Approximation Algorithms for Independent Set via Semidefinite Programming Hierarchies and Randomized Rounding Jason Altschuler Matthew Brennan [email protected] [email protected] Abstract In this literature review, we present two recent papers of Bansal, Gupta and Guruganesh in STOC 2015 and SODA 2015 on approximating the Independent Set problem on graphs with maximum degree d. We present proofs that a certain randomized rounding algorithm for the standard SDP relaxation yields a O(d log log d= log d)-approximation, and that the integrality gap of this relaxation is O~(d= log3=2 d), where O~(·) holds log log d factors. We also present algorithms using polylog(d) and d levels of the SA+ hierarchy to achieve approximations factors of O(d= log d) and O~(d= log2 d), respectively. Contents 1 The History of Approximating Independent Set 2 1.1 Notation . 2 2 Preliminaries: Formulations and Relaxations of Independent Set 3 2.1 Standard ILP, LP and SDP Relaxations . 3 2.2 Tighter Relaxations via the Sherali-Adams Hierarchies . 3 3 The SDP Relaxation and Halperin’s O(d log log d= log d)-approximation 5 4 The Integrality Gap of the SDP Relaxation is O~(d= log3=2 d) 7 5 O(d= log d)-approximation with polylog(d) levels of the SA+ hierarchy 9 5.1 Pre-processing . 10 5.2 Iterative Thinning Procedure . 10 5.3 Only using poly(log(d)) levels of SA+ ............................. 12 6 An O~(d= log2 d)-approximation d levels into the SA+ Hierarchy 12 7 Future Directions 13 References 13 Appendix A. Technical Lemma for HALPERIN-SDP in Section 3 15 Appendix B. Ramsey Theory Background and Results for Section 4 15 Appendix C. Technical Lemmas for Section 5 18 1 1 The History of Approximating Independent Set As one of the original problems shown to be NP-complete by Richard Karp in 1972 [12], independent set was one of the first combinatorial optimization problems that algorithmists sought to approximate. In 1979, Lovasz famously introduced the #-function, a semidefinite programming relaxation of independent set, and used this relaxation to prove several results in extremal combinatorics [16]. In 1996, Hastad proved the seminal hardness result that assuming NP 6⊆ ZPP, there is no n1− approximation algorithm for any constant > 0 for the general independent set problem [11]. This hardness result has since been improved by Khot 3=4+ and Ponnuswami to show that there is no n=e(log n) approximation for any > 0 [13]. Because of these inapproximability results for the general independent set problem, the research commu- nity has shifted its focus to approximating independent set on subclasses of graphs. In particular, significant attention has been given to graphs with maximum degree bounded above by d. This problem is the focus of this literature review. For this restricted problem, a simple greedy approach already provides a decent approximation ratio. Given a graph G of maximum degree d, consider greedily coloring the vertices of G using d + 1 colors by iteratively choosing a vertex and coloring it with a color not used to color any of its neighbors so far. This yields a (d + 1)-coloring of G. Outputting the vertices in the largest color of G yields an independent set of size at least n=(d + 1), which gives a simple O(d)-approximation algorithm. It turns out this is nearly optimal. In 2009, Austrin, Khot and Safra showed that assuming the Unique Games Conjecture, this problem cannot be approximated with a ratio better than Ω(d= log2 d) for d mildly increasing as a function of n [3]. Until 2015, the best ratio achieved was O(d log log d= log d) by Halperin’s SDP relaxation [10]. In 2015, Bansal, Gupta and Guruganesh released two papers further analyzing the SDP relaxation of Halperin and higher levels of semidefinite programming hierarchies in [4] and [5]. These two papers are the topic of this survey. The structure of our paper is as follows. In Section 2, we outline the standard relaxations of independent set and the SA+ semidefinite programming hierarchies. In Section 3, we prove that Halperin’s randomized rounding algorithm for the standard SDP relaxation yields an O(d log log d= log d)-approximation; and in Section 4, we prove that the integrality gap of this relaxation is O~(d= log3=2 d), where O~(·) hides log log d factors. In Section 5, we present an algorithm of Bansal using polylog(d) levels of the SA+ hierarchy that sacrifices quasipolynomial time in d, to achieve an O(d= log d)-approximation. In Section 6, we present an algorithm of Bansal, Gupta and Guruganesh using d levels of the SA+ hierachy that sacrifices exponential time in d, to achieve an O~(d= log2 d)-approximation. The proofs of some technical lemmas are deferred to the appendix. We also include in Appendices B and C a short introduction to the Ramsey Theory used in the paper. 