2.2 Complexity
2.2.2 Approximability and Approximations
Intractability, i. e. , N P-hardness, of a problem almost automatically creates need for good approximation algorithms and simultaneously poses a new question: How close to the optimum solution can we get?
Let us start with the bad news: For general graphs, Kann [Kan92] showed that the FEEDBACK ARC SET problem is MaxSN P-hard, which also implies APX -hardness.
Consequently, unless P = N P, no polynomial-time approximation scheme for FAS can exist, which would allow to construct a solution in polynomial time that is at most(1+ε) times bigger than the optimum, for any ε >0. As a matter of principle, APX -hardness still leaves the possibility to find a constant-factor approximation algorithm, i. e. , an algorithm that guarantees that the quotient of the size of the delivered solution divided by that of the optimum is bounded by some constant c. Unfortunately, no such algorithm for FAS on general graphs is known presently. Furthermore, as VERTEXCOVER⪯ FAS (cf.Section 2.1.5) via an approximation-preserving reduction, the fact that it is N P-hard to approximate VC better than to a factor of10√5 − 21 = 1.3606... [DS05] also applies to FAS. However, VERTEX COVER has a relatively simple factor-two approximation
[GJ79,PS98] and is thus APX -complete, whereas APX -completeness is still an open question for FAS.
It is important to note here that the closely related ACYCLICSUBGRAPHproblem has
a constant-factor approximation with ratio 1
2: Take an arbitrary linear ordering of the vertices of the input graph. If the number of backward arcs induced thereby exceeds the number of forward arcs, reverse the ordering. Then, if the graph has m arcs, the forward arcs form an acyclic subgraph containing at least m
2 arcs. As the optimum solution is at most m, we obtain a guaranteed approximation ratio of 1
2. Thus, AS is APX -complete. Due to a result by Papadimitriou and Yannakakis [Pap91], AS is also complete for MaxSN P.
In the context of hardness of approximation, a conjecture postulated by Khot [Kho02] in 2002, which is widely known as the Unique Games Conjecture, plays an influential role. One of several equivalent formulations is based on the so-called LABELCOVERproblem:
The Unique Label Cover Problem
Instance: undirected, edge-weighted, complete bipartite graph U with edge set
E, a set of vertex labels M, for every edge e ∈ E a bijection pe: M → M
Question: What label assignment maximizes the total weight of those edges
e= {u, v} with the property that pemaps the label of u to the label of v?
The Unique Games Conjecture now states that for a pair of constants(c1, c2) sufficiently small, there is a constant k depending on c1and c2such that it is N P-hard to distinguish whether an instance of UNIQUELABELCOVERwith |M| = k has an optimum value of at
least1−c1or at most c2. If this conjecture could be confirmed, it would prove a number of classic N P-hard optimization problems to be inapproximable below some specific bound [Kho10]. Among these problems is not only VC (with a bound of2 [KR08]), but most notably also the FEEDBACKARCSETproblem. In particular, Guruswami et al. [GMR08]
prove that on condition of the Unique Games Conjecture, FAS is inapproximable within a constant-factor. They also show that if the conjecture is true, ACYCLICSUBGRAPH
cannot be approximated within a factor better than1 2.
On the positive side, Even et al. [ENSS98] showed that FAS (in fact, SUBSET-wFAS) can be approximated to a factor of O(min {log τ∗log log τ∗,log n log log n}) in polynomial time, where τ∗denotes again the optimum fractional solution (cf.Section 2.1.2) and n the number of vertices. Their algorithm is based on a result by Seymour [Sey95], who proved that the integrality gap, i. e. , the maximum ratio between the optimum integral and the optimum fractional solution, is at most O(log τ∗log log τ∗) in the unweighted case. Alternatively, the authors provide an O(log2|X|)-approximation algorithm for SUBSET-wFAS, where X identifies the set of interesting cycles (cf.Section 2.1.5). For FAS, Even et al. improve a result by Leighton and Rao [LR88], who obtained a factor of O(log2n) via the computation of approximately sparsest cuts using linear programming. Klein et al. [KST90] replaced this dependency with a randomized algorithm. Demetrescu and Finocchi [DF03] were able to show that the straightforward approach of identifying and destroying cycles step by step yields an O(m · n)-time algorithm which guarantees a solution that exceeds the optimum by a factor of at most the length of the longest cycle in the graph.
There are further encouraging results if we consider FAS again on special classes of graphs. For tournaments, Ailon et al. [ACN08] presented a randomized3-approximation algorithm called KwikSort that constructs a linear ordering in a similar manner as
2.2 Complexity 23 the well-known QuickSort algorithm sorts numbers. The authors also extended this approach to tackle wFAS on tournaments with various kinds of weight constraints, which led to different approximation ratios. Van Zuylen and Williamson [vZW09] were able to obtain respective derandomized versions while preserving approximation guarantees. Shortly after the conference version of [ACN08], Kenyon-Mathieu and Schudy [KS06,KS07] designed a polynomial-time approximation scheme (PTAS) for WEIGHTEDFEEDBACK ARCSETon tournaments. Their algorithm is based on earlier
results by Arora et al. [AFK02] as well as Frieze and Kannan [FK99], who obtained polynomial-time approximation schemes for the ACYCLIC SUBGRAPHproblem on n-
vertex graphs whose number of arcs is inΩ(n2). To complete the picture, we note that for FVS on tournaments, a3-approximation algorithm has been known since 1989 and is due to Speckenmeyer [Spe89]. In 2000, Cai et al. [CDZ00] were able to improve the ratio down to2.5 and, only recently, Mnich et al. [MWV15] even achieved 7
3.
Finally, there are also approximation results for bipartite and multipartite tourna- ments: Gupta [Gup08] presented both randomized and deterministic4-approximation algorithms for FAS on these classes of graphs; van Zuylen [vZ11], however, pointed out an error in the correctness proof. Nevertheless, she was able to confirm the findings for wFAS on bipartite tournaments by designing a different algorithm. Her work also includes a2-approximation algorithm for wFVS on bipartite tournaments, thereby im- proving on earlier results by Cai [CDZ02] and Sasatte [Sas08a], who achieved a factor of 3.5 and 3, respectively.