THE CIS PROBLEM AND RELATED RESULTS IN GRAPH THEORY RYAN ALWEISS, YANG LIU Abstract. In this survey, we will show that there are instances of the CIS problem on n vertices which cannot be solved deterministically in Ω(log~ n) time. Until recently, the best known lower bound for this prob- lem was only linear in log(n). The approach will be to instead lower cc bound the conondeterministic complexity coNP (CISG) through a re- duction to a problem in decision tree complexity. Interestingly, this result has an application to graph theory, the negative resolution of the Alon- Saks-Seymour conjecture. 1. Introduction Given a communication matrix M of a function F , define χ0(F ) to be the minimal number of rectangles necessary to partition the 0s of M, χ1(F ) similarly, and χ(F ) = χ0(F ) + χ(F ). Clearly the deterministic communication complexity of F is always at least log χ(F ), and it is always at most log2(χ(F )). It was unsolved until recently whether some problems required superlogarithmic amounts of com- munication. Here, we present some results about the complexity of the clique versus independent set problem (CIS) that imply there are problems with superlogarithmic communication complexity. In this CIS problem, there is a graph G on n vertices which Alice and Bob both see. Alice has vertices that form a complete subgraph, and Bob has vertices that form an independent set. Alice and Bob wish to see if their sets of vertices intersect or not. A solution of CIS in some time t extends to a solution of any communication problem in the same amount of time. In the communication matrix, blocks of 1 can be realized as vertices, connected if they share a common row. Then the 1-rectangles that form some column are an independent set, and those that intersect some row form a complete graph. If there is some algorithm to figure our whether the complete graph and independent set intersect in time t, then the problem can be solved in time t. As such, proving lower bounds for log(χ1) is equivalent to proving lower bounds for CIS in much the same way that solving NP is equivalent to solving SAT. In this survey, we will show there are instances of the CIS problem with conondeterministic complexity Ω(log~ 1:128(n)), where the graph has n vertices. As such, there is a CIS problem with deterministic complexity Ω(log~ 1:128(n)), ~ 1:128 or Ω(log (χ1(F ))). In fact, a later result shows that some problems require ~ 2 ~ 1:5 complexity Ω(log (χ1(F )) and Ω(log (χ(F )) [GPW 2015]. 1 2 RYAN ALWEISS, YANG LIU Furthermore, the CIS problem is related to a conjecture in graph theory, the Alon-Saks-Seymour conjecture. Given a graph G, let χ(G) denote its chromatic number and bp(G) denote the minimal number of bipartite graphs into which its edges can be partitioned. The initial conjecture was that χ(G) ≤ bp(G) + 1, but it turns out that showing lower bounds for the CIS problem shows that it is possible to have χ(G) be more than polynomial in bp(G). In other words, we have χ(G) ≥ bp(G)m for any fixed m > 0 and for infinitely many graphs G. 2. General Structure of this Survey The main objective of the first part of this survey will be to prove the cc cc result due to Goos, that coNP (CISG) and therefore P (CISG) are at least Ω(log~ 1:128(n)). First, we will develop the theory of decision trees. Then, we will proceed to prove this results. To conclude the first part, we will briefly discuss more recent improvements and examine the results from the literature that we cite. We will then turn our attention to graph theory, discussing the relations between the CISG problem and the Alon-Saks-Seymour conjecture. We will examine how these conjectures from communication complexity statements translate to graph theory. As such we will highlight the rich connection between these two areas of mathematics. 