induced by V. Step 4) For each cycle in C remove a maximum-weighted edge to form a set of vertex disjoint paths P. Step 5) Patch together the resulting set of paths in P and T to produce a tour T. 12
An Example C T T Figure 2: Constructing a tour using a partial cycle cover, and a tour. Not all edges are shown. T is the tour constructed by combining C and T. 13
Running Time In the worst case, step 2 reduces the size of the graph by one. That is, only one 2-cycle is found. So the algorithm recursively calls itself at most n = V times. If a b-good cycle cover can be computed in polynomial time, then the generic recursive cycle cover algorithm will run in polynomial time. Theorem 1 If the b-gpcc problem can be solved in polynomial time for some 0 < b 1, then there is a polynomial time (b log n)-approximate algorithm for ATSP. The rest of the paper concentrates on showing that a b-gpcc can be found in polynomial time. 14
Improved Approximation Ratio This paper improves the approximation ratio of Frieze, i.e., 1 log n, by solving a relaxed LP with an additional constraint. In addition to the previous constraints in the LP formulation, a 2-cycle constraint is added This constraint eliminates 2-cycles. x uv + x vu 1 for all u v However, the solution matrix X is no longer totally unimodular. So this means we have to deal with fractional solution values x uv. 15
Decomposing a Fractional Solution The rest of the paper is devoted to showing how to compute a 0.999-good partial cycle cover given a fractional solution X. The method is involved so I am just going to give an overview of the main ideas. 16
Some Definitions and a Lemma doubly stochastic - A matrix S is called doubly stochastic if all its entries are non-negative, and the entries in each row, and in each column, sum to 1. permutation matrix - A matrix P is called a permutation matrix if each entry has value either 0 or 1, and it is doubly stochastic. Lemma 1 (BIRKHOFF - VON NEUMANN) Every doubly stochastic n n matrix S is a convex combination of at most n 2 permutation matrices. Such a decomposition can be found in polynomial time. Since the solution matrix X is double stochastic (because of the in-degree and out-degree constraints), we can decompose it: X = t α i P i i=1 where t n 2, the α i are non-negative reals such that t i=1 α i = 1, (convexity), and the P i are permutation matrices. 17
A Permutation Matrix Notice that every permutation matrix induces a cycle cover of G. 1 2 3 4 5 6 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 7 8 PERMUTATION MATRIX 2 CYCLE COVER Figure 3: A 2-cycle cover and its corresponding permutation matrix. 18
Reducing the Number of Cycle Covers Choose a constant B, such that B is the smallest integer such that B α i is an integer, for each i. Define γ i = B α i. Bläser claims that all the α i are rational. So we can find such a B which will be polynomial in the problem input size. Notice that B can be quite large. So now we can treat each permutation matrix P i as a cycle cover C i, which has a multiplicity γ i. That is, there are γ i copies of the cycle cover C i. The point is that we can work with a set of t cycle covers C i instead of working with all the B cycle covers explicitly. 19
Normalization So now lets treat each P i as a C i. (Note C i is a set of edges, P i is a matrix) Let C i C j denote the graph (V, E i E j ). This is the graph formed by the set of the edges in the two cycle covers. ( ) t For each of the possible cycle cover pairs C 2 i C j, we transform some particularly shaped strongly connected components, that are formed by 2-cycles in C i C j, into larger length cycles. This process is called normalization and can be done in polynomial time. 20
Normalization Figure 4: A strongly connected component formed by 2-cycles in C i C j. Green(dashed) denotes a 2-cycle in C i, blue(solid) for C j. 21
Computing Good Partial Cycle Covers When we normalize the union of two cycle covers, we are eliminating 2- cycles in each of C i and C j, and replacing them with larger length cycles. After we normalize all pairs of cycle covers, we compute a b-good partial cycle cover, where b = 0.999, from the normalized set of cycle covers. Bläser gives a case based analysis, and in each case he shows that we can compute a 0.999-good partial cycle cover. So we have the following: Theorem 2 There is a polynomial time algorithm for the 0.999-GPCC problem. Corollary 1 There is a (0.999 log n)-approximate algorithm for the minimum ATSP problem with polynomial running time. 22
Main Results Presents a recursive generic algorithm which has a 0.999 log n performance ratio. Shows that the approximation ratio of 1 log n shown by Frieze, et. al. [2] is not tight. This contrasts with the the set cover problem which has a tight threshold of 1 lnn as shown by Feige [5]. The algorithm does not appear to be practical because its running time depends on the value of B. 23
Homework Do one of the following problems: 1) Write the minimum ATSP problem in the 4-tuple notation < I Q, S Q, f Q, opt Q > 2) In a previous slide, we said that if we could construct a minimum weight n-cycle cover for an input graph G(V, E) (for the ATSP problem, where 2 n = V ), we could find an optimal tour directly. Q) - Why is this so? Assume n is even. Hint: There are two cases, 1. There are exactly two n-cycles 2 2. There is exactly one cycle with size greater than n 2 24
References [1] M. Bläser. An 8 -Approximation Algorithm for the Asymmetric Maximum 13 TSP. Journal of Algorithms, 50(1):23 48, 2004. [2] A.M. Frieze, G. Galbiati, and F. Maffioli. On the Worst-Case Performance of Some Algorithms for the Asymmetric Traveling Salesman Problem. Networks, 12:23 39, 1982. [3] M. Lewenstein and M. Sviridenko. Approximating Asymmetric Maximum TSP. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 646 655, 2003. [4] M. Bläser. A New Approximation Algorithm for the Asymmetric TSP With Triangle Inequality. In Proceedings of the 14th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 638 645, 2003. [5] U. Feige. A Threshold of ln n for Approximating Set Cover. Journal of the ACM, 45:634 652, 1998. 25