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**Determinant Sums (and Labeled Walks) for Undirected Hamiltonicity**

Andreas Björklund

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**Undirected Hamiltonicity**

Instance: Undirected graph G=(V,E) on n vertices. Question: Is there a vertex permutation v1,v2,…,vn such that vivi+1 in E for all i, including vnv1?

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Examples Yes No

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**History Introduced by Kirkman 1855. Popularized by Hamilton 1857.**

Special case of traveling salesman problem studied since 1930. Exact algorithm for TSP in O*(2n) time in 1962 by Bellman, Held and Karp, and Gonzalez. Proved NP-complete in Karp’s 1972 paper. Polynomial space O*(2n) time algorithm by Kohn, Gottlieb, and Kohn 1977, Karp 1982, and Bax 1993.

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**Gerhard Woeginger’s 2003 Survey**

Open problem: Construct an exact algorithm for the travelling salesman problem with complexity O*(cn) for some c<2. In fact, it even would be interesting to reach such a time complexity O*(cn) for some c<2 for the closely related, but slightly simpler Hamiltonian cycle problem.

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Our Results There is a Monte Carlo algorithm that detects if any input n-vertex undirected graph is Hamiltonian or not running in O*(1.657n) time. In bipartite graphs, O*(1.414n) time. Small weight TSP in O*(1.657nw) time, where w is the sum of all weights.

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**Old Idea 1: Dynamic Programming Across the Vertex Subsets**

(Bellman, Held and Karp, and Gonzalez.) Grow the path one vertex at a time, remembering which vertices were previously visited (but not the order in which they were traversed).

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**Old Idea 2: Inclusion-Exclusion Counting Across the Vertex Subsets**

(Kohn, Gottlieb, and Kohn, Karp, and Bax.) For every vertex subset, count the number of closed walks on n vertices in the induced graph. Sum up the results with alternating signs determined by the parity of the vertex subset.

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**Too Good? Our algorithm here…**

Uses exponential space, (but we can get rid of it) Works only in undirected graphs, Is randomized, And cannot even approximately count the solutions. The inclusion-exclusion algorithm… Uses only polynomial space, Works also for directed graphs, Is deterministic, And counts the solutions.

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Common Obstacle Both algorithms keep track of the vertices visited by explicitly enumerating all vertex subsets. We have to figure out how to bookkeep visited vertices in a cheaper way. First idea: Restrict the input space.

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**Previous Restricted Input Results**

O*(1.251n) time for cubic graphs (Eppstein’07, Iwama and Nakashima’07) O*(1.715n) time for graphs of degree at most four (Gebaur’08) O*((2-e(d))n) time for graphs of bounded degree d (B.,Husfeldt, Kaski, and Koivisto ’08) O*(1.682n) time for claw-free graphs (Broersma, Fomin van’t Hof, and Paulusma ’09)

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Bipartite Graphs A Hamiltonian cycle (when it exists) visits vertices from the two color classes in the bipartition alternatively.

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**Labeled Hamiltonicity?**

1 A 2 1 BF B B 3 A B C 3 BD B C C D A F 2 BE D 5 AE CE 4 D E 5 4 E E E F 6 F 6 n’=n/2=6 vertices n=12 vertices |L|=n/2=6 edge labels

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**Labeled Hamiltonicity**

We have to keep track of visited vertices and used labels. Seems like O*(2n’2|L|)=O*(2n). Have we really gained anything? Yes, We will describe an O*(2|L|) time randomized algorithm based on counting in characterstic two.

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**The Tutte Matrix + + … 1 T 1 2 3 4 5 6 x12 x13 x15 x16 x23 x24 x25 x26**

x12 x13 x15 x16 x23 x24 x25 x26 x34 x35 x45 x46 x56 3 2 5 4 6 + + …

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**Special Asymmetric Vertex**

1 T 1 2 3 4 5 6 x12 x13 x15 x16 x12’ x23 x24 x25 x26 x13’ x34 x35 x45 x46 x15’ x56 x16’ 3 2 5 4 6 + + …

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**Labeled Cycle Covers Labels L={A,B,C,D,E,F}**

