# Small Maximal Independent Sets and Faster Exact Graph Coloring

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1 Small Maximal Independent Sets and Faster Exact Graph Coloring David Eppstein Univ. of California, Irvine Dept. of Information and Computer Science

2 The Exact Graph Coloring Problem: Given an undirected graph G Determine the minimum number of colors needed to color the vertices of G so that no two adjacent vertices have the same color We want worst-case analysis No approximations No unproven heuristics

3 Isn t it impossible to solve graph coloring exactly? It seems to require exponential time [Garey and Johnson GT4] but that s very different from impossible So why study it? With fast computers we can do exponential-time computations of moderate and increasing size Algorithmic improvements are even more important than in polynomial-time arena Graph coloring is useful e.g. register allocation, parallel scheduling Approximate coloring algorithms have poor approximation ratios Interesting gap between theory and practice worst-case bounds and empirical results differ in base of exponent

4 Register Allocation Application Problem: compile high-level code to machine instructions Need to associate code variables to machine registers Even if code has few explicltly named variables, compilers can add more as part of optimization Two variables can share a register if not active at the same time Solution: Draw a graph, vertices = variables, edges = simultaneous activity Color with k colors, k = number of machine registers Fast enough exact algorithm might be usable at high levels of optimization

5 Previous work on exact coloring Lawler, 1976: 3-coloring O( n ) For each maximal independent set, test if complement bipartite k-coloring (unbounded k) O( n ) Dynamic programming Schiermeyer, 1994: Beigel & Eppstein, 1995: Schöning, 1999: Eppstein, 2001: This paper: 3-coloring O(1.415 n ) Transform graph to increase degree until degree = n 1 3-coloring O( n ) Reduce to more general constraint satisfaction problem Complicated case analysis to find good local reductions General constraint satisfaction algorithm Random walk in space of value assignments No improvement for coloring 3-coloring O( n ), 4-coloring O( n ) More case analysis, simple randomized restriction k-coloring (unbounded k) O( n )

6 Lawler s algorithm Dynamic programming: For each subgraph induced by a subset of vertices compute its chromatic number from previously computed information for S in subsets of vertices of G: ncolors[s] = n for I in maximal independent subsets of S: ncolors[s] = min(ncolors[s], ncolors[s-i] + 1) Outer loop needs to be ordered from smaller to larger subsets so ncolors[s-i] already computed when needed

7 Lawler s algorithm analysis Key facts: n-vertex graph has O(3 n/3 ) maximal independent sets [Moon & Moser, 1965] MIS s can be listed in time O(3 n/3 ) [Johnson, Yannakakis, & Papadimitriou 1988] Worst case example: n/3 disconnected triangles Time: sum 3 S /3 = sum ( n i ) 3 i/3 = O(( /3 ) n ) Bottleneck: listing MIS s of every subset of vertices of G Space: one number per subset, O(2 n )

8 First refinement: When the loop visits subset S, instead of computing its chromatic number from its subsets, use its chromatic number to update its supersets for S in subsets of vertices of G: for I in maximal independent subsets of G-S: ncolors[s+i] = min(ncolors[s+i], ncolors[s] + 1) Why is it safe to only consider maximal independent subsets of G-S? We need only correctly compute ncolors[s] when S is maximal k-chromatic but if I is not maximal, neither is S+I Analysis Same as original Lawler algorithm

9 Second refinement: Only look at small maximal independent subsets for S in subsets of vertices of G: limit = S / ncolors[s] for I in maximal independent subsets of G-S such that I limit: ncolors[s+i] = min(ncolors[s+i], ncolors[s] + 1) Why is it safe to ignore large maximal independent subsets of G-S? If X is maximal k-chromatic, let I be its smallest color class Then S=X-I is maximal (k 1)-chromatic and I will be below the limit for S So, the outer loop iteration for S will correctly set ncolors[x] = k

10 Small Maximal Independent Sets To continue analysis, we need facts and algorithms analogous to Moon-Moser and Johnson-Yannakakis-Papadimitriou Theorem: For any n-vertex graph G and limit L there are at most 3 4L n 4 n 3L maximal independent sets I with I L All such sets can be listed in time O(3 4L n 4 n 3L ) These bounds are tight when n/4 L n/3: G = disjoint union of 4L n triangles and n 3L K 4 s

11 Proof idea: Show set of MIS s = union of MIS sets of multiple smaller graphs Combine smaller graph MIS counts to form recurrence First case: vertex with degree three

12 If given vertex is part of MIS Then rest of MIS is also an MIS of G neighbors(v) Subgraph has four fewer vertices, smaller bound on remaining MIS size

13 If given vertex is not part of MIS Then it is also an MIS of G v Subgraph has one fewer vertex, same bound on MIS size Not all MIS s of subgraph are MIS s of original graph but overcounting doesn t hurt

14 Details of Case Analysis If G contains v of degree 3: split into MIS s containing v or not containing v #MIS(G) #MIS(n 4, L 1) + #MIS(n 1, L) If G contains v of degree = 1: Every MIS contains either v or its neighbor #MIS(G) 2 #MIS(n 2, L 1) If G contains v of degree = 0: Every MIS contains v, #MIS(G) #MIS(n 1, L 1) If G contains chain u-v-w-x all of degree = 2: Each MIS contains u, contains v, or excludes u and contains w #MIS(G) 2 #MIS(n 3, L 1) + #MIS(n 4, L 1) Remaining case, G consists of disjoint triangles has 3 n/3 MIS s, all of size n/3 Prove by induction that each expression is at most 3 4L n 4 n 3L Easily turned into efficient recursive algorithm

15 Analysis of second refinement to coloring algorithm Still not any better than Lawler Problem: If S has chromatic number at most 2 then limit = S /2 and small MIS bound only an improvement for S 2n/5 Doesn t cover the the worst case sizes of sets S for the algorithm

16 Final refinement: Handle low-chromatic-number subsets specially for S in subsets of vertices of G: if S is 3-colorable: compute ncolors[s] using 3-coloring alg if ncolors[s] 3: limit = S / ncolors[s] for I in maximal independent subsets of G-S such that I limit: ncolors[s+i] = min(ncolors[s+i], ncolors[s] + 1)

17 Analysis Each set S has limit S /3 So time to find small maximal independent sets of G-S is found by plugging G-S and S /3 into small MIS formula: time to process S = O(3 4 S /3 G S 4 G S 3 S /3 ) Sum over all S simplifies to O((4/ /3 /4) n ), approximately n Additional 3-coloring test per subset only adds O( n )

18 Conclusions Improvement to Lawler s exact coloring algorithm Reduced base of exponent means can solve problems larger by some constant factor New algorithm still simple enough to possibly be useful (BE95 gives simple 2 n/2 alg for 3-coloring step, good enough here) Space O(2 n ) may be a bigger problem than time Ideas for possible further improvement Reduce 4-coloring time below 2 n/2 would allow algorithm to assume ncolors[s] 4 Can worst case number of small MIS s happen for many inner loop iterations? Generalize small MIS bound to be tight for L n/4 but doesn t affect worst case of current alg without further refinement

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