1.1 Notation In this paper, G = (V; E) denotes an arbitrary undirected graph on the vertex set V = [n] with maximum degree d and average degree d¯. Without loss of generality, we assume throughout that G is connected, since otherwise we can restrict to the connected components of G. We denote the number of vertices by n := jV j, and for succinctness write [n] := f1; : : : ; ng. The independence number (also called stability number) of G is the cardinality of the largest independent set of G. As is standard, we denote this quantity by α(G). For notational shorthand, we often abbreviate “positive semidefinite” by “PSD”. The O~-notation denotes O-notation with suppressed poly (log log d) factors. 2 2 Preliminaries: Formulations and Relaxations of Independent Set 2.1 Standard ILP, LP and SDP Relaxations ILP Formulation. The independent set problem can be exactly reformulated as the following integer linear program (ILP) on n Boolean decision variables: n X n α(G) = max xi s.t. xi + xj ≤ 1; 8(i; j) 2 E and x 2 f0; 1g i=1 LP Relaxation. The standard linear program (LP) relaxation optimizes the same objective over the convex hull of the ILP’s (non-convex) feasible set. More concretely, the LP relaxation allows x to vary over [0; 1]n. Let OPTLP denote the optimum value of this LP. SDP Relaxation. Introducing an auxiliary unit vector v0 to represent the value 1 yields the following SDP relaxation of independent set n X OPTSDP = max v0 · vi s.t. kv0k2 = 1 i=1 vi · vi = v0 · vi 8i 2 [n] vi · vj = 0; 8(i; j) 2 E The integrality gap of this relaxation is the ratio between the SDP value OPTSDP and the desired inde- (n+1)×(n+1) pendence number α(G). Define the Gram matrix Y 2 R with entries Yij := vi · vj for each i; j 2 f0; : : : ; n + 1g. Then the above SDP can be written in the following slightly more conventional form: n X OPTSDP = max Yii s.t. Y00 = 1;Yii = Y0i 8i 2 [n] and Yij = 0 8(i; j) 2 E i=1 Y 0 (2.1) The above SDP is equivalent to the celebrated Lovasz´ #-function. This can be checked by taking the dual of the above SDP and arguing that there is no duality gap by Slater’s condition (for details see e.g. Lemma 3.4.4 of [15]). As such, we will henceforth write #(G) to denote OPTSDP. While there is significant literature on the #-function (see e.g. [7, 9, 15]), we will only need the following inequality which shows that the SDP relaxation is tighter than the LP relaxation (both are upper bounds on α(G) since they are relaxations of a maximization problem) α(G) ≤ #(G) ≤ OPTLP (2.2) 2.2 Tighter Relaxations via the Sherali-Adams Hierarchies As suggested in Section 1, it is well-known that the LP and SDP relaxations of independent set can have large integrality gaps. A common approach to obtain tighter relaxations is through convex-programming hierarchies. The rough intuition is to create a sequence of mathematical programs that have the same objective, but with progressively smaller feasible sets that all contain the original ILP’s feasible set. A standard way to construct such hierarchies is by successively adding valid inequalities, which are constraints 3 for which every point in the original ILP is feasible. Clearly the addition of valid inequalities can only restrict the feasible set of the relaxed program to a smaller superset of the feasible set of the ILP. Moreover, if this restriction is strict, then we have obtained a tighter relaxation. A standard such hierarchy for the Independent Set problem is the following. Sherali-Adams (SA) Hierarchy. The idea is simply to add clique inequalities, which are valid inequalities P of the form j2S xj ≤ 1 for some clique S ⊆ [n]. Intuitively, a clique inequality simply ensures that an independent set can contain at most one node from any clique. The kth level (k 2 [n − 1]) of the Sherali- Adams (SA) hierarchy, denoted by SAk, is then defined to be the LP relaxation with clique inequalities for all subsets of [n] of size at most k + 1: n X n X OPTSA(k) = max xi s.t. x 2 [0; 1] and xj ≤ 1; 8S ⊆ [n] clique; jSj 2 f2; : : : ; k + 1g i=1 j2S Observe that the number of constraints grows exponentially in the level of the hierarchy. However, de- spite the fact that the relaxations get tighter throughout the hierarchy, the optimal value #(G) of the SDP relaxation is always a better approximation [14].