3. Decision Trees Throughout the remainder of this paper, we will be using results about decision trees. A decision tree is a directed binary tree with a single root whose out-edges are labeled 0 or 1. Leaves are also labeled with 0 or 1. All other vertices are labeled xj for 1 ≤ j ≤ n, and have two out-edges, labeled with 0 and 1 respectively. When on some vertex labeled xj, one queries xj and traverses the out-edge from xj corresponding to its value. As such, a decision tree induces a Boolean function f on n variables. We say that such a decision tree is a decision tree for f. The queries are analogous to bits ex- changed in a communication protocol, and thus are important for understand communication complexity. This is made explicit in the appendix. We construct complexity classes for decision trees. Denote by Pdt(f) the minimal possible height of a decision tree for f. This is called the deter- ministic query complexity of f, because it is the minimal number of queries necessary to always compute f. We define NPdt(f), the nondeterministic query complexity, to be the minimal k such that for for any string x of length n with f(x) = 1, there exists some k bits of the string that provide enough information to show that f(x) = 1. In other words, any y that agrees with x just on those bits has f(y) = f(x) = 1. We say these k bits are a 1-certificate for f. Indeed, NPdt(f) is also the minimal k such that we can write f as a k-DNF, since the terms of the DNF correspond to 1-ceritificates. THE CIS PROBLEM AND RELATED RESULTS IN GRAPH THEORY 3 We define coNPdt(f) similarly, as the minimal k such that whenever f(x) = 0 there is an analogous 0-certificate for f. We have coNPdt(f) = NPdt(:f). We define UPdt(f), the unambiguous query complexity, to be the minimal k such that for any string x with f(x) = 1, there is a unique 1-certificate of size k. This is also the minimal k such that there exists a k-DNF for f where exactly one clause is satisfied when f(x) = 1, and no clauses are satisfied otherwise. Again, this is because the DNFs correspond exactly to 1-certificates. We can generalize the notion of certificates. An f(x)-certificate of size k for a string of x of length n, are a set of k bits of the string that provide enough information to determine the value of f(x). The certificate complexity of f is the minimal k such that there is an f(x)-certificate of size k for every string. In this paper, we will be concerned with relationships between the above quantities. It is an easy result that NPdt(f) ≤ Pdt(f). If there exist a tree of height k computing f and if f(x) = 1 the values of the at most k variables on the path taken on input x are sufficient to conclude f(x) = 1. Replacing f with :f, coNPdt(f) ≤ Pdt(f). Proposition 3.1. Pdt(f) is at most quadratic in UPdt(f): More explicitly, we have the inequality Pdt(f) ≤ UPdt(f)2: Proof. Consider a specific decision tree with a unique certificate of size k = dt UP (f) for any x with f(x) = 1. For any C1 and C2 there exists xj such that in one certificate xj = 0 and in the other xj = 1. Else, some assignment would satisfy both C1 and C2, contradicting uniqueness. Now, take some certificate C0 (say, the first lexicographic one) and ask for all k of its associated variables. After these variables are determined, any assignment to the remaining variables satisfies exactly one certificate C of size k of the original problem. But because C intersects C0, to determine whether some assignment satisfies C it now suffices to query at most k−1 variables. Indeed, this resulting problem f 0 has UPdt(f 0) ≤ UPdt(f)−1 = k−1 and we reduced f to f 0 by querying k variables. We rename k − 1 as k and repeat. Through this reduction argument, in fact we find that in fact UPdt(f)2 X UPdt(f)2 + UPdt(f) Pdt(f) ≤ k = : 2 k=1 We do not need the full strength of this result. We have that UPdt(f)2 + UPdt(f) coNPdt(f) ≤ Pdt(f) ≤ ≤ UPdt(f)2; 2 so coNPdt(f) ≤ UPdt(f)2: We will show that 2 cannot be replaced by 1:128. 4. Reductions to Decision Tree Complexity The goal of this section will be reduce our problem of communication complexity to a problem relating to query and decision tree complexity. First, 4 RYAN ALWEISS, YANG LIU cc note that UP (CISG) = log n, as the set of vertices constitutes a set of 1- certificates. Because it is known that the CISG faily of problems is complete for unambiguous communication, we can focus on constructing a family of functions F that exhibit an exponent of α > 1:128 separation between UPcc cc cc 1:128 and coNP . This would imply that coNP (CISG) ≥ O((log n) ): We rephrase this in the following result. Theorem 4.1. There is an infinite family of functions F such that coNPdt(F ) ≥ UPdt(F )α for some constant α > 1:128: We claim that it suffices to show the corresponding theorem for query complexity by using a result of [GLM+14].