On The Complexity of the k -Colorabitly Problem

1. On The Complexity of the k-Colorabitly Problem Danny Vilenchik Based on a joint work withJulia Böettcher, Amin Coja-Oghlan andMichael Krivelevich

2. The k-Coloring Problem Given a graph G=(V,E), • Properk-coloring of G is • Goal: find f with minimal possiblek • Such k is called the chromatic number of G 1 2 4 3

3. Colorablilty – Some Background • Determining the chromatic number is NP Hard • No polynomial time algorithm approximates(G)within factor better than s (unless ) [FK98] How to proceed? • Hardness results only show that there existhard instances • The heuristical approach - relaxes the universality requirement • Typical instance? • One possibility: random models Heuristic is a polynomial time algorithm that produces optimal results on typical instances

4. Random Models for k-Colorability • The most popular random graph model • Every edge is included independently of the others w.p. p • For the “interesting” values of np, • The model is thus not suitable for the study of k-colorable graphs, k is fixed • A popular adjustment is • Partition the vertex set into k sets of size n/k each • Include every edge respecting the partition, independently of the others w.p. p • Now the graph is surely k-colorable V1 V2 V3 Thm [AlonKahale 97] There exists a polynomial time algorithm that finds whp a proper k-coloring for with np>Ck2, C some sufficiently large constant

5. Random Models for k-Colorability • Another possible model – the uniform distribution over k-colorable graphs with exactly m edges • Main difference from the planted case: edges do depend on each other • In addition, a characterization of the structure of the solution space of a typical instance in both the uniform and the planted distributions • Single-cluster behavior Thm [Coja-Oghlan, Krivelevich, V. 2007] The result of AlonKahaleextends to the uniformsetting for the same edge-density regime

6. Using Random Models as Benchmarks (?) • Suppose that we have an algorithm that works whp over some random distribution • Is there any hope that the algorithm will still work when changing the distribution? • How about “real-life” instances?

7. Limitations of Random Models • Do they really capture “real life” instances? • Random instances tend to have a very particular structure • Algorithms designed with random inputs in mind might over-exploit properties of the random instance • A desirable property from an algorithm: robustness for changes in the input distribution • Might give hope that the algorithm will work “in practice” • How to achieve this? One possibility – use semi-randomness • Blending random and adversarial decisions • Mediating between the “easy” random case and the over-pesimistic worst-caseD • Drive us to design more robust algorithms and more “accurate” analysis

8. Semi Random Model for the k-Colorability problem • Step 1:Generate • Step 2:Adversary may add edges between planted color classes of • We shall denote it • Adversary in some sense should help the algorithm • However, one can show that Alon-Kahale fails in this model • On the positive side: • [BS95] Algorithm that whp colors for • [FK01], [CO04] Algorithm that colors whp for with • On the negative side: • [CO04] No efficient algorithm for unless • How to “challenge” algorithms for sparse inputs? V1 V2 V3 Proof outline: plant a hard instance on the set of isolated vertices

9. Semi-Random models for sparse graphs [Krivelevich, V. 06] • We suggest an adaptation of for the sparse case • Let A be a subset of vertices (may depend on the input graph) • Define to be the following model: • Step 1: Generate • Step 2: Adversary adds edges that respect the planted partition whose endpoints lie in A • In particular, • when then • when then • Optimal: require that the algorithm keeps working efficiently whp for • First step: require that the algorithm works (disregarding efficiency) for • Is there an “interesting” set A? E.g. all vertices with degree 10, or vertices 1,…,logn easy hard

10. Semi-Random model for sparse graphs • Yes! • Take A to be the maximal set of vertices that induce a “regular” graph, H • Regular in the sense that every vertex has neighbors in every color class • One can show that (using similar arguments to AK97) • H is typically large (all but, say, 1% of the vertices) • The algorithm of AK fails (regardless of efficiency) for • [KV06] present an algorithm based on SDP that works whp for • Building on some ideas from [AK97] and [CO04] • Using the semi-random setting the weakest links in AK were identified • Replacing spectral approach with SDP (pioneered by [FK01]) • Leaving out a re-coloring step

11. Towards finding the “threshold” A [Böettcher, V. 07] • We saw that has a jump in complexity as A varies • Finding A s.t. is hard, but for every is easy would shed light on the intrinsic difficulty of the k-colorability problem • First step: can one find a less restrictive (more natural?) set than H • Yes! • Define Jc={v: v has at least c neighbors in Jc in every other color class} • Easy to see: there exists a unique maximal Jc • Note that for every constant c, H is contained in Jc • Jc is basically the natural adaptation of the well known c-core notion • Define Fc={v: v has at least c neighbors}

12. Towards determining the “threshold” A Thm[Boettcher, V. 2007] For every c>1, there exists d=d(c) s.t. is easy for For k>3, is hard for and • A constructive way to interpret this statement: • Think of k=4, np=2000 • Setting A to be the maximal 2-core of the graph – problem remains easy • Setting A to be the set of vertices with degree at least 400 makes it hard • How about setting A to be all vertices with degree at least 2 in every other color class?

13. Algorithm and Analysis – Quick Facts • Suppose A isthe maximal 2-core of the graph • We call H a super-core of if every v in H has neigbhours in every other color class (w.r.t. ) • Such H exists whp and contains all but vertices • Proved using “standard” expansion arguments • Using our definition H need not be unique • The (unique maximal) 2-core of the graph contains any set H • In , whp there are no sets of size with minimal degree 2 • This is not necessarily true though for • The size of the connected components in when plucking out the 2-core is logarithmic

14. Algorithm and Analysis • Run SDP for max k-cut • Group the vertices according to assigned vectors • Color only the k largest groups (different color for every group) • Repeatedly uncolor vertices with less than 2 neighbors in some color class • Exhaustively color the graph induced by the uncolored vertices Blackboard proof outline

15. Going beyond the 2-core • Where does the argument might break? • When the SDP step terminates, the set A need not necessarily be colored correctly • It is not clear how to proceed form here (as the adversary may add edges between vertices of A) Thanks!