1 Labels L={A,B,C,D,E,F} B B 3 C + C F 2 E F D 5 A D E 4 A We can use inclusion-exclusion counting to compute the labeled cycle covers. 6

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**Labeled Tutte Matrices**

1 TC 1 2 3 4 5 6 x23,c x24,c x34,c BF B 3 A B BD B C C D A F 2 BE 5 CE AE D E 4 E E F 6

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**Algorithm Input: Bipartite graph G=(U,V,E) on n vertices.**

Reduce to labeled Hamiltonicity instance G’=(V,E’) and labels L=U. Fix field F of characteristic two of size >>n’. Assign values from F to xij,l for all ij in E’ and labels l in L, uniformly and independently at random. Compute If H nonzero return Yes, else return No.

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Analysis Runtime O*(2|L|) = O*(2n/2) since computing a numerical determinant is a polynomial time task. No false positives: if H is nonzero there must be a Hamiltonian cycle. False negatives with exponentially small probability of failure (in n): H is an n-degree multivariate polynomial and is zero in at most n/|F| points by the Schwartz-Zippel Lemma.

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**General Graphs vs Bipartite Ones**

We don’t have a natural a priori way of partitioning a graph’s vertices in labels and vertices of a labeled Hamiltonicity instance. First idea: we can use subsets (not just singletons) of a label set to label the edges in a labeled cycle cover. We will use a random equipartitioning of the vertices. Problem: We cannot label edges with the empty set. Second idea: we add a few labels on every direct edge.

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**Subset Labeled Hamiltonicity**

1 A B L’={a,b} {a} {b} {AD} {a} {b} {ADE} {AB} 1 2 3 2 3 C {C} {BADE} 5 {B} {C} D 4 {a} {b} {DE} 4 5 {C} {E} {BADE} E n=10 vertices n’=n/2=5 vertices |L|=n/2+|L’|=7 edge labels

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**How Many Direct Edge Labels are Needed?**

Consider a fixed Hamiltonian cycle H. If we choose an equipartition of the input graphs vertices and use half of them as vertices in a subset labeled Hamiltonicity instance, we get n/4 direct edges in expectation along H. We need in total n/2+n/4=3n/4 labels. Can we still find a O*(2|L|) time algorithm for Subset Label Hamiltonicity?

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Going Exponential Instead of one matrix per label, we imagine one matrix TX per label subset X. Now gives us what we want.

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Computing We can use a recursion like the Bellman-Held-Karp-Gonzalez dynammic programming to compute all matrices TX in O*(2n-n’) time. We can use the fast zeta transform to compute and tabulate the inner sum for all X’s in O*(2|L|) time.

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**Analysis and Extensions**

We still have O*(2|L|) time algorithm, and remembering |L|=3n/4 we get O*(1.682n) time. We can trade a few labels for a modest exponential number of runs to get an O*(1.657n) time algorithm. We can use walks instead of paths in the labels’ construction getting a polynomial space algorithm with the same running time. By adjoining a new indeterminate we can embed a min-sum semiring in the polynomial and solve for TSP.

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**Another Related Technique**

(Joint work with Husfeldt, Kaski, and Koivisto) Idea: Count labeled walks instead of labeled cycle covers. We don’t need determinants. Previous reductions from Hamiltonicity to Subset Labeled Hamiltonicity still in play.

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Labeled Walks 2 B C B C F F A B,D,E 1 5 3 4 A E D D E 6 +

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Why Label at All? 2 1 5 3 4 6 Counted only once!

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**Aha! Just don’t Backtrack**

2 1 5 3 4 6 1->6->3->4->6->3->1 1->6->4->3->6->3->1

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What about k-Path? Given undirected graph G=(V,E) and positive integer k, determine if there is a simple path of length k in G. Just count labeled walks of length k’ instead of closed labeled walks of length n’. Then we only need a k’/n’ fraction of the labels! Poly(n,k)1.657k time algorithm.

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**Open Questions Could 1.657 really be the optimal base?**

Can we find other k-vertex subgraphs faster with labeling techniques? Can we derandomize the bipartite Hamiltonicity algorithm? Is characteristic two essential?

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**Thank you for listening!